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Search the Wayback Machine Search icon An illustration of a magnifying glass. Mobile Apps * Wayback Machine (iOS) * Wayback Machine (Android) Browser Extensions * Chrome * Firefox * Safari * Edge Archive-It Subscription * Explore the Collections * Learn More * Build Collections Save Page Now Capture a web page as it appears now for use as a trusted citation in the future. Please enter a valid web address * About * Blog * Projects * Help * Donate * Contact * Jobs * Volunteer * People * Sign up for free * Log in Search metadata Search text contents Search TV news captions Search radio transcripts Search archived web sites Advanced Search * About * Blog * Projects * Help * Donate Donate icon An illustration of a heart shape * Contact * Jobs * Volunteer * People Full text of "Adrian C. Melissinos, Jim Napolitano Experiments In Modern Physics Academic Press " See other formats EXPERIMENTS IN MODERN PHYSICS Second Edition Adrian C. Melissinos UNIVERSITY OT ROCIICSTER Jim Napolitano RENSSELAER POLYTECTINIC INSTITUTE 53 08 MEL GbRy ACADEMIC PRESS 3 5 0O when the drop moves up, ands < U when it is moving down (recall that ¢ is negative). +V¥ FIGURS5 1.1) Forces on u chapped ail drop berween the plates of a Milliken apparams. 4 1 Experiments on Quantizatioa A convenient method of analysis 1s to wiite Eq. (1.4) as l= tAn~B Aes vie oe " 6srrand | (1.5) Lip pa2tenon ny 9 na so thar A and B can be easily determined. Indeed a plot of 1 ny against a reveals the lincar relationship and the fact that only integer values of n appear, proving that the drop has acquired one, two, three, or more electne charges of value ¢, and never a fraction of that value. Thus we have clear evidence that the ionic charge picked up by the oi! drops is quantized. Furthermore, the absolute value of this minimal clectnic charge is in good agreement with inferred measurements of the charge carried by the atomic electrons,! and therefore is accepted as the most accurate value of the charge of the electron. 1.2.2. The Experiment The appatatus used in this laboruory (Fig. 1.2) consists of two parallel brass plates 1/4 in. thick and approximately 2 in. in diameter, placed in a lucite cylinder held apart by three ceramic spacers 4.7 mm long, This assembly is in fim enclosed jn a cylindrical brass housing with provisions for clectrical connections and containing two windows, one for illumina- tion of the drops and one for observation. The top plate has a small hole in its center for the admission of the oil drops, which are produced by spraying oil with « cegular atomizer. To charge the plates, a 300-V DC power supply and a reversing switch are used, the plates are shunted by a 50-M22 resistor to prevent them from remaining charged when the switch is open. For observations a 10-cm focal fenpth microscope is used (Ceaco 72925), while iNumination is provided by a Mazda )017-W Lamp and condensing lens. To avoid convection currents inside the apparatus, o heat-absorbing filter (Corning infrared-absorbing) is placed in the iuminating beam. The plates should be made perpendicular to the gravilatonal field by means of the three leveling screws at the base of the apparatus and a level LAs in e/m caperiments, shot noise meusuremcnls, ¢tc. ee se Fe 1.2 The Millikan OI Brap Experiment 5 a a 1 8 . a * Maria 1017 + Burgess U-320 S00 V (approx.} rans "are “ans “a aCe Se Ae a | a8 aoe h a = em va Ss 8 @ #8 s Fe b © hb Bo 8 6 tor 4 or © FIGURE 1.2 Millikan oil drop experiment schematic of the apparans. placed on the top plate. Being a cosine error, the deviution introduced hy an angular displacement of the gravitagonal component (rom perpendicular 7. by 8° is 1%. A value for the plate spacing s may be obtained by using the §lape micrometer. The micrometer should be focused on a wire inserted in the oil holein the center of the lop plate, and the cross hair of the micrometer should be moved along the length of the wire. Several measurements should be taken and their results averaged. The velocities are determined by measunng with a stopwatch the me ‘) required (or the droplet to cover a specified number of divisions of the “2 microscope scale. Core must be taken to avoid drafts and vibrations in the it) Vicinity of che apparatus: for that reason and because of Brownian mation, the drop may wander or be displaced cut of the field of the microscope. It may chen be necessary to reposition che microscope between measurements on a single drop. Moreover, the drop should be kept in focus to avoid parallax errors. Both the microscope and the light source may be adjusted by viewing a sroull wire ingerted in the ail hole. The light should be adjusied so that the focal point is somewhat ahead or behind the wire and the wiee is mare or less evenly illuminated. To Heht the scale, a small light is placed next to the skt just ahead of the eyepiece of the microscope. The actual distance to which a scale division comesponds may be found by using a microscope slide pp pe eee ee 7 - - = = a . + ce 7 © r 7 7 . - . . - - - + a - a - . a oR ee ee ee ee soe 6 1 Experiments on Quantization on which a subdivided millimeter scale has been scratched.* The eyepiece focus of the microscope should not be changed Curing a mn, since moving the eyepiece changes the effective distance of Uke scale. (To bring the drop back into focus the entire microscupe should be moved.) . itis important lo he tparing in the amountof oil sprayed into the chamber. In addition to gumming up the interior more quickly, large quantities create so Many particles in the microscope field that without excessive eyestyain it is virtually impossible to single out and follow a single droplet. Under the influence of gravity, droplets will fall at vanous limiting specas. If the plates are charged, some of the drops will move down more rapidly, whereas others will reverse their direction of motion since in the process of spraying some drops become pusitively charged and others nega- dvely charged. By concentrating on one drop that can be controlled by the field, and manipulating the sign of the electric field so that this particular drop is retained, it is pussihle to remove all other drops from the Aci The limiting velocity is n-ached very quickly and the measurement should be started near the top or bottom of the plate. Measurement should be completed before the drop has reached a point in its travel where appli- cation of the reverse potential is insufficient to save the drop from being “gobbled up.” The densily in air of the oil used was? 0.883+0.003 g/cm?. It is desirable to take measurements in the shortest possible time since, as previously mentioned, the mass af the drop changes dirough evaporation. It is also important to make measurements on as many different charges on the same or different drops as possible. Thus after four or ive measure- ments of yr) cand ty have been taken, the charge on the drop must be changed; this is accomplished by bringing close to one of Ihe windows a radioactive source (1010 100 Ci of Co™ will do).4 The draplet should be brought close to the top plate and allowed to fall with the feld off; on its way down it will sweep up a [ew ions created by the source. This can be checked by occasionally (uming the field on to sec whether the charge has changed; rarely will a drmp pick up any charge when the field ts an, The power supply voltage should be checked with a 1% digital multi- meter (DMM): microscope calibration should be checked before and after 2 Note that the focal length of the micruscope oust not be changed, bul instead the slide should be brought into the focal plane. >This may be found by u simple measurement, 4c) = Cune = 3,7 x 1919 Jisinteysralions per sccond. a eh hk = b & Od ERD RED - a - t r . t an 1.2 The Millikan O11 Drop Exporiment 7 7 (10-8 N sims 10 15 rd) 34 30 ls 40 Tempatatura (°C) FIGURE 1.3 Viscosity of dry air as a funclitm of fempcramic. The data pois are taken from D. Poucli and C. Gutfinger, Fluid Mechonies, Cambridge Univ, Press, Cambridge, OK, 1992, Table B-1. These points are ited to a second-order polynomial io intezpalaic io the termperature in the laboratory. the meusurements, The same holds true for air temperature and pressure, which are needed for a correction to Stokes” law. Indeed, when the ciameter af the drop is comparable to the mean free path in air, the viscosity 7 in Eq. (1-2) should be replaced by? b 7]! nT) = no(T | 1+ =] , (1.6) where no(7) is the viscosity of air as a funetinn of 7 (Fig. 1.3), b = 6.17 x 1075, P is the air pressure in centimeters of mercury, und a is the radius of the drop in meters (on the order of 107° m). In analyzing the dati itis convenient to calculate ao by letting 4 = no(T) in the second of This fonmula, altematively parameterized with b/ P = Ad, where £ is the mean-fice path of the cor molecules, was dhe subject off much research by Millikan and many others. Sec, forcxample, R.A. Millikan, Pays. Rev. 22, 1 (1923). Our value for d is taken from ¥. Ishida, Pays. Rev. 20, 550 (1929), Table I. 8 1 Experiments on Quantization Eqs. (1.5); a9 is then inserted in Eq. (1.6) to obtain n(7)} and thus a more accurate value for @. 12.3. Analysis af the Data Table 1.1 is a sample af data obtained by a student. Two dmps were used and scveral charges were measured, for each chiurge six measurements were performed and averaged, with the results shawn in Fig. 1.4. The drop Tadius @ was determined from the average values of 1 /ij. The viscosity 4 uses the correction from Eq. (1.6), Values of # that give consistent values for A = [C1 ft.) — (ift-)]/2n were identified. The pertinent parameters for these data were Distance of fail ¢=763x 107m Temperacure T= 2c Pressure FP = 76.01 om He Density o =p—oa = 882ke/m" Potential Y = $00 V Plate scparation :24.71~ 10-7 m TABLE L.4) Data from the Milhkan Oil Drop Experiment iy jm im A Wier) Drop | 27.9 +8.69 5.65 } —0.146 —29.4 +136 1.38 § —~0.158 ~28.2 $3.66 —3.00 2 ~0.152 —29.3 40.75 —0,716 gy -O,13? 29.4 +235 —1.97 3 —0.155 =a = 4.66 x 10°? m = 158 x 1075 N- sim? Drop 2 24.22 43.9% 3.071 2 ~0.144 —25.75 +9.73 -5,65 t ~0.140 —25.4 +25 —2.12 3 ~O.145 ~25.29 $9.67 —5.42 { ~0.144 25,22 44.1 ~3,07 2 ~0.143 —24.4 +173 ~1.73 t 0.144 -24.4 49.95 6.02 l ~0.433 =>o=$504x 107? m n= 1.60 « 1079 N. sim? eal on on San | . » © © 8 E68 SE + © 2 bok 1 Gf ened - @ 8 1.2 The Millikan Oil Drop Exparimant 3 15 Field Opposing Gravity a bP FP ee eS eee oe Lite” *) = “05 Field Aiding Gravity —10 -§ 0 § 10 n (Number of aleclron charges} Orap 2 Field Opposing Cravity “=5 0 5 no (Number ai alactron charges) FIGURE 1.4 Plots of 1/5 and L/t. versus «1 where nis an integer. Negative valucs of n ae used io represent dhe dada taken with the electric field pointing downward (i.e. 44). The data arc from Table 1.1. 10 2861 «Experiments on Quantization Averaging the appropriate columns in Tahle 1.1] (See Eq. (1.5)) we find that Ay = —0.1526 + 0.0046 37! BR, = 0.0346 + 0.0009 5! lef = (1.524 0.05) x 107" C Ar = -0.541940.00425°' By = 0.0401 + 0.001057! je} = (1.55 + 0.05) x 1077 C, where the valucs of e¢ are calculated using the value of A and the drop radius a§ obtained from the value of #. They are in good agreement® with the aceepted value le} = 1.602 x 107'? C. Errors on A and 8 are simply taken to be the standard deviation of the set of measurements. (See Chagter 10.) The data are plotted in Fig. 1.4 along with the straight lines predicted by Eq. (1.5) using the values of A and & derived above. The realization chat the elementary (hadronic) parucies are composites of quarks that have electric charge of - or 2 of the electron’s charge led to a revival of the Millikan experiment. Automated versions’ of the experiment have been built and nin for aloag time without revealing any such fractional charges. ).3. THE FRANK-HERTZ EXPERIMENT 1.3.1. General From the carly spectrascopic work it was clear that atoms emitted radiation at discrete frequencies; from Bohr’s model the frequency of the radiation v iS related to the change in energy levcis through AZ = hy. Further experments demonstrated that the absorption of radiation by atomic vapors also occurred only for discrete frequencies. ®f1 is Seen that in this special case (partly because of the law voltage), the diameter of the drops is so small that the correction ta the Stokes equation, t.¢., Eq. (1.6), is considerable (about 795). 7See, for example, N. Mar ef af, Phys. Rew D $3, 6017 (1996}. a b 1.3 The Frank-Hartz Experiment it Sn Ati is then to be expected that the aznsfer of energy to atomic electrons by “any mechanism should always be in discrete amonats® and related to the “glomic spectsum through the equation given above. One such mechanism meh - h | oe ees aut hoe - ~". "Of energy transfer is by the inelastic scattering of electrons from the entire »”--atom. Lf the atom that is bombarded does not become ionized, and since oer ttle energy ts needed for mamentum balance, almost the entire kinenic ana anergy of the bombarding electron can be transferred to the atomic system. sac... J, Prank and G Hertz in 1914 set out to verify these cohsiderations, , namely that (a) it is possible to excite atoms by low-energy electron bom- - hardment, (b) that the energy wansfeited from the electrons to the atoms +) ‘always hau diserete values, and (c) that the values so obtained for the energy *-t: fevels were in agreeinent with the spectroscopic resulle. . The necessary apparanis consists of an clectron-emitting filament and *:an adequate structure for accelerating the electrons to a desired (variable) -! potential. The accelerated electrons are allowed to bambard the atomic -, vapor under investigation, and the excitation of the atams is studied as a -) function of accelerating potential. For detecting the excitation of the atoms in ihe vapor it is possible to * observe, for example, the radiation emitted when the atoms retum to the “ground slate, the change tn absorplion of piven spectral line, or s0me other related phenomenon; however, a much more sensitive techniquc consists "of observing the electron beam itself. Indeed, if the electrons have been ". accelerated to a potential just egual to the energy of the first exciusd level, 5 some of them will excite atoms of the vapor and a5 a consequence will . lose almost all their energy; if o small retarding potential exists before the .'. collector region, electrons that have scattered inelastically will be unable to overcome it and thus will not reach the anode. "These conditions are created in the experimental wirengement by using “two gnds between the cathode and collector, When the potentiuls are dis- ". ibuted as in Fig. 3a, the beam is accelerated between the cathode and *. prid 1; then it is allowed to drift in the interaction region between the two .' grids and finally must evercome the retarding potential between grid 2 and . the anade. When the threshold for exciting the first level is reached, a sharp -. decrease in electron current is observed, proportional to the number of col- ‘, sions that have occurred (product of the atomic density and cross section). .. When the threshold of the next level is reached, a further dip in the collector current will be observer These current decreases (dips) are superimposed i F tee bee bit -_ - L hd ad tan hehe tal “tal halter L] oe be bb Pe oe os es eo Fk bh & . . Ql tel L ee on ee . a | | a ad at hel ad tad had tal had ee had Datel lhl adhd hl dl ee | . 7 . . a . . . bh bee kh kh hk ek eek ] oeaerunraearhaeenephk BH @h FS Pee =. 8 6 a . = o . a a bh kh eh kh 8When they remuin bound alter the collasion, b eee ee ee eee L 12 1 Experiments on Quantizatian {a} (b) {c) FIGURE 1.5 Different contigurauons of the potential in a Frank—Hertz arrangement: (2) For observation of a single excitation, (hb) for observation of a multiple excitation, and (c) for measuring the ionization potential. on a mobotomically nsing curve; indeed the qumber of electrons reaching the anode depends on V,,,. inasmuch as it reduces space charge effects and elaslic scattering in the dense vapor. In addition, the dips are not perfectly sharp because of the distribution of velocities of the thermionically emitted electrons, and the energy dependence of the excitation crass section. An alternate distribution of potentials 1s shown in Fig. 15b, where Vag. is applied at grid 2 so that an electron can pain further energy after a col- lision io the space between the two gnds. In this case when Vee reaches the first excitation poicntial, inelastic collisians are again possible and the decrease in electron current is observed at the anode; when, however, Vace reaches a vajuc twice that of the lirst excitation potential, itis possible for an electron to excite an atony halfway between the grids, lose all its energy, and then gain anew enough energy to excile a secand atom und reach gnd 2 with practically zero energy. Thus it is not able to overcome the retarding potential to teach the anode, giving rise to a secand dip in the current. ee 1.3 The Frank-Hertz Experiment 13 SN ‘The advantage af this seep i8 (hat the current dips are much more pro- Se aced, and it is easy to obtain fivelold or even larger multiplicity io. as _ ee “ibe excitation of the first level. However, it is practically impossible to fn. -abserve the excitation of higher levels. As before, a sitpht retarding poicn- a. ‘tial; is applied between grid 2 and the anode, and an accelerating potential fat between: the cathode and grid |, sufficient to overcome space charge effects oe, a oe -'and to provide adequate clectron current. Itis evident that the densicy of the fe ‘atomic vapor through which the electron beam passes greatly affects the ee. -gbserved results. Low densities result in large electron currents but very Be ‘small dips; in contrast, high density has as a consequence weakel Currents > but proportionally larger dips. When mercury vapot is used, adjustment of fe “the. tube teniperacure provides control of the density, ae | Another important poiot is that in principle the experiment must be an , performed with a monatomuc Fas; since if a molecular vapar is bombarded, “42-7 is possible for the electrons to tansfer energy to the molecular energy “27 tevels which form almost a continuum. Some of the preferred elements for #.:) the Frank-Henz experiment are mercury, neon, and argon. cae _ The same apparatus can be used for the measurement of the ionization en : patential—that i is, che energy required to remove an electron completely (fie) ‘fram the atom. In this case, instead of observing the bombarding clec- a ‘tron beam, it is easier 10 detect the ions that are formed. The distributian .".".of potentials is as shown in Fig. i.5c, where the anode is made slightly fis, negative with respect lo the culthode; oo electrons can then reach the <-". anadle, which becomes an inn collector. The accelerating potential is "4° inercased until a sharp cise in the ion current measured at the anade is 2". observed. oe a In both hypes of measurements the values obtained lor the wccelerating i . : potential have to be corrected for the contact potential diffcrenee (cpd) Moe, between cathode and anode.” If in the excitation experiment the same level ee! thas been observed two or more times, bowever, the potential difference wine, Dbeiween adjacent peaks is an exact measure of the excitation energy, since “2. the contact potential difference shifts the whole voltage scale. Once the me excitation energy has becn found the contact potential difference is given alee. by the difference between unis true value and the first peak: in tum the wen oe eS _ eae . Briefly this is because the “work fiincd oo” for dhe metal of which the anode 15 made is ae 7, usually higher than (hat of the cathode, The work function is a measure of the “ionization --Botantal” of the rocta), that is, of the energy aceded ¢o extract an efectron from it 14 «861° «Experiments on Quantization contact polential difference so found can be used to correct the ionization potential measurement. 1.3.2. The Experiment In this laboratory a mercury-filled tube made by the Leybold Company (55580) was used, the clectrode configuration is shown in Fig. 1.6, and the circuit diagrams for the measurement of excitation and of ionization potential are given in Figs. 1.72 and 1.7b, respectively. As seen in the circuit diagram, grid 1 1s operated in the neighborhood of 1.5 V, and the retarding potennal is of the same order. The anede ¢cur- rents are on the order of 107° A and are measured either with a Keithley O10B electrometer or with a high-input impedance digital multimeter, for instance, Hewlett-Packard 3440) A; adequate shiefding of the leads is required to eliminate AC pickup and iuduced voltages. The diagram of Fig. 1.7a uses the distribution of potentuls shown in Fig. 1.5b, and the accelerating voltage can be measured with a DMM in steps of 0.1 V. The Frink—Hertz tube is placed in a smal! oven, which is heated by jine voltage through 9 variac; ut should be operated io the vicinity of 200°C for the excitation curve and between 100 and 150°C for the ionization curve. Ta measure the temperanire a copper-constantan thermocouple should be FIGURE 3.4 Sketch of a cylindrical Frank—Heriz tube. ae Bee kL ’ 8 s ££ 8 ee ee ee ee ee oF 4s mB 4 mw kot 4 8 t.3 The Feank-Hertz Experiment To Keilhly alacirameter ey 1144 V Dr, call et tH10Vde { 10K _ OQ Helipot sou a eh 10Vde 1OK Heapal 10K _ DOW 14 (ee FIGURE 1.7 Wiring diagram for the Frank—Henz caperiment (a) for observation of 27-7 2xcilation, and (b) for observation of ionization. As inserted through the small hole of the furnace, The junctlivn should be 2--- ‘positioned on the side of the tube near the electrodes. The othes junction is ro eay a.--immersed in athermos of ice and water bath, The potential developed across ws". the thermocouple is measured with a DMM; Fig. 1.8 gives a calibration "curve for the copper-constantan thermocouple. ée.".'. The resolution and definition of both the excitation and ionization curves fends a function of atom density (temperature) and electron beam density (fila- wa. ment and grid | voltage) and the experimenter must find the optimum “.---condilions. However, for large beam densities a discharge occurs, which, S-nabviously, should be avoided. ae) A suggested adjustment procedure is to set grid 2 at 30 V and then a “u- advance grid 1 unit the discharge sets in, as evidenced by the immediate 16 1 Experiments on Quantization = rit oft 2AM WoOkronanw7yoeowoo — rf my across thermooouple juncilons o~ nn MD FOO WY & oa © 20 44> 666 680 61D0 6180 «6140 «6160)6180 COD Ue20 Junction température (°C) FIGURE L.& Calibration of copper-constantian thermocouple using tee standart build-up of the anode current. Grid 2 should then be quickly retumed to O'V and gnd 1 set slightly below the discharge voltage; a reasonable filanvent voltage is between 4 and 6 V. To determine whether the tube is overheated it can be taken cut of the oven for about 30 s; the collector current will then inctcase and maxima may appear if such is the case. If the tube is toe cool, the emission current will be large, and the maxioia, particulasly those of higher order, wil} be washed out It is possible to use an oscilloscupe far a sumullaneous display of the electron or io0 current against accelerating potential. The sweep generator (sawtooth) ouinut Is fed to the accelerating grid, while it synchronously drives the honzontal sweep; the output of the clectrometer is fed to the Vertical japut, An excitation curve and an ionization Curve oblained by a. student in this fashion are shown in Fig. 1.9. Altemately a simple ramp circuit can be built to dnve the accelerating grid and the digitized output of the electrometer read directly into a computer. 1.3.3. Analysis of the Data Two sets of data obtamed by a student for the excitation potenhal are shown in Fig, 1.10; both curves were obtained al a temperature of 195°C and with +1 V on grid !. The filament voltage was 2.5 V for curve C and 1.85 V for Lis LL Lk Bn ne be ae os kb = »& b Oe . . » FoF DE | L moa . k & & & eke & = Shhnenhnannnes ee So o bo o@ a os. 8 te = = bor f Ff Flociron baam currant (nA} 1.3 The Frank-Yertz Experimant 17 sD "HIGURE 1.9 Oscilloscope display of a Frank—Hertz experiment: (a) Beam curtent vs 21.4202 21.6401 14-320.1 166201 11AgO3 11.640.4 Brite Bato! a 10 14 20 G, Accelerating (V} ASCEIRING POLE it] “ldecelerating potential. (b) Ton current vs accelerating potential. JtHi09 25 rH 3 ae! FIGURE L.fO Fiat of beam current versus accelerating voltage tn Frank—Hertz expen- eee Ment Dats (or curve € (upper points) wre ubtained wilh the filament set ut 25 Y while data aca for curve BD (loser points) are obtained with filament at 1.85 V. wie. Readings are taken for 1-¥ changes on gnd 2 with smaller steps in the Za = 22-: Vicinity of the peak. A significant decrease in electron (collector) current 18 0S ts s«Expesiments on Quantization is nonccd every time the potential on grid 2 is increased by approximately 5 V, thereby indiculing that energy is transferred from the beam in bundles (“quanta”) of 5 eV oaly. Indecd, a prominent linc in the spectrum of mercury exists at 253.7 om, comespondiog to 1237.8/253.7 = 4.86 eV, arising from the transition of the 6s6p7P; excited state (9 the Gs6s 'Sg ground state. Our interpretation is that the electrons ia the bean excite the mercury atom from the ground state to the 3 P, state, thereby losing 4.86 eV in the process. The location of the peaks ts indicaled in Fig. }.10 and was measured in this case with a DMM. The average value obtained for the spacing between peaks is 5.02 +01 ¥, to be compared with the accepted spectroscapic value fur the energy level difference (as wiready Mentioned) of 4.86 eV. Using the value found for the spacing between peaks and the Jucation of the first peak, we obtain the contact potential (6.65 -£ 0.15) — (5.02 £0,1) = 1.63 + 0.18 V. As discussed in Section 1.3.1, with the configuration of potentials used (Fig. 1-56) it is mone probable thal the same energy level will be excited twice rather than chat several diifcrent levels will be excited: indeed, this is the way in which the data in Fig. 1.10 have been interpreted. This is not surpnsiog if one considers the excitation probabilities for the energy levels lying closest to the ground state of mercury. It is possible, however, by using different grid and voltage configurations (for example, Fig. ! 5a) and improved resolution, to observe the excitations to ather levels, namely; 6 Py, 6 Po, and 6' Py. For the ionizalion potential, data obtained by a student ar: shown in Fig. J.11. A word of caution is to be added to the interpretation of such ionization curves, which seem strongly dependent on Alament voltage and Vapor pressure; indeed, the very sharp increase observed in jon curtent 1s due to an avalanche (regenerative effect) of the ejected electrons ionizing more atoms, the thus-ejected clectrons ionizing soli more atoms, and so on. This avalanche does not necessarily uccur as suon a8 the ionization threshold is crossed. Lf the vapor is too dense, the ions recombine before teaching the anode, thus masiang the effect watil complete breakdown sets 1m. | The curve shown was taken at a temperature of 155°C with a filament voltage of 2.6 V. If, then, the onsct of ion currentis taken (o be at 11.4+0.2 V, . oS see se. 1.3 The Frank-Hertz Experiment 19 . < a** 's es * ‘ — 12] ‘ eee *ft * “eee Se sa *ee ee *_e * »- * a a i’ lon Currant (707-8 A) a - sae ee . . . haba! es tehe ‘eo. . . w tae ‘ *4e ew tweweeeee SBaeé »*-*F 4 ' o § 10 15 G,G, Aoceleraling (Vv) 4a.4 ‘. ‘ * 8 t we ‘ a e sian oN i ‘ ivsnnnaenneenren GURELIL loncurreni versus sepelerating voltage in the Frank—Herw experiment. The nce“ ar 8 V 1s due to the photoelectric effect. => oe ee ow le ee “x d using the value for the contract potential previously determined (from excitation curve), 1.63 40.18 V, the iontzalion potential is obtained as (11.440.2) — (1.63 + 0.18) = 9.77 40.25 eV ah # ‘ ess - * oon “nly in Fair agreement with the accepted value of 10.39 eV. -.. An additional feature of the curve Fig. 1.1) is a “knee” in the ion cur- =“-fent, setting in at approximately 8 Y; the observation of this “knee” as well “is: sirongly dependent on the tomperalure and current density, but can be = oonsistently reproduced over a considerable cange of these parameters. In = order to understand this behavior we remember that the arrival of ions at =" the anode is equivalent to the departure of electrons; indeed, the observed +: behavior is due to a photoelectric effect produced at the anode, by short- 4. wavelength light quanta (tbe electrons are further accelerated by grid 2). ~: When the electron beam reaches $ V, it can excite the 61 P; level (lying at ~°6.7 eV above the ground state, plus 1.63 V for contact potential difference), Aa. 80 the mercury atoms radiate the ultraviolet line at 184.9 nm when retuming e2-*-fo-the ground state. These quanta are very efficient in ejecting pholwelec- “- trons from the anode, and the cylindrical geometry of the anode is most - favorable for this process. ss Siri RS 20 3861 Experiments on Quantization 1.4. THE HYDROGEN SPECTRUM The hydrogen atom 1s the simplest quantum-mechanical system. It consists of an electron bound, due to the Coulomb force, to a proton. It is character- istic of bound quantum-mechanical systems that their total energy cannot have any value, but that the system is found in one of a discrete set of energy levels, or states. Transitions of the system between these statcs may occur. Such transitions must satisfy the basic conservation laws of electric charge, energy, momentum, angular momentum, and the other relevant symmetries of nature. Transition from a higher energy state to a state with less energy can occur for an isolated system, and the larger the probability for this transition, the shorter the “lifetime” of that excited state. During such spontaneous transitions of a quantum-miechanical system to a lower energy state, a quantum of radiation, or one or more particles, can be emitted, which will carry away the encrgy lost by the system (after recoil effects have been taken into account), In the presence of a radiation field the quantum-mechanical system can etther gain energy from the field and change into a state with higher energy, or lose energy to the field and revert to a lower energy state. For ail quantum-mechanical systems there exists a lowest energy state, called the ground state. By observing the quanta of radiation, or the particles emitted during such transitions, we gain information on the energy levels involved. The typical example is optical spectroscopy, which consists of the accurate determination of the energy of the light quanta emiited by atoms. Infrared spectroscopy deals mainly with the quanta emitted by molecules, nuclear spectroscopy with the quanta emitted in nuclear transitions, and so on. In nuclei, however, the separation between energy levels is much larger, so that the emitted quanta of electromagnetic radiation lie in the gamma ray region; thus different techniques are employed for detection and measure- ment of their energy. It is also very common for nuclei to decay from one energy state to another by the emission of an electron and neutrino (beta decay) and for certain heavier nuclei by the emission of a helium nucleus (alpha particle). Similar processes take place in the interactions or decay of the elementary particles. The idea of energy levels and their structure for the hydrogen atom was first introduced by Niels Bohr m 1913. However, a complete theoretical interpretation had to wait until the introduction of the Schrédinger equation in 1926. Even then, for theory to agree with observation it is necessary to . : *0Fy ae 1.4 The Hydrogen Spectrum 21 = *, .) SAEON GAS tate ss include additional smal! effects such as the fine and hyperfine structure, relativistic motion, and other higher order corrections. These corrections = are derived using the theory of quantum electrodynamics (QED) so that E oday we can theoretically calculate the energy levels of the hydrogen atom zB “to the amazing accuracy of | part in Lo"! ZF We will use the apr theory to predict the hydrogen energy levels, Sa es Rh hy Pook RES wetenatete? ——+ += ‘jum is aed in integral units of Planck's constant (divided by 2yr); ; . pannel, pr=mur = n(h/22)=nh; and (b) that the electron 1 in this orbit ¥ ean then calculate the radii of these orbits and the total cnergy of the system, D te potential plus kinetic energy of the electron. The attractive force between g Be: ame electron (charge —e) and the Proton (charge +e) ora nucleus (of wee Sa a tates! % ae ~ "The total mechanical eneray of the electron is % Be ei E=T+V ae pe I 1 Ze Hes = ~ . °P tes sa Se OOS x —13.6 eV n=1 "FIGURE 1,12 Energy-level diagram of the hydrogen atom according to the simple Bohr Zot theory. a SS “with Sok —_ Roa eas owns Be dog = — — =~ = 0.5291772 x 107" m, m ie #5, correspond to transitions between these levels; this is shown in Fig. 1,13, ==: where arrows have been drawn for all possible transitions. The energy of Haters") tate taty te Mate cate ty fe MO aS + >*, . tar ahs + 1 ] AE =h R ay ee ’ 1.14 if C Keo n2 ( ) Satay tataMs hes Mote tata ye Me tete ‘ oterateeoepcies g ! o > 144) nm co a 5. a=] B mS, Vv B o. ey 3. 5 os a th z g =! & & A < NA, = oO a) = § a © a > m 3 cr. =) = | o° : Oo : 5 a a e as — oO —_ oO 5 =F < '- -f ‘“"* 5 - - * “ane * > om on CAR Ss | {| > < SS : = one finds that SSS Se 4 <- 24 #1 Experiments on Quantization a 7 yy Pa 1.097 x 10° cm=' FIGURE 1.13 Transitions between the energy levels of a hydrogen atom. The lines Lg, Lg, etc., belong to the Lyman series, Be, By, etc., to the Balmer series, and Fy, Pg, elc., to the Paschen series, and so forth. ] ] ] — =Re|+z-s. ).15 My 00 (5 -} (1.15) Indeed, the simple expression of Eq. (1.15} is verified by experiment to a high degree of accuracy. From Eg. (1.14) (or from Fig. 1.13) we note that the speciral lines of hydrogen will form groups depending on the final state of the transition, and that within these groups many common regularities will exist; for example, in the notation of Fig. 1.13 v(Lg) — v(Ly) = ¥( Ba). If Hi f = 1, then 9 He amas (2 ) nm; = 2 and all lines fall in the far ultraviolet; they form the (so-cailed} Lyman series. Correspondingly if a = 2, then n2 iz = 364.4 5 nm. my = 3 and 1.5 Experiment onthe Hydrogen Spectrum 25 = and all lines fall in the visible part of the spectrum, forming the Balmer series, Forn ¢ = 3 the series is named after Paschen and falls in the infrared, 1,3, EXPERIMENT ON THE HYDROGEN SPECTRUM - 15.1. General . ee” To measure the frequency of the radiation emitted by atoms one can use f i using @ prism, one exploits the variation, with wavelength, of the refractive Mz: index of certain media, Prism spectrometers are limited to wavelength a —, ~ =: 4n the infrared, special fluoride or sodium chloride prisms and lenses are =: used. In the ultraviolet, the optical elements are made of quartz. Also, the SCNT ste ~ SKeees eats bs Be sensitivity of the detectors varies with wavelength, so thal different types ees are used in each case (thermopile, photographic emulsion, phototube, etc.). fee-:: In this laboratory a small constant-deviation prism spectrograph and a #:--2-in. reflection grating spectrometer were used. We will consider in detail a 25° brief discussion of prism spectrographs is given in Section 1.5.4. =: From Fig. 1.14, it is evident that the path difference between rays ] and =. 2 after reflection is ‘ BD — AC = CBsin®, — CBsin&, =<‘ where CB is the grating spacing d. The angles 6; and 6, are both taken as =: positive when they lie on opposite sides of the normal. Since for coustruc- =5:-tive interference the path difference must be a multiple of the wavelength, =: we obtain the condition ae nh =d (sin & —sin 6). (1.16) ge It can be shown!! that the resolution of the grating is given by were eS oS ae A gre —=nN, Bee Ar where n is the order of diffraction and N the total number of rulings. The = Same considerations apply to a transmission grating. NN oe USee Chapter 5, Section 5.5, So SS *. *' * * =n SN S 7 Woriiken ~*, ne 2 4606 ots Experiments on Quantization FIGURE 1.14 Schematic diagram of a reflection grating. A paralle] beam of radiation is incident along the rays 1 through 4 at an angle 6, with respect to the normal; the reflected radiation is observed at ai angle @;. The spacing between the grooves of the grating is d. Focusing lens Collimator lens Telastope position 2 FIGURE 1.15 Diagrammatic arrangement of a grating spectrometer. The grating is mounted on a goniometer table in the general arrangement shown in Fig. 1.15. A sit and collimating lens are used to form a bear of parallel light from the source, and a telescope mounted on a rotating arm is used for viewing the diffracted lines. It is obviously necessary to ensure ee ee a. en a 1.6 Experiment an the Hydragen Spectrum 27 meee ae ae may me pe se moe ye a we mo Oo . ©) From now on one may have to work in dark, or by draping the - pparatus with a black cloth. mec (f) The grating is placed in position and aligned for normal incidence #2°(, = 0). This can be done by “autocollimation’’; a strong light is focused poe the slit and a cardboard mask with a narrow slit is placed on the in the telescope in position I, ==" With any reasonable grating it is possible to observe the visible lines of : “the spectrum in several] orders; thus we expect the measurements for 4/d z=: to be self-consistent, since (1.17) Rul . X SiN Om.) — SING = (7m + 1) a i =:independently of angle of incidence 6;, or order.'!? The grating spacing 2d is usually stated by the manufacturer; for example, the grating in this =->laboratory had rulings on the order of 7000 to the inch (¢d = 3.629 x ==3076 m). However, d can be obtained by using one or more standard lines pce OF known wavelength. a — ee ae fa siz: a ae ae ae 12 Provided that both 6,, and Om+4 arc taken on the same side of the normal. 28 42=«o 1 sSC«Experiments on Quantization The following data were obtained by a student using the grating spec- trometer. The source was a low-pressure hydrogen discharge tube (Cenco type 87210) operated at a few thousand volts; a 5-k¥V transformer and variac were used to provide the variable voltage. The useful life of these discharge tubes is limited because of the appearance of strong molecular bands after some hours of operation. 1.5.2. Determination of d To obtain the erating spacing d, sodium (Na) was used as a standard, and measutement on three lines (for the shorter wavelength of the doublet) gave the results shown jn Table 1.2. Since for all the above measurements 4; is the same, it follows that d—(nzhz) + sin & = sin 6 and a least-squares fit to the linear relation Bx + @ = y can be made; we have LN Diag sin 6) — Di(sin &) Vers) 7 5 3 : (1.18) N Yong An)? — Pong] TABLE 1.2 Diffraction Angles from a Sodium Source Aim om Order ?1 Gn A = 19°12’ 615.43 1 29°42! 2 41°27! 3 59°58" 589.00 l 29° 14’ 2 49°21/ 3 53°49" 4 75°15! 308.27 2 99°32’ 3 §2°12/ 4 70°48? Moa na her 1.6 Experiment on the Hydrogen Spectrum 29 where the sums are over k, k = 1,2,..., N and N is the total number of measurements. From the data of Table ].2 we obtain! l 5 = 2.7085 + 0.009 x 10° m7! (1.19) in good agreement with the manufacturer’s specification. oe Some care must be exercised when comparing wavelengths, since they = do depend on the refractive index, n, of the medium in which they are Ee measured, , e{vacuum) c — % n hence he A (vacuum) Ht The wavelengths listed in most tables are given for dry au at a pressure of 760 mm mercury. However, any theoretical calculation, such as in Eg. (1.15) predicts the vacuum wavelengths. The refractive index of air at stp is n(air} = 1.00029, (1.20) 1.5.3. The Balmer Series Measurements on the first four members of the Balmer series, which lie in the visihle region, can be made with the spectrometer described above. The data obtained by a student and their reduction are given in Table 1.3. We observe that the obtained values for the wavelengths of the Balmer series are in agreement with the accepted values at the level of 1/1000. We can now test Eq. (1.15) and obtain the Rydberg wave number, We note that J_ pis rn HI4 n2 |- So that from a least-squares fit _ yo Ry = = —_., y AGe: 131m reaching this result we have constrained 6 = 19°12’. 3000S 1ss«Experimants on Quantization TABLE 1.3 Data on the Balmer Series of Hydrogen as Obtained with a Grating Spectrometer Calculated ~ Accepted Balmer series Color Oy sin@, — sing Order X a identification Violet 33°12' 0.22199 2 410.7546 410.17 Hg mi = 6 41°15’ 0.33378 3 Blue 26° 16/ 0.11698 1 34°06" 0.23483 2 433,82+8 434.05H, nj=s 42°42! 6.35259 3 Green = 27°10’ 0.13001 1 36°Q4’ 0.26316 2 485.75 410 486.13 Hg nj =4 46°09" 0.39559 3 Red 30°11" 0.17720 | 42°57 0.35579 2 657.9414 656.28 Hy n=3. §9° 29? 0.53532 3 Note. All wavelengths sre in om. These measurements used d = 3692.1 + 30 nm as determined by the previous measurements on the sodium standard lines, and sin 6; = 0.32557. where giving Ry = (1.09601 + 0.003) x 107 m7! in good agreement with the accepted value!* M Ry = Min Roo = 1.096776 x 107 m7!. Here M is the mass of the proton and m the mass of the electron, 1.5.4. The Prism Spectrograph Long before gratings became widely available, prisms were used as the dispersive element in spectrographs. Prism spectrographs are handy for Viewing a large span of the spectrum and come in various ingenious optical \4The difference berween Ry and Royo is due to the motion of the electron about the center of mass rather than about the proton. ON SSN ode ok MS ope eye . “ 4 AN ™ orate Sos Sy Sees ~~ SS zie Fal ee pC le eee ~~ oe < SAM Witenteten ny SS SES 1.5 Experiment onthe Hydrogen Spectrum 31 ea FIGURE 1.16 Diffraction of a ray at minimum deviation through o prism of apex angle A a arrangements. The dispersion of a prism is a function of the refractive index; thus it cannot be used for absolute measurements without careful =: calibration. In the case of a simple prism at minimum deviation (see Fig. 1.16) the #2 diffraction angle @ is given by het ore ay sy! idee * sin F I = pi) 6 = 6, — 0: sin 0, . , rr > = thus A+@ _A sin (“=*) =nsin z' (1,21) BS where 6; and @, are the angles of incidence and refraction, respectively, “* and A is the apex of the prism. In Fig. 1.17 the refractive index of flint glass as a function of wavelength is given. We note that in the determina- =: tion of wavelength from the diffraction angle the relation is by no means linear and is in general of serious complexity. Further, most modern prism == spectrographs do not consist of a single dispersive element, but of some =. combination of prisms. The instrument used in this laboratory was of the =. “constant-deviation™ type, and Fig, 1.18 gives the optical paths for an inci- =... dent ray. It may be seen that the angle of incidence and the angle of exit can =. ‘remain fixed for al] wavclengths by an appropriate rotation of the prism; 2. this has obvious advantages for positioning and alignment of source and == detector. The rotation of the prism is calibrated to give rough wavelength indi- =. cations, but measurements are made on the exposed photographic plate 32 87 Experiments on Quantization 2.2 2,0 fi NaBr TW £18 2 i Flint glass = 1.6 9 o 1.4 Crown glass 2000 4000 Bodo Wavelength (A) FIGURE 1.17 Refractive index of various materials as a function of wavelength. FIGURE 1.18 A constant-deviation prism and the diffraction of a tay passing through it. or film. A known spectrum is superimposed on the spectrum that is to be investigated, and an interpolation between the known lines is used. The general arrangement of the spectrograph is shown in Fig. 1.19. Source, lens, and slit should be aligned and the source focused on the slit. By viewing through the eyepiece and varying the prism position, one can get a feeling for the dispersion and the range of the instrument, To obtain photographs of a spectrum, the telescope is replaced by the camera assem- bly. Several exposures can be had on the same plate; to distinguish different spectra superimposed at the same location on the plate, the “fishtail,” which controls the length of the slit, can be used. Ge 1.6 The Spectra of Sodjum and Mercury 33 = = oe Constant-deviation prism fe Focusing lans i ie eh ae oe. Source j es Poa Slit a a ' ! eae ! . ge | es Camera lensx—<>~-» Bellows: | © . Plate holder —— D ' | E2—}—+ F FIGURE 1.19 Schematic arrangement of the constant.{-+ 1; when the potential that the electron sees is exactly of the Coulomb type as in the case of hydrogen, where V = (—Ze*)/r the energy eigenvalues mZ*e4 ] E, = - | —-—— | S 1,22 " Ee a | n? Oe!) are independent'> of /, and agree with the Bohr theory. However, the screened polcntial that the free electron secs is no longer of the simple Coulomb type, and the cnergy of the level depends on both # and J. Orbits with smaller values of i are expected to come closer to the nucleus and thus be bound with greater strength; as a consequence their energy will be lower (more negative). The energy level diagram of sodium is shown in Fig. 1.21, where the levels have been grouped according to their! value. The customary notation is used, namely, / = 0 > S stale, i = 1 > P state, / = 2 — D state, {= 3-> F state, and so on, alphahetically. The last column in Fig. 1.21 gives the position of the levels of a hydrogen-like atom. ‘This is the so-called Coulomb degeneracy: a peculiar coincidence for the Coulomb potential when used in the Schrédinger equation. er ee nae ee et ee ee ee ate ane a a Pat a ahr ee et eS a ee HT A AL Ae ee ee a ~ fate eh ee aC ot a War ha a SN, Ce aL a 4 1 Lats 1.6 The Spectra of Sodium and Mercury 3 i=0 #£Sstate /=1 Pstate /=2 Dstate /=3 Festate 0 FIGURE 1.2! The energy-level diagram of sodium, grouped according to the orbital angular momentum, The Jas! column gives the corresponding position of the levels of hydrogen. The left-hand scale is in 10° m7!, referred to 0 for the singly tonized sodium atom; the right-hand scale is in electron volts referred to 0 al the ground state of the sodium alom. We note that the higher the value of /, the smaller the departures from the hydrogen-like levels (as suggested qualitatively previously), and that for given / the energy levels for different n's follow the same ordcring as the hydrogen-like atom, but with an effective charge Z*, which for sodium is as follows: § states Z* ~ 11/9.6; P states Z* ~ 11/10.1; D states Z* ~ 1; F states Z* ~ 1, 1.6.2. Selection Rules The spectral lines that we observe are due to transitions from one energy State to a lower one; however, in analyzing the spectrum of sodium, it 350s 1 « aad ier pe 10° m- ety * 5 CN) . ate Tataatacate ts ett ts tata ~ ad a el ad | Ps 30 40 n=3 FIGURE !.22 The “allowed” transitions between the energy levels af sodium. The wave- lengths in angstroms (10 A = I nm) of some of the principal lines are indicated. Note that ss. the P states haye now been shown in two columns, one referred to as Py/2 the other as 22. P72; the small difference between their energy levels is the “fine structure." BRERA RR re) see Pare es aaa" 6 4 sae he a 3860S] Experiments on Quantization FIGURE 1.23 Photograph of the visible spectrum Go nm) of sodium as obtained with a constantdevialon spectrograph. TABLE 1.4 Data on the Fine Steneture of Sodium as Obtained with a Grating Spectrometer Line Red Yellow Green Order Be & wh Wb ay 41°27" 55°58" 40°21! §3°49' 75°15 39°39! 79°49" ea) 41°29! 56°00" 40°23’ 53°52" 75°23! 30°33/ 70°56’ Aé (radians) 58x 1074 5.8 53.8 B.? 23.2 2.9 2-2 ante tat tatet atten steele AURSALAL GIADA Tessas bese! ee ii Cr ee nr ee ee fet eat : aa el el od ar ee NE i SEUSS TERETE SATE SES USANA RES HLERAL TEMA CSA wna, Para 1.6 The Spactra of Sodium and Mercury 39 = To reduce the data we note that : npA = d(sin & — sin 4), 2°: where 6; is the angle of incidence. Also : Oy = 6) + AB = By letting sin Ad, © Ab, cos AO * 1, a ny Ad = d cos 6 Ab,. : (1.24) pe Using d = 3,692.1 nm and averaging over orders within each line, namely = writing 2 COs H% AG, cos O Ab AA=d= e at (1.25) we obtain for Ad: . Line AA (Experiment,nm) Ad (Exact value, nm) Red 0.57 0.651 Yellow 0.63 0.597 Green 0.59 0555 == The expenmental data are thus in ~10% agrecment with the exact values. ==": This splitting of spectral lines was named “fine structure” and must = feflect a splitting of the energy levels of sodium; if we express the wave- i lengths of the sodium lines in wave numbers (D = 1/A = w/e, i.c., = in a scale proportional to energy since AE = hcAj), it becomes evi- === dent that the spacing in all doublets is exactly the same and equal to =: AD = 1,73 x 107 m™. Indeed, the doublet structure of all the above 22°: Hines is due to the splitting of only the 3P (n = 3, / = 1) level as can be =: seen by referring back to Fig. 1.22. The splitting of the 3P Jeve) is due fo the effect of the electron “spin” and its coupling to the orbital angular momentum (designated by 7). According to the Dirac theory, the electron #2 possesses an additional degree of freedom, called “spin,” which has the g properties of angular momentum of magnitude s = f/2 (and therefore two 2. possible orientations with respect to any axis, m, = +5 orm, = —4). ge The spin s can then be coupled to 1 suocurling, U2 tip quanturs-mechanical 2 Momentum of magnitude j = / +5 or j=l— t and the energy of the Bee § State will depend on j. In the case of sodium, the 3P level splits into two ee “levels, with j = 5 pm j=3 5 designated as 3 Pj,z and 33/2 separated by Ye si = = 1.73x 10? m Spe: | 409 1 Experiments on Quantization 1.6.4. Electron—Electron Coupling; the Mercury Spectrum The mercury atom (2 = 80) has 80 electrons. These fill the shells n = 1, n= 2,n = 3,andn = 4 completely (60 electrons}, and in addition, from the n = 5 shell, the ! = 0, 1, 2 subshells account for another 18 electrons. The remaining two electrons instead of occupying the / = 3 and/ = 4 subshells are in the x = 6 shell with / = 0, giving rise to a configuration equivalent to that of the helium atom. We thus have an alom with two electrons outside closed shells in contrast to the one-electron systems of the hydrogen and sodium type. In the two- electron system, we can hardly speak of the x number of the atom, since each electron may be in a different shell; however we can still assign a total angular momentum J to the system, which will be the resultant of the values of each of the two electrons, and (as we saw in the previous section) of their additional degree of freedom, their spin. The addition of these four angular momenta, 11, Ie, 81, $2, to obtain the resultant J can be done in several ways. For the helium or mercury atom, the Russell Saunders coupling scheme holds, in whieh k; and lb are coupled into a , resultant orbital angular momentum L and s, and sz into a resultant spin S; finally L and § are coupled!’ to give the total angular momentum of the system J. Since s; and s2 have necessarily magnitude Be the resultant § has magnitude § = Oor S = 1. It is customary to call the states with S = 0 singlets, those with § = 1 tnplets, since when S$ = O for any value of L, only a single state can result, with J = - + S = L; when S = |, however, three states can result with J = £+ 5, L, EL — 8, namely J=L+4+1, L,£ —1 (provided L # 0). In systems where energy states have total angular momentum J, the selection rules for optical transitions are different, namely AL=x] (1.26) AJ —0,+1 but not J=Q— J/=0, and in principle no transitions between triplet and singlet states occur. 171n the ensuing discussion the quantum-mechanical rules of addition of angular mormen- tum are used, Even if the reader is not familiar with them, he can infer them from following the development of the argument. eaten a eer ae eee ed et iti sea bk ts Me oDaTADpEn ESSERE CESSGSSESAEESRSRSSS SALES LESSER SSAC TSGHRESECESESOGGSLRGELELASU tag CLTALATOG GOGOL ECECLAC A SEATAEMEEESC TESLA RBS SBSBOtabaECECORARSSSESEG EGRESS CAGES ty am 1.6 The Spectra of Sodium and Mercury 41 * With these remarks in mind we consider the energy-leve] diagram of =: mercury. Since there are two electrons outside a closed shell, in the ground =: gtate they will both be in the n = 6, / = 0 orbit, and bence (due to the Pauli 7: principle) must have opposite orientations of their spin, leading to S = 0; Be: the spectroscopic notation is ' So. For the excited states one should expect both a family of singlet states and a family of triplet states; the singlets, B'S = Q, will be - be AYA for L = 0, and necessarily, J = 0 Bes 'P, for L = 1, and necessarily, J = 1 ee 1D, for L = 2, and necessarily, J = 2 ete. ee Note the spectroscopic notation, where the upper left index is 28 + 1, = indicating the total spin of the state; the capital letter indicates the total L = of the atom (according to the convention); and the lower right index stands = for J. For the triplets, S = 1, and the states are ye *So for L ='0,J = 1 fe 3 Po.1,2 for L = 1, J =0, 1 J =), ses se. nee 3D 123 forL=2, =. The energy levels for mercury are shown in Fig. 1.24 with some of the #5: strongest lines of the spectrum. It is seen that the selection rules on AL ==> and AJ always hold, but that transitions with AS ¢ 0 do occur. It is also == -and since for nonrelativistic velocities 2 2pidp; dN (w; a aN (ui) =n = == 2m w;. (2.2) dw; hy Ni Wy — WE bis ig jexn ( +7 ) + | ' (2.3) B “where k is the Boltzmann constant, 7 is the temperature of the system, and wp is a characteristic energy, called the Fermi energy or Fermi-level s aS ~ SE Se se a Ss Ss ‘* SS ~ . SS RNARAN NSS a Ss “Tis interesting to note the properties of this function, graphed in Fig, 2,1: =(a) It is properly bounded, so that it can represent a probability 0 < N;/2n < I. 4s 2 Electrons in Solids Nz FIGURE 2.1 Probability of occupancy of a state of energy w; as derived from Fenmi—Dirac statistics. (b) For large values of w; it assumes the form of the Boltzmann distribution Const x exp(—w;/ kT). (c} For T = itis a step function, with N;/2n = | W; < WE N; /2n = 0 Wy > WE. (d) For T # 0, wp has the property that N(wF) = 5; and as many states above wp are occupied, that many states below wp are empty. ¢e} In solids and for average T + 0, the distribution function is only slightly modified from its shape at 7 = 0 (for solids wp is on the order of a few electron volts, while 1/k7 = 40 cV—! at T = 300 K). Combining the Fermi—Dirac distribution (Eq. (2.3)) with the energy density of states (Eq. (2.2)) it 1s possible to obtain any desired distribution. For example, the number of elecirons per unit volume (density) at an energy w in the interval dw is given by ; g - —[ N(w) dw = <5 Vimy {exp (A) 4 | dw. (2.4) If we express Eq. (2.4) in terms of the Cartesian coordinates of the velocity, Vx, Vy, and v,, and integrate over v, and vy, we obtain the number of electrons per unit volume with a given velocity in the z direction, v, (in the stiches cata ich tear a Mer tah caceneleetceesla heeteachca ca alil araatintatcanelittn tate SNELL ESSERE SD SSD ASAE ORAS BNR ec AS SLSR CHAN SS NHS State ete gta taty Sorts et td +. me REAR Eisenstein oN hi SAAS 2.) Solid Materials and Band Structure 49 eed fellas nn. Sefeledeelapyye} ay hs 4 Po ate EME et ot an eae! Sh ele the BEDE Bt bse BOREL Bat Niw) oe ee (a) (b) Be FIGURE 2.2 (a) Number of electrons with an energy wi in the interval dw. (b) Number ; a of electrons with z component of velocity v, in the interval du;. Te bs fe interval dv,). The result of this integration is* a Sn m2kT e Ni\due = Se iy ie, aa a) kT seth = 2 — (Se) fan a5) The two distributions given by Eqs. (2.4) and (2.5) are shown in Fig, 2.2. @: Even though the majority of the electrons in a solid are not free (as we = originally assumed), Fermi—Dirac statistics are applicable, especially to = metals, In metals at least one electron per atom has several states available fe (is in the conduction band), so that it can be considered free; since there == will be 6 x 10°? free electrons per gram mole, statistical methods are well =. jusified. p21 .2. Elements from the Band Theory of Solids = Up to now, no account has been taken of the interatomic or intramolecular oe forces that might act on the free electrons. Indeed, we expect (from previous = experience) that the consideration of some potential in the region where =the electrons move will result in the appearance of energy levels; however, pe because of the periodic structure of this potential, instead of energy levels, e2 energy bands appear, and only the states contained in these bands can be es ee 24 Sommerfeld, Thermodynamics and Statistical Mechanics, p. 285, Academic Press, New York, 1956. So ve SSO oes.” 50 2? Electrons in Salids FIGURE 2.3 Apcriodic potential that may be considered as an idealization to the actual potential of a crystal lattice. occupied (with any significant probability). In the following paragraphs we will sketch two approaches toward the understanding of the physical origin of the energy bands. Consider first the one-dimensional problem? of an electron moving in a potential consisting of an infinite sequence of “square” wells of depth Vo and width & and spaced at a distance / from one another (Fig. 2.3). The solution of the Schrédinger equation for such a potential gives for the electron wave function We =upye™ (2,6) with k = 27/4 = p/h the wave vector of the electron. This wave function consists of the plane wave part e*, and ux(x), which must have the periodicity of the lattice, namely, u,(x &/) = u,(x). If there arc N lattice sites, the length of the crystal is NJ and we impose the periodic boundary condition W, (x + NJ} = W(x). This leads to 2"! = 1, or KNi =n27 k=Hn2n/NI n=0,+1,+2,.... (2.7) Equation (2.7) determines the allowed values of &, which form almost a continuum because of the very large integer value of N. Note that for N = ] one obtains the familiar “particle in a box” energy levels, with —_ p 7 2 Fe 7 n2 he 3m om ml? SE, Merzbacher, Quantum Mechantcs, third ed., Wiley, New York, 1998. ec ace LSA nS sata acheter cues cue at acaat cls anata aut ateratttateatetutautaeeachughugy ata aniagpaeiselstarghehteatenetd tata altabthntaitaltettnatecdsestentattaadeett teat stained teas wept ttle etetata eae etna eee eee eat tata a ae a! RCRA RH ERG ORCS COLLECT CS Shite ch ei cht 2.1 Solid Materials and Band Structure 45) Having determined the wave function, itis possible to solve the Schrédinger equation for the energy eigenvalues H\x)=E(K)|\%) or (WEA |Wy) = E(k), (2.8) where H is the one-dimensional Hamiltonian operator 2 2 he. gS and V(x) is now the potential of Fig. - The solution of Eq. (2.8) is given in graphical form in Fig. 2.4. We note the following: (a) Even though all values of k are allowed, discontinuities anse at k = ss _ as /l (note that for this particular electron wavelength, Bragg reflection i from the lattice will occur with a half-angle @ = 90°; nA = 2/ sin @, hence Pe \ = 2L/n.and since A = 2m/k, it follows that k = nx/1). * . ., ¥ at Ue ON Ot a ' + ee ee eee oe see abe e uty? * anaes! apetepetecele ecegerecertat stay aa at atet atta tele te ath a ntatetiteteldie . . *eeee . . rere es i + si wot Da WEF OA . . nleTere eo W's 8 'a7e . - i pee = SSN » SSS ) &, A 4e That 440 SN AeSatasatetete lech nttatatatatacatates ote eles tele Phere oe eta %e%e Pan yy Se wes = e's’ = be ee be waa oe . Py Palate — . = « FPP Ft F ys . we ‘. . oe ry, eet aNa tate Syl eate ape nal ran) et eat eee! i oo” vveteerere te Sh : SE a ny S S *S a vet ietstetys eee wih ge; (b) Not all values of the energy are allowed, but only certain “bands”; fe: other bands of energy are forbidden, z: (c) The relation between E and p (or k) is no longer the familiar = parabolic 2 262 p kk E — — +— ——=Ps ~ 2m 2m (2.9) Allowed energy bands MM SUI ALLL LLLLLPLILLLLLLLL LL) (b) “FIGURE 2.4 Results of the solution of the simplified one-dimensional lattice problem. : *fa) Plot of energy E versus wave number k = p/fi for an electron in a crystal lattice. (b) The Be “Allowed and forbidden energy bands. HR? 32s 2s Electrons in Solids 2g Interatamic spacing FIGURE 2.5 Energy levels of a system of six similar atoms placed in a lincar array. We can, however, retain this relation if the mass mm is assumed variable and a function of k, namely, sh (d* FE f/dk*) The same formalism is carried over into three dimensions, but now the bands are replaced by allowed (Brillouin) surfaces and the axes of symmetry of the crystal must be taken into account. A different approach is to start with a molecular wave function and study its behavior as the number of identical atoms is inereased. In Fig. 2.5 are plotted the energy levels against interatomic distance for the ls and 2s states of a linear array of six atoms (after Shockley). If, then, in the limit the (almost infinite) array of the crystal is considered, the energy levels coalesce into bands, This is shown in the left-hand side of Figs, 2.6 and 2.7, where the energy bands plotted against interatomic spacing are given for diamond which is an insulator (after Kimball), and for sodium (after Slater), which is a conductor. If the lattice spacing for the particular crystal is known (from experiment), il is possible to read off from the graphs the limits of the energy bands. This is done diagrammatically on the nght-hand side of Figs. 2.6 and 2.7; also indicated is the position (in electron volts) of the Fermi level (as it can be calculated, for example, from Eg. (2.4) and the electron density within each band), m*(k) = (2.10) wianaat ete cecas rete cetepe aac atatebetaadaa ot Sats eanted Sega nena gegeae eceledoCaLLcaUaLALagacefesadaselalaLeSagutatisasaegetogetetatata’atatatapasataslsarayssurataseteneescassssertactecetegasyezatetatatad taf. cAtltocces att atatetatettetatalalateto cel etaratatelatutathtetitatates ste atecoQotati coegetettatvte,ofatatathteotnacitiecte PAE SERSER EERE IE Teac te occ CS ERE RAR LESS nese cclgche sh ashenoasg gta pagan etat ae 2.1 Solid Materials and Band Structure 3 Diamond C (1s)*{2s)*(2p)* Energy E Remaining 4 Stales per atom 6 stales per atom é he 2s 7 7 aaa 4 ay ge Galena tard Lo Valence band (2s)*(2p) 2 of BS | 1 *— Observed lattice spacing ae Lattice spacing Diagrammatic sketch = FIGURE 2.6 The energy band structure of diamond (insulator) as a function of lattice °° spacing. The observed lattice spacing is also indicated. = = *.' . Ae pNsatytaty acetacecatecarsh sa athe . Ne 8 te SS See by ihe eettene Sees paleo bl be be bet bet bbe | phrtel telatet stele ' peer etes ‘ Sodium —Na(1s)*(2s)*(2p)%(3s) Fermi level Valence band Lattice spacing Diagrammatic sketch ee FIGURE 2.7 The energy band structure of sodium (condactor) as a fonction of latice feo: spacing. The observed lattice spacing and position of the Fermi level are also indicated. ge From these considerations it is possible to understand the difference eg between conductors, insulators, and semiconductors. For diamond, for He example, the valence band is completely filled (this fact follows also from = the atomic structure of carbon and the deformation of the energy levels). 54 2 Electrons in Solids The next available states are approximately 5.4 eV higher and hence can- not be reached by the electrons, with a consequent inhibition of their mobility; diamond therefore behaves as.an insulator. For sodium, in con- trast, the Fermi level lies in the middle of an energy band, so that many States are available for the (3s) electron, which can move in the crys- tal freely; sodium behaves as a conductor. Pure semiconductors, such as permanium, have a configuration such that the valence band is com- pletely filled, but the conduction band lies fairly closely to it (0.80 eV). . At high enough temperatures (that is, on the order of a few thousands of degrees), the electrons in the valence band acquire enough energy to cross the gap and occupy a state in the conduction band; when this hap- pens the material that was previously an insulator becomes intrinsically conducting, Both the electric and thermal conductivity of a solid depend on the density and mobility of the free electrons. Completely analogous to the motion of electrons is the motion of “holes”; holes can be thought of either as “vacancies” in an almost-filled band, or as electrons with negative effective mass.* Due to their thermal energy, the carriers have a random motion characterized by (3/2)kT = E = m*u*/2. When an electric field is applied, a drift velocity is superimposed on the random motion of the Catmiers, resuliing in a steady-state current flow. 2.2. EXPERIMENT ON THE RESISTIVITY OF METALS In this experiment we will! explore the physics behind electrical resistance in metals. What’s more, we will do it with a novel technique that measures the resistivity of the metal, a property only of the type of material and independent of the size or shape of the conductor. This technique, in fact, can make measurements of the sample without actually touching it, and has found a lot of use in modem applications. It is based on the paper C. P. Bean, R. W. DeBlois, and L. B. Nesbitt, Eddy current method for measuring the resistivity of metals, J. Appl. Phys. 30, 1976 (1959). First, we make the connection between resistance and resistivity. We assume that Ohim’s law is valid, that is, ¥V = JR, where & is independent ‘This can be seen from Eq. (2.10) and the negative curvature of some parts of the E(k) curve of Fig, 2.4a, Sa ae SS LAT SSS SANS SE —1 a sat 5 we Ger ne" ra a=: “ Sonoe : a NHR HRHY UP) 8 2.2 Experiment on the Resistivity of Metals 55 Area A L FIGURE 2.8 An idealized resistor. of voltage or current. Consider the idealized resistor pictured in Fig. 2.8. The resistor has a length L and a cross-sectional area A. A voltage V is applied across the ends of the resistor. A current J of electrons flows from one end to the other, against a resistance R, which ts due to the electrons interacting somehow with the atoms of the matenal. Consider Ohm's law on a microscopic level. The magnitude of the elec- tric field setup across the ends of the resistor is just E = V/L. The electrons that carry the current will be spread out over the area A, so at any point within the resistor the current density is (magnitude) ; = J /A. Therefore Ohm's law becomes E= jp, (2.11) where L R= aa" and p is the “resistivity,” a property of the material that is independent of the dimensions of the resistor. Equation (2.)1) can be derived from the theory of electrons in metals. The resistivity arises from collisions between the electrons and the atoms of the material. In a metal, the electrons are essentially tree, so without any collisions they would continually accelerate under the applied field with an acceleration a = eE/m, where e and m are the electron charge and mass. However, the collisions cause the electrons to stop and then start up again, until the next collision, If the time between collisions is called r, then the “drift” velocity vy is just Vg = at = Po. (2.12) Now if there are n electrons per unit volume im the resistor, then a tota! charge gq = (nAL)e passes through the resistor in a ime t = L/vy. 5 2? Electrons in Solids TABLE 2.1 Electrical and Therinal Properties of Metals Electrical Temperature Thermal resistivity coefficient conductivity ap Name = Z A (u.2- cm) (1073 /K) (a4) (K) Al 13 26.98 2.65 4,29 0.53 393 Fe 26 55.85 9.71 6.51 0.18 420 Cu 29 63.55 1.67 6.80 0.94 333 zn 30 65.38 5.92 4.19 0.27 300 Sn 50 118.69 11.50 4.70 0.16 260 Pb $2 207.19 200.65 3.36 0.083 86 Bi 83 208.98 106.30 — 0.020 118 Therefore, the current density is I lq 1 nALe —_—_—— j= AATI™A Lin — neva, (2.13) and therefore, pal (2.14) ne-t Often the “conductivity” o = 1/p is used instead of the resistivity. Electrical resistivities are listed? for various metals at room temperature in Table 2.1. Also included are some thermal properties, which are closely related to the resistivity through the underlying physics.© One of these is the temperature coefficient of resistivity, defined as (1/p)dp/dT. This quantity is in fact temperature dependent as we shall see, and the quoted numbers should be valid near room temperature. Clearly, the fundamental physics of resistivity lies in the values for the collision time 1. The interaction of the quantum-mechanical electron waves and the quantized latuce of the metal crystal accounts for the collision time "Values for Z, A, resistivity, and thermal conductivity are taken from L. Montanet et al. Review of particle properties, Phys. Rev. D 50, 1241-1242 (1994). The temperature coefficient of resistivity, and all data for Zn and Ri, is from D. R. Lide, CRC Handbook of Chemistry and Physics, 56th ed., p. F-166, CRC Press, Boca Raton, FL, 1975. The Debye temperature is from E. U. Condon and H. Odishaw (Eds.}, Handbook of Physics, 2nd ed., Part 4, Tables 6.) and 6.3, McGraw-Hill, New York, 1967. An interesting exercise #6 to plot the electrical conductivity 1/p agamst the thenmal conductivily (sce Exercise 30 in Appendix G), i ee a ee a eee Seber eat rt woe teats Se he heer ACP oo eee be a og Safe al a a et mete BCAA H Haaeserth eget apeta te SSS aA aaa La esac acne usc aria ceca tae cages PRR Stee RRS SEAS SDN EAR scacasueaccentseicacsascesetetneegegenenugan tage SSSR ae . oes CSS att crac tcte ate td tat ataattct estas ltgetagesegteneete et Soa sts at onacan sce oc cue RRC cae id ' hee Bo eth het a is" tain on "herbal alta a ad * ' ' ih 4 a . 2.2 Experiment onthe Resistivity of Metals &? in a pure metal crystal. If there are impurities, then the scattering will contain an additional contribution. We can write I ] ] ee tT TCRYSTAL TIMPURITY The scattering from the crystal depends crucially on the vibrational energy stored in the crystal lattice, and therefore on teraperature. The impurity scattering is essentially independent of temperature. The technique we use measures resistivity directly. The idea 1s based on Faraday’s law, which gives the EMF (i.e., voltage) induced io a coil that surrounds a magnetic field that changes with time. That is, we measure a signal V(r) that is proportional to some dB/dt. The magnetic field B is generated by the “eddy currents” left in a metallic sample when the “:- gammple is immersed in a constant magnetic field that 1s rapidly switched “ off. Figure 2.9 shows how this is done. In Fig. 2.9a, a cylindrical metallic bar is placed in a constant magnetic field whose direction is along the axis of the cylinder. We assume the bar is not ferromagnetic, so the magnetic «<<. field inside is essentially the same as it is outside. Remember that the bar A on a ee Se et ee ee ee ee AMORA SMM ESOS “ig filled with electrons that are essentially free to move within the metal. Now we shut the field off abruptly. By Faraday’s law, the electrons in the metal will move and generate a current that tmes to oppose the change in the external magnetic field. These so-called eddy currents are loops in the plane perpendicular to the axis of the sample, and they generate a magnetic in (a) Field on (b) Field shut off = FIGURE 2.9 The eddy current technique for measuring resistivity. (a) A magnetic field = Bg permeates a cylindrical metal sample. (b) Eddy currents set up when the field is shut off a. generate a field B of their own. The eddy currents, and therefore 4, decrease with time at ceracans Set acalecarectDalaeS CHEERS ehiganunnia nummer SeeMM ANNU tetitatytetetateteteressrenetersteretetetetetereretinerermrerene eget sh. arate that depends on the resistivity. HR OE sC?)s« Electrons in Solids field of their own. See Fig. 2.9b, However, as soon as the external field is gone, there is nothing left to drive these eddy currents, and they start to decay away because of the finite resistivity of the metal. The tume it takes for the currents to decay away is directly related to the resistivity, as we shall see. We again use Faraday’s law to detect the decaying eddy currents. The mapnetic field set up by the eddy currents also decays away with the same time dependence as the currents, Therefore, if we wrap a coil around the sample, Faraday’s law says that an induced EMF shows up as a voltage drop across this coil. This voltage drop is the signal, and the rate at which it decays to zero is a measure of the resisitivity of the metal sample. In order te determine the voltage signal as a function of time, one needs to solve Maxwell's equations in the presence of the metal. The derivation is complicated, but outlined in Bean ef ai. (1959), where a series solution is obtained by expanding mm exponentials. For a cylindrical rod, this series takes the form Vit) x y- exp (—A?at), i=l where @ is proportional to p and the A are roots of the zero-order Bessel function, i.¢., Ay = 2.405, Ax = 5.520, A3 = 8.654, und so on. Since the dX increase with cach term, for long cnough times, only the first term is significant because all the rest die away much faster. That is, the falloff of V(t) with time will look like a single exponential if one waits long enough, but will be more complicated at shorter times. For a cylindrical meta! sample where the external magnetic field points along the axis of the cylinder, the result is Vit) = Voge tite, (2.15) where 9 2-5 r2 te = 2.17 x 107? | —— | —, (2.16) cm | 6 Vo = 1ONp Bo, (2.17) and ¢ = O is the time when the external field is switched off. In this equation, r is the radius of the cylinder, expressed in centimeters, and p is the resistivity of the metal, expressed in chms-centimeters. Also, N is were woe ae a es ah Ce ek Lr gt am ey Ce oe se ‘ re ri rr ae Sook ity SNS Anat eae St co etal cece alo atta AR SS SESS HR Re uu ET rhe Pata atat ee Tne MO BL earn Gis ot Ae oo oo er er NaC SU at he oe pe ratahte want Nitabat ten r F i oe ee ee oe es i re 2.2 Experiment on the Rasistivity of Metals 458 the number of turns in the detector or “pickup” coil and By = join Cin ST units) gives the magnetic field Bg set up by a solenoid carrying a current i through x turns. This equation is only valid for times t on the order of te or larger. At earlier times, there are transient terms left over that cause V(r) to fal] off more rapidly than given by Eq. (2.15). 2.2.1. Measurements . The lifetime t_ given by Eq, (2.16) is on the order of tenths of milli- seconds. Therefore, the magnetic field must be switched off considerably more rapidly than that. This ts hard to do mechanically, so we will resort to an electrical switch, using a transistor.’ The circuit that produces the switching magnetic field is shown in Fig, 2.10.8 A garden variety 6-V/ 2-A power supply puts current through the solenoid, creating the magnetic field Bo. However, after passing through the solenoid, the current encoun- ters a transistor (321/TIP 122) instead of passing directly back to ground. The lead out of the solenoid is connected to the collector of the tran- sistor, and the emitter is connected to ground, The base is connected through a 1-k& resistor to the 600-2 output of the HP 331/1A wave- form generator, The waveform generator 1s set to produce a square wave, oscillating between around —I0 V and +10 V with a period of a few milliseconds. Consider the current through the solenoid. First, the DC power supply is connected so that the solenoid is always positive with respect to ground, thus the collector voltage is always above the emitter voltage. Second, the base-emitter acts like a conducting diode, so there will be a voltage drop across it of around 0.6 V when it conducts. Also, if there ts no current through the base, then the base-collector junction is reversed biased and no cuirent flows through the transistor, or therefore through the solenoid. That is, the switch is off. Now when the waveform generator is at +10 V, the current through the base is ig * 10 V/I kK = 10 mA. This tums the switch on atid lets the current flow through the solenoid pretty much as if the transistor wasn't there, so long as [Ir < Blg = 10 A. You might want to measure the resistance in the solenoid coil to make sure it does not 1 This ransistor is actually a “Darlington pair,” which effectively gives a single transistor with a gain parameter Ape = § = 1000 orso. Vop = 6 V does notexceed the specifications. ®For students with minimal experience in laboratory electronics, Sections 3.1, 3.2, and 3.3 should be consulted. 60 2 Eteactrons in Solids HPasi14 600 Ohm = Ground at a HP33i1A Test point (Probe to scopes channel 2) FIGURE 2.10 Switching circuit for turning the magnetic field on and off, It is a good idea to check the current through the solenoid by measuring the voltage at the testpoint, timed against the HP33114 square wave generator. draw a lot of current, but since you are using a 2-A power supply, it is a good bet that you are in the clear. So, when the square wave generator is at +10 V, the solenoid conducts, However, when the generator switches to —10 V (or presumably anything less than around 0.6 V), the solenoid and the magnetic field shut off. This is, t = 0 in Eq. (2.15). The pickup coil is wound on a separate tube, which can be inserted inside the solenoid. One can then introduce and remove different metal samples from inside the pickup coil. By connecting the terminals of the pickup coil to a digital oscilloscope, we record values of V (¢) corresponding to Eg. (2.15). There is one complication. The magnetic field shuts off so fast that the imstantaneous induced voltage in the pickup coil is very large. That is, Af is so small that dB/dt =~ AB#/Atr and therefore also V are very large. An oscilloscope would typically have circuitry that protects it, but one should take some care to avoid damaging the equipment. To fix this problem, the simple circuit shown in Fig. 2.11 is used to connect the pickup coil terminals to the oscilloscope input. The two diodes are arranged so that any current is taken to ground, so long as the voltage is bigger than +0.6 V or smaller than —0.6 V, for diodes with Ve = 0.6 V. That is, the circuit “clamps” the input to the oscilloscope so that it never gets more negative, but still big enough to make the mcasurement. Ce ae ae he a + ee * wee 1 a ™ er ae ek . ee A rae See wee he ee Be bt ee LLC ned be aie ae . oT te . 1 ek a ek ee ae Le ee ete tal Be hr) Lo 4 vor A te Si he ale rr TL srutatatinatitatocrectisistecenegete! pet rae Ber hel el ha he hla fat al Sta Te rsa SAI Co ee PCN a ee ae ei Le Sao oO a te SRR RTRs use nn uk unmnnnannneniuneanunennnsnuasnnnumaiarememn eee sneer remem rama terre namnn ry ranean 2.2 Experiment onthe Resistivity of Metals 64 In (from pickup coil) Out (fo scope) FIGURE 2.11 Clamping circuit for the oscilloscope input Sometimes we see the signal “ring” just as the switch shuts off. That is, we see the decaying exponentia! but a rapid oscillation? is superimposed on it, and this gets in the way of measuring the decay time. If the ringing goes away while the signal is still decaying exponentially, just use the data past the point where the ringing is gone. Otherwise, a resistor should be attached in paralle] with the scope input. It is best if you can get a variable resistor, and play with the values so that the exponential decay is unaffected but the ringing is thoroughly damped out. Before measuring the resistivity, one should know what the solenoid circuit is doing. Connect a probe to the junction between the solenoid and the transistor collector. View this on the other channe! of the oscilloscope, and confirm that you see what you expect. That is, when the square wave is high, the solenoid is conducting and the voltage at this point should be around -+-1.2 V,1.e., the sum of the two forward voltage drops for the CB and BE diode equivalents for the transistor. On the other hand, when the square wave is low, the solenaid should not be conducting and there is no voltage drop across it, so the voltage at this junction should be around +6 V, i.e., the voltage of the DC power supply. This probe should now be removed since the oscilloscope channel is needed to make the resistivity measurements. Next, connect the pickup coil to the clamping circuit and plug it into the second channel of the scope. Do not put any metal sample in just yet. You should see a voltage spike, alternatively positive and negative, when the magnetic field switches on and off, clipped by the diode clamping circuit. Now insert a sample into the pickup coil. Watch the pickup coil signal on the scape as you do this. The effect of the decaying eddy currents ? The circuit has lots of “loops,” each of which is essentially an inductor. Any capacitance “somewhere wil] cause osciljacions, but the exact source can be hard to pin down. One should lake care to wind the pickup coil in a way that minimizes the inherent capacitance. A good way to do this is to crisscross the windings of each layer. f2 2? Electrons in Solids 10° Coil signal (V) ia”! 0 0.2 0.4 0.6 0.8 Time (ms) FIGURE 2.12 Resistivity data taken with a high punty aluminum rod as the sarmple. The decay is clearly not described by a single exponential at the earlier times. should be clear. You may see some transient oscillations of the signal right after the field shuts off, bui there should be plenty of time left after these oscillations die away for you to get a smooth curve. Figure 2.12 shows data acquired with a hin. diameter high-purity aluminum rod!° at room temperature as a sample. The data poinis are the output of a digital oscilloscope displayed using MATLAB, Note that at the earliest times, there are higher order contributions to the signal (as described by Bean ef ai.), and one must choose a suitable range over which the data are indeed described by a single exponential, The fit shown in Fig. 2.12 yields a decay time te = 3.051 x 10 s. Then, from Eq. (2.16} we find for the resistivity 217 x 10-7 x r? (cm*) = 2.87 x Jo-° a. cm, te (Ss) where we used the fitted value of tg and r = 0.635 cm. This compares well with the value listed in Table 2.1. ———e 10From the Alfa Aesar company, http//wew.alfa.comy/. aoe Cea Sei ete tochantecatatade stata? Hh ee ee a a es valet DS A ee ee i ee Oe ie ek en a a or) week Be etre eb er ee we ey a Pte tae a a hints gate sgatinetepe cones: Sa a ae a NAN Ne ot PNT ae REECE partners x Renee ee ce a] Pet ne aT oF ee | ty! cae wate to. Oe ee ee ee ee ee oe ee ee ae - es ae ee ee eee ee Be tL settee ba as PaO A a al A a ee Pa at at ba a hs tah 4 Lit et er | be hal tal eal he ee es ae bet SC 1 ieetipeaatal tay wpataat 2.3 Experiment on the Hall Effect The main source of systematic uncertainty is likely to come from the ° times over which the decaying voltage signal is fitted. At short times, the : decay is not a pure exponential because the transient terms have not all died away, so we want to exclude these times when we fit. At long times, Es there may be some left over voltage level that is a constant added to the 7 exponential, and again, a pure exponential fit will be wrong. Varying the “. gpper and lower fit limits until we get a set that gives the same answer as g set that is a little bit larger on both ends is one approach. One should be "convinced that the results are consistent. For example, use aluminum alloy “rods of the same composition but different radii, and check to make sure * that the decay lifetimes tg scale like r?. This should certainly be the ease ’ to within the estimated experimental uncertainty. “ Having learned how to take and analyze data on resistivity, We can now " investigate the temperature dependence. It is best to start simply by com- ‘ paring the two samples of Anin, diameter aluminum rods, one an alloy and ~ the other a (relatively) pure metal. Vary the temperature by immersing the ' samples in baths of ice water, dry ice and alcohol, and liquid nitrogen. :: Boiling water or hot oil can also be used. These measurements are tricky. . One must remove the sample from the bath and measure the eddy current :- decay before the temperature changes very much. Probably the best way to * do this is to take a single trace right after inserting the sample, stop the oscil- * loscope, and store the trace. Then one analyzes the trace offline to get the = decay constant, One might also try to estimate bow fast the bar wanns up by : making additional measurements after waiting several seconds, e.g., after ” saving the trace. This would best be done with a sample whose resistivity, : and therefore tz, can be expected to change a lot with temperature. Pure : aluminum is a good choice. Remember that the temperature dependence =: will be much different for the pure metal than for the alloy. Try to estimate : the contribution to the mean free path of the electrons due to the impurities. = 2,3, EXPERIMENT ON THE BALL EFFECT . In Section 2.2 we saw how collisions of electrons with the crystal lattice : lead to an electrical resistance, when those electrons are forced to move “under an electric field. If one also applies a magnetic field, in a direction = perpendicular to the electric field, then the electrons (and other current * Carriers) will be deflected sideways. As a result an electric field appears in : this direction, and therefore also a potential difference. This phenomenon 64 2 Electrons in Solids is called the Hall effect, and has important applications both in identifying the current carriers in a material and for practical use as a technique for measuring magnetic fields. | Let us rewrite the microscopic formula for Ohm’s law, but this time taking care to indicate current density and electric fields as vectors, and to also note the negative sign of the charge on the electron. Following ; Eqs. (2.12) and (2.13) we write ae j= —nevg = ne*tE/m (2.18) or aw = ~eB. (2.19) It is clear that in Bq. (2.19) we have made an approximation, replacing & the time rate of change of momentum, i-e., dp/dt = mdv/dt, with an = expression that uses the average acceleration vqg/t. This is how we have FE taken into account collisions with the crystal lattice. . li is swaightforward to modify Eq. (2.19) to take into account the elect of a magnetic field B. We have 3 aS = —e(E+¥3 «x B). If we assume that the magnetic field lies in the z direction, and define the cyclotron frequency w, = ¢8/m, then we can rewrite this equation as et Vd, = —— By - We TUd, ie) : ET : Ud, = ~7 Ey + @ Tt Udy (2.20) | eT Vd, = —— #;. rh Consider now a long rectangular section of a conductor, as shown in | Fig. 2.13. A longitudinal electric field E, is applied, leading to a current density flowing in the x direction. As this electric field is initially cumed on, the magnetic held deflects electrons along the y direction. This leads to a buildup of charge on the faces parallel to the xz plane, and therefore an electric field Ey within the conductor. In the steady state, this clectric field cancels the force due to the magnetic field, and the current density is strictly a ‘ . Wore ae ‘* . * . 2.3 Experiment onthe Hall Effact 65 Magnetic flatd 8, | y x ot es “ee a4 TS Tt : Section -- perpendicular =. parpandicular ~ 102 axis; . = - _ drift velocity Benne Srna ; x eat ayia * * - The appearance of the electric field Ey is the Hall effect. A convenient experimental quantity is the Hall coefficient Ry, defined as Ey y= - iB The quantities E,, j,, and 8 are all straightforward to measure, and in our i222: Simple approximation for electrons in conductors we have (from Eq. (2.18)) (2.21) D je = ne* TE, /m; therefore, eBrE,/m ] we 2.22 A (net E,/m)B ne ame) 65 2 Electrons in Solids That is, the Hall coefficient is the inverse of the carrier charge density. In 3: fact, the Hall effect is a useful way to measure the concentration of charge | carriers in a conductor, It is also convenient to define the Hall resistivity as = the ratio of the transverse electric field to the longitudinal current density, = that is, pu = Ey/jz = BRu, (2.23) 2 which depends (in our approximation) only on the material and the applied 2 magnetic field. 2.3.1. Measurements In order to measure the Hall effect, one needs a sample of a conductor, :: but not an especially good conductor. This is because one also needs a “3: relatively low carrier density ne in order to get a sizable effect; this of =: course leads to a relatively high resistivity, As seen in Table 2.1, bismuth “2 is a good candidate metal, and we describe such an experiment here. !! The setup uses a bismuth sample with rectangular cross section, mounted Ee on a probe with attached leads for measuring current and voltage. A ther- 2:2 Seat totagat bent mocouple is also attached to the sample so that temperature measurements “2 can be carried out, The magnetic field is provided by an electromagnet = capable of delivering a ficld up to ~5 kG over a volume roughly 1 em?. The bismuth sample probe is shown in Fig. 2.14. The width of the bismuth sample is w = 6.5 mm and its thickness, measured with a micrometer, 2% ist = 1.65 x 10~* m. The effective length of the sample is the distance 2 between the leads used to measure the current (“white” and “brown,” as “= shown in Fig. 2.14). In our case, this distance is 2 = 7 mm. Current is & supplied by a DC power supply, connected to the sample through the “red” “2 and “black” leads. The Hall voltage is measured with a digital multimeter,’ using the “green” lead and the output of a potentiometer used to balance & the voltage on the “white” and “brown” leads. A separate bundle of wires S are connected to leads that carry current to the heating resistor, and to a 4 thermocouple that measures the temperature of the bismuth sample. Begin by determining the Hall coefficient at room temperature and for a relatively high magnetic field. Turn on the electromagnet power supply to “2 1) Semiconductors also make good candidates, with a very low carrier density compared a to a metal. For a description of such a setup, see A. Melissinos, Experiments in Modern Physics, First ed., Academic Press, New York, 1966. orn EEE RBA Donn Dn ~CASOSATATALOL ALASALE LOC aTeDecapaceocetaetecusereret = 2.3 Experiment on the Hall Effect 67 oF) (Fy _ ss 8 $44.4 - Sa a, an ae SASS D R g ~ a Se = BIGURE 2.14 Schematic of the probe used to make measurements of the Hall effect “in bismuth, Electrical connéctions are made to the bismuth sample using copper leads, A =". thermocouple, as well as a resistor which acts as a heat source, is also attached to the sample. =.Fwo separate bundles of wires emerge from the probe, one of which is used exclusively for =. heating the sample and for measuring its temperature. . Pid Gata te ie Be a ire ta kok OE hd had hee See Wyn RD Aaa SENS ox aay we a ™ =" around 4 kG, It will likely need an hour or so to stabilize. In the meantime, ge with the sample probe removed from the magnetic field, run about 3 A ‘=: through the bismuth sample, and adjust the potentiometer so that the Hall == voltage is zero, Return the current through the sample to zero. The sample = can get quite hot while it is conducting so much current. Be careful not to = touch it, or to touch it to anything else. When the electromagnet is stabilized, measure and record the magnetic = field using a gaussmeter, or by some other technique. Now, place the sample =. probe in the center of the magnetic field. Quickly raise the current / through =the sample to 3.0 A, and record the Hall voltage Vy. Then, quickly, reduce =the current by 0,25 A, and record the Hall! voltage again. You should carry = kT, (2.25) the Fermi distnibution degenerates to a Boltzmann distribution. (Here F =. ts the energy of the electrons as measured from the top of tbe conduction * band; obviously it can take either positive or negative values.) With this -. assumption the integration is easy, yielding 3/2 3/2 n= (=) pERs/kT x (=a) eg Eg/Ar. (2.26) 121 the effective masses of p- and n-type carriers are the same. LA 74 =62:sCElectrons in Solids similarly, 2/9 3/2 | ase _ (Pamuk AE ED IET pe (Fo) Fu (2.27) | h? he It is interesting that the product np is independent of the position of the | Fermi level" especially if we take m. = mj, <= np=231 x 10! T3e7F e/KT From the analysis we expect that as the temperature is raised, the density of the intrinsic carriers in a semiconductor will increase at an exponential” ate characterized by £,/2kT. This temperature is usually very high since se 0.7 V (see Eqs. (2.29)). “Ne have already meotioned that impurities determine the properties of a semiconductor, especially at low temperatures where very few intrin- 3 sic Carriers are populating the conduction band, These impurities, when = in their ground state, are usually concentrated in a single energy level:: lying very close to the conduction band (if they are donor imputities) or ® very close to the valence band (if they are acceptors). As for the intrinsic’ carriers, the Fermi level for the impurity carries lies halfway between the: 2 conduction (valence) band and the impurity Jevel; this situation is shown in, = Figs. 2.19a and 2.19b, [If we make again the low temperature approximation . 3 of Eq. (2.25), the electron density in the conduction band is given by 9) Ty 3/2 : me n= Ng (=) oe Bal2kT (2.28) 2 he where Ng is the donor density and Eq the separation of the donor energy 2 Jevel from the conduction band. In writing Eq. (2.28), however, care must- 3 be exercised because the conditions of Eq. (2.25) are valid only for very: 3 low lempcratures. Note, for example, that for germanium E,=O7 eV, andforkT =O7cV, TF =8000K whereas &g=O.0leV, andforkT =O0leV, TFT =—120K. (2.29 Thus at temperatures T ~ 120 K most of the donor impurities will be in the: : conduction band and instead of Eq. (2.28) we will have n = Ng; namely. : 'SThis result is very general and holds even without the approximation that led ws Egs. (2.26) and (2.27). wa pate te rat BODE EAn " 7 ata utegesnaiisriesen wee sian Aa Ronnsterentct atti wien wt Parte SERRE Peta ey eecetatehh essay CCU anratyerageancbeey REN Seas Lk Xs me re st ahh be cai i : I Ree one ees : es ee ‘. se a My nee SSB ae ee na tte bk rt, -_ ae ence ere a RRC ties a oy Sta ete uty ne Hat L wey Sis eteletatetet te . reletatetinety chahtgitaetsitdntacet SEAR Ree er “eee an a ht 2.4 Semiconductors 75 WLLLLE (a) =. FIGURE 2.19 Same as described in the legend to Fig, 2.18 but with the addition of “> impurities. (a) The impurities are of the donor type and lie at an energy slightly below ==. the conduction band. (b) The impurities are of the acceptor type and lie slightly above the = yalence band. Note the shift of the Fermi level as indicated by the dotted Line. = the density of impurity carriers becomes saturated, Once saturation has =: been reached the impurity carriers in the conduction band behave like the =: free electrons of a metal. = 2.4.2. Sketch of p~n Semiconductor Junction Theory ==: Semiconductor materials with high impurity concentration, when properly =: combined, form a transistor. Junction transistors consist of two junctions of =: dissimilar-type semiconductors, one p type and one jn type; the intermediate region, the base, is usually made very thin. We will bricfly sketch the zcmaterials with dissimilar band structure are joined, itis important to know “at what relative energy level one band diagram lies with respect to the Roeicrcnenencties Se ees a eee ey = behavior of such a p~n junction and then see how the combination of = two junctions can provide power amplification; for this we will use our = knowledge of the band structure of semiconductors and the position of “other: the answer is that the Fermi levels of both materials must be at the “same energy position when no extemal ficlds are applied; this is shown ain Fig. 2.20. From the energy diagram of Fig. 2.20, it follows that only electrons with UE > AW, will be able to cross the junction from the n material into the p y fegion and only holes with Z;, > AW, from the p region into the 7 region. 76 2 Electrons in Solids Increasing @ Downhill potential a a = . £ 3 _ & _. rs ir Freterred Increasing -V direction of | potential AW, moton (downhill FIGURE 2.20 Structure of the energy bands at the junction of an n-type and a pype semiconductor. Minority cartlars Reverse bias (a) to-—of Minority carrlars to + of A P Battery Batle Y -V (AW, a a tae sais Qs SN watt ey at he nl aa a 2a wy See se eect SSeS FIGURE 2.21 Structure of the energy bands at a biased n—p junction: (a) reverse bias and : ie (b) forward bias. The solid dots represent electrons, whereas the open circles holes. Holes in the n region or electrons in the p region are called “minority 23 carriers.” Indeed, there will be diffusion of some minority carrters ACTOS. the junction, but since no electnc field is present these carriers will remain : in the vicinity of the junction. '* If now a reverse bias is applied—that is, one that opposes the further motion of the minority carriers—the Fermi levels will become displaced ™: by the amount of the bias, as shown in Fig. 2.21a. We see that the barriers '4The resull of such diffusion is the buildup of a local charge density, which prevents: further diffusion. Throughout the present analysis, however, we will neglect the local effects: at the junction. naeretats ue pat tet we vate . Preah tes : ie ee ah meee Re As Ss Re = as a hy 4 att hh er hr at Pen tr& oe OO 2.4 Semiconductors TV? 2A W, aud A W;, are increased by almost the full voltage, making any motion of minority carriers across the junction very improbable. Figure 2.21b, on =4he other hand, shows the situation when forward bias is applied (favoring ==the motion of minority carriers). The Fermi levels are now displaced in the = opposite direction so that the barriers are lowered. However, the full bias “voltage does not appear as a difference between the Fermi levels because =“qdynamic equilibrium prevails. There is a continuous flow of minority car- =-tiers in the direction of the electric field (holes obviously moving in the =cgpposite direction from electrons) and as a result a potential gradient exists =< along the material; thus the entire bias voltage does not necessarily appear -at the junction itself. ~- We will now consider two junctions put together; in Fig. 2.22a, p-type, ‘n-type, and again p-type material are joined. When neo bias is applied, we “expect the Fermi levels to be at the same position, with the resulting config- =-“uration shown in the diagram; in agreement with our previous conclusions from the consideration of a simple junction, we see that barriers exist for f> 1. Therefore, we present the V—/ curves for Vg > 0, on a semi-log plot in Fig, 2.24a, From the fit we find the slopes T=24C=297K = e [2kT = 21.3 V" 319K = 19.6 V7! 342 K = 18.9 V7, ee Note the onset of saturation for biases Vz = 0.5 V and also the different "> intercepts at V = 0. We observe that the measured slopes do indeed scale with temperature as expected and if we average the three results we obtain e/2k = (6.28 + 0.19) x 10° K/V. ‘SG. W. Neudeck, The p-n Junction Diode, Addison-Wesley, Reading, MA, 1983. sahunnnnsimn senate 80 2 Elactrons in Solids (@) 103 [yaesec o T=37°C e T=24°C 10° as _ 40° Current (WA) 0 0,1 0.2 0.3 0.4 0.5 0.6 Bias voltage (V_) Currant (pAj cn l -_s Lair O8 OF -O6 O05 44 03 -02 -01 0 Bias voltage (V,) FIGURE 2.24 Measurements of the current through a diode as a function of bias voltage, S Z for different temperatures, (a) is for positive bias, plotted on a semilogarithmic scale, Exponential fits are indicated. (b) 1s for negative bias voltage, plotted on a linear scale. Rronirentan Ss Van a el er a stat Seb hng ee 2] ae cage eee ath Fa odo od ad nd dod bd od 0d bd 0d eo uated aT LT ete tata eeete OM A ee Es ee nen he me br oeteee 25 High 7¢ Superconductors 81 Thus, using the value of the Boltzman constant k = 1.38 x 10-79 J/K we find that e=(1,73+0.05) x 10°" C in good agreement with the value of the electron charge. The different intercepts are an indication of the vanation of /g with temperature. (Of course at Vg = 0, J = 0 but this point cannot be reached on the logarithmic plot.) A better way of determining /p is by applying negative bias. From the negative bias data (Fig. 2.24b) we find that 1=297K h=3.9pA 303 K =44pA 310K = 6.7 pA 319K = 11.6 pA, The reverse current is proportional to the minonty carrier density. As the temperature increases, the population density increases as no oe F/2kT where Ey is the energy gap between the valence and conduction bands. From the data we find thal E, = 0.84 eV. This is in reasonable agreement with the energy gap in silicon (1.1 eV at room temperature), Systematic error can come from a number of sources, including contact potential differences and the extent to which the negative bias data of Fig. 2.24b has reached its asymptotic value. 2.5. HIGH T. SUPERCONDUCTORS 2.5.1. Introduction In 191] it was discovered that certain metals completely lose their electrical resistance when cooled to very low temperatures, typically less than 10K. The loss of resistivity sets in sharply when the critical temperature T, 1s crossed. This is analogous to a phase transition between different states of matter, as for instance from ice to water. The phase diagram for the f2 2? Electrons in Solids FIGURE 2.25 Typical phase diagram of the superconducting stale in the 7, H plane. 7 D> qT, T superconducting state depends on the temperature and the external magne- a tizing field 4 as shown in Fig. 2.25. Values of 7, and Bo for some common . = metallic superconductors are given below Te Ho Niobium (Nb} 946K §.1944T Lead (Pb) 7TBIK 0.0803T Mercury (Hg) 4.1K 0.0411 T By now many materials have been found to become superconducting at = low temperature and such behavior can be explained by the BCS theory.!® 23 The basic mechanism is that at low temperature electrons bind in pairs with =: opposite spins. The net spin of the pair is zero so the pairs obey Bose statis- =: tics and can move through the lattice without scattering, namely without = resistance, This gives rise to a supercurrent, which once started contin- “= ues to circulate even after the external electromotive force (i.e., the applied =: voltage) is removed. In fact, all of the pairs occupy the ground state andcan be described by a wave function that extends over macroscopic dimensions. <3 Superconductors not only exhibit zero resistance (when below T,) but °: also have the property that no magnetic field can exist inside the super- Re cadcanhiccaat eee cnn SSS SS Gs - a Se conductor. Inside an idealized “perfect conductor” the magnetic field = canpot change, dB /dt = 0, This is because any change in the external field =: induces, by Faraday’s law, surface currents that exactly cancel the effect of 16}, Bardeen, L. Cooper, and J. Schrieffer received the 1972 Nobel prize for their © microscopic theory of superconductivity proposed in 1957. We! eae tates Diy ‘ee oe * wietat tet ~O) anes 25 High Te Superconductors 83 LAND AN B.=0 Room - tamperature 8, (a) cial (e) > Cooled wee (b) Bee Low Soe By tamperature , sree pooner ae (c) ae 0 B-= G (d) (9) = FIGURE 2.26 Behavior of a superconductor when placed in a magnetic field. (a-d) The | field is switched on after the sample is cooled below 7-. (¢,f) The field is applied before éooling the sample. In either case the flux is expelled from the superconductor and no field tg trapped in its interior. Pehavavenatalasatateatatatntylanetetace tnt site! the external field inside the conductor. Ina superconductor, however, B = 0 in the interior region, irrespective of whether the field is applied before or after the superconductor is cooled below 7,. This is shown in Fig. 2,26. ‘The exclusion of the magnetic field (flux) from the interior of a super- conductor is called the Meissner effect and can be easily demonstrated by levitating a small permanent magnet above the surface of a superconduc- era Ye: or. This is shown in Fig, 2.27 where the solid lines are the magnetic field guns of the permanent magnet. Since the superconductor must expel the flux from its interior, the induced surface currents produce the field shown a by the dotted lines, me ee cancel ue external field in the interior of ere er Po « 84 2 Electrons in Solids Permanent magnet 4 Superconductor FIGURE 2.27 A permanent magnet is placed above a superconductor. The solid lines : 3 are the flux produced by the permanent magnet. The dotted lines are the flux produced by = the induced surface currents and completely cancel the externa! flux in the intcrior of the 2% superconductor. In the exterior they give rise to a lift force. ee eee Aa Sans tat bone Roca ao Cr a een kok ge ee L Ss ae FIGURE 2.28 Levitation of a smal] permanent magnet above a YBCO pellet cooled by E liquid nitrogen. Courtesy Colorado Superconductor, Inc. . ae distance increases, the magnetic lift force decreases; equilibrium is reached Fe when the lift equals the gravitational force on the magnet. Figure 2.28 is: an actual picture of levitation duc to the Meissner effect using the high 7, superconductor discussed in the next section. a gee 25 High 7g Superconductors 85 sec Superconductors are widely used for the construction of high-field mag- pets. They are also extensively used in some of the most sensitive scientific e'instruments; finally, they display fascinating quantum-mechanical effects ae ‘io a macroscopic system. !? pe 25. 2. Observation of the Superconducting Transition : in YBCO ! AN ene ta ale eee Se ke * *, * + Sa s ~~ mH eee ax An 1986 Bednorz and Miiller reported superconductivity at temperatures in excess Of 40K in certain samples of La~Ba—Cu-O. It was soon dis- ‘covered that the YBazCu307 ceramic (YBCO) undergoes transition to =the superconducting state above 90K. Pellets of YBCO can be manu- = factured i in the laboratory by mixing the chemicals in powder form and = compressing them in a steel die using a hydraulic press (to approximately 15, 000 psi). The pellets are then heated in a furnace to about 900°C in an = “oxygen atmosphere and allowed to cool. However, itis by far more conve- Bs ion to order 1-in.-diameter disks of YBCO from a commercial supplier, eae =A reliable source is Colorado Superconductor, P.O, Box 8223, Fort Collins, 7 Ze0 80526. . To measure the resistivity of the sample, a four-point probe, as well ss thermocouple leads, are attached to one side of the disk as shown in a | Fie 2.29, The probes can be fastened using conductive epoxy. 18 * The whole ~* te Baca nca ens ee *y! Ws rs ‘ * Se ae ue rere Coo Ge = Data can be taken as the sample cools or, as was done for the data presented : es =here, by first cooling the sample for 2 min. and then removing it from the Paid Nz bath, Temperature and resistivity are recorded as the sample pe ‘: "The four connections (see Fig. 2.29) are spaced equidistantly, separated “y a distance s, typically ~1 mm. A high-impedance source Bibe a ? ferminals 2 and 3 is measured. For a flat temple of thickness ¢ < s, as tb the present case, Current rings emanate from the outer tips, so that the en = ee oe in| x 2s as R=] p= ws] 32. aie gy 2ext dnt g 22t 2 Ze gee R. P. Feynman, The Feynman Lectures, Vol. Ul, Lecture 21. ie ate The commercial pellets can be obtained with all the leads atlached. 36 2? Electrons in Solids SADE ea PaO ia ee a YBCO peilet oe THEE eo pase" mee at ee ee Ee a . orn tiatitete ities et eee rect MOM BeAtabeb Thermocouple + bay teats al aa nh Be ae ae tats a ice water bath FIGURE 2.29 Schematic of the connections to the four-way probe and of the measuring ee apparatus. oor Furthermore, due to the presence of two outer tips, V23 = 2/ R, so that for : z a thin sample peas ni {V poe p= ina (7) . (2.32) 2 Note that the probe spacing s does not enter Eq. (2.32). 7 In these measurements, the constant current source provided / ays 500 mA to terminals | and 4. Typically, in the normal conducting state“ V23 < 1 mV, whereas below the transition, V3 is at the noise limit of: the HP 34401A meter used for the measurement (V3 ~ 10 wV). Since: the transition occurs rapidly it is important to use a computer to record the: data. In the present case data were recorded every 0.33 s. The HP meter was) connected to the computer through an RS232 serial port, the thermocouple: voltage and source current through an ADC card. as tine OSG was nh Molde a eae a) ae 25 High Te Superconductors 87 — <---> - ST 390 95 100 105 110 115 120 125 130 Temperature (i) 7 IGURE 2.30 Plot of V2, vs 7. Below the mansibon at 7 = 98°K the voliage on the ria rons terminals is conspatible with zero. The transition width AT < 1°K bi SNA NNSR ERRAND mors SE PN? SS SORE = . ~ Oo ent eae ee DS ipo seas o ooo e ete se ete” eeeeee ree © Bo -’ Results obtained by a student are shown 1n Fig. 2.30, which is a plot of = Voy vs T. It is clear that a phase transition occurs at T = 98 K. The width =: of the transition is AT < 1K. Note that the voltage V23 below Tz is too D peereall to measure. For T > Ty, the resistance across terminals 2 and 3 of the probe is of 7 es 777 Ro = V3/21 ~7 x 10-4 V/1 A=0.7 mQ, p=3.15 x 1074 Q-cm. SS ’ ‘ Shi cr ee This: is two orders of magnitude higher than the resistivity of metals (see | y the resistivity with temperature for T > T, is also expected since the eB BB “normal” electrons scatter from the lattice thermal vibrations as discussed mein Section 2.2. Ce ee > ~ i Rete | j $8 2? Electrons in Solids 2.6. REFERENCES For the material covered on semiconductors, the reader may also consult 2 the following texts: W.C. Dunlap, In, Aa introduction to Semiconducturs, Wiley, New York, 1957, Brief but clear treatment. R.A. Dunlap, Experimental Physics: Modern Methods, Oxford Univ. Press, New York, 1988. Detailed discussion of semiconductors, their physics, and device applications, a C. Kittel, lutraduction to Solid State Physics, 7th ed., Wiley, New York, 1996, A mute general treatment “2 of the solid state. O€ W. Shockley, Electrons and Holes, Van Nostrand, New York, 1950. A thorough presentation of the. subject. : aa oy Po a 4, The lecture on superconductivity in the Feynman lectures, Vol. H-21 is highly recommended. Also highly recommended is the text nam + sega 2 inne SSS A. (. Rose-Innes and E. H. Rioderick, /ntroduction to Superconductivity, Pergamon, Elmsford, NYSE 1969, r aie For a practical account including information on high T, materials one: can consult ae peices Dy. Prochnow, Superconductivity: Experimenting in New Technology, Tab Books, Blue Ridge Summit, 2! PA, 1989. SA ie Sans snes arate SERRE wig ie ates Ne he LS + es ~ teatatarerara at Se eas Electronics and Data Acquisition a : Up to ths point, we have descnbed measurements that require only -eidimentary laboratory equipment. Before conuinuing, however, we will =-discuss a broader range of topics in electronics and data acquisition, rf 3 1. ELEMENTS OF CIRCUIT THEORY - Nearly every measurement made in a physics laboratory comes down to ze-determining a voltage, so it is important to have at least a basic understand- eed dng of electronic circuits. It is not important to be able to design circuits, or “even to completely understand a circuit given to you, but you do need to now enough to get some idea of how the measuring apparatus affects your “yesult. This section introduces the basics of elementary, passive electronic circuits. You should be familiar with the concepts of electric voltage and : furrent before you begin, but something on the level of an introductory 9 3 Electronics and Data Acquisition physics course should be sufficient. It is helpful to have already leamed F something about resistors, capacitors, and inductors as well, but we will - review them briefly. : 3.1.1. Voltage, Resistance, and Current Figure 3.1a shows a DC current loop. It is just a battcry that provides the electromotive force V, which drives a current / through the resistor R. © This is a cumbersome way to write things, however, so we will use the | shorthand shown in Fig. 3.1b. All that ever matters is the relative voltage ~ between two points, so we specify everything relative to the “common” = or “pround.” There is no need to connect the circuit loop with a line; it : if Understood that the curtent returns from the common point back to the % terminals of the battery. % The concept of electric potential is based on the idea of electric potential - energy, and energy is conserved. This means that the total change in electric ; potential going around the loop in Fig. 3.1a must be zero. In terms of ¢ Fig. 3.1), the “voltage drop” across the resistor R must equal V. For ideal - resistors, V = J R; that is, they obey Ohm’s law. The SI unit of resistance - is voltsfampetes, also known as the ohm (£2). : Electric current is just the flow of electric charge (J = dq/dt, to be : precise), and electric charge is conserved, This means that when there isa ; “junction” in a circuit, like that shown in Fig, 3.2, the sum of the currents ° flowing into the junction must equal the sum of the currents flowing out. © In the case of Fig. 3.2, this rule just implies that J; = Io + 43. It does not {a} (b} +V FIGURE 3.1 The simple current loop (a} showing the entire loop, and (b) in shorthand, 3.1 Elements of Circuit Theory 94 i c—- 1s as i so ‘. Prof oe SS > * oo a ee . eee Sn" re es ws Ase = -* a a ras - (Loe Of | Say on Sa ooo ” =~ tf * AP t =" A . — ‘ a | -_* ee ¥ Ss ' i <= . =. : Kase = « — sa sf . ns Paes y, Pes « wae SS AAS Ses AX a gE oes - hy Ss SNS eS SO patty FIGURE 3.2 A simple three-wire circuit junction, A Se a .,* A — » an (b) x ‘ ‘3 . ee ee itateeeetep ety ee Al 3 FIGURE 3.3 Resistors connected (a) in series and (b) in parallel. 22°" matter whether you specify the current flowing in or out, so long as you are =. consistent with this rule. Remember that current can be negative as well as = positive. ' These rules and definitions allow us to determine the resistance when = resistors are connected in series, as tn Fig. 3.3a, orin parallel, as in Fig. 3.3b. =: In either case, the voltage drop across the pair must be / R, where J is the oe current flowing through them. For two resistors R; and R2 connected in :° series, the current is the same through both, so the voltage drops across #\ them are / Ry and / Rp, respectively, Since the voltage drop across the pair é:. must equal the sum of the voltage drops, then /R = 7 R, + J Ra, or ers SAN ’ he Se besebeaes vets peeStoess os . . a . '* . ' the R= R;+ R2 Resistors in series. + If Ry and R2 are connected in parallel, then the voltage drops across each =) are the same, but the current through them is different. Therefore JR = ‘Ty Ry = InRz. Since J = J; + hh, we have — = — + — Resistors in parallel. = Remember that whenever a resistor is present in a circuit, it may as well *: be some combination of resistors that give the right value of resistance. sASeTNCNNNNTUSHTNNRUNN sprinter siebtetar ators’ yeoteteceeatgtaratesiaret a tomes tan mmr a Sarnta MAL Norns at neyE Uritnsehooe * Pe el Pier esee eae ie NAS . ‘abet se ' ee 92 3 Electronics and Data Acquisition GQ A very simple, and very useful, configuration of resistors is shown in: os Fig. 3.4. This is called a “voltage divider” because of the simple relationship: between the voles labeled Vout and V;,. Clearly V, ‘in = I (Ri + Ro) ae ere) ere | Vout = Ving a Gay That is, this simple circuit divides the “input” voltage into a fraction deter-" mined by the relative resistor values. We will see lots of examples of this. sort of thing in the laboratory. Do not get confused by the way circuits are drawn. It does not matter: which directions lines go in. Just remember that a line means that all points: along it are at the same potential. For example, it is common to draw a voltage divider as shown in Fig, 3.5. This way of looking at it is in fact an easier way to think about an “input” voltage and an “output” voltage. = “ Seer agentes arenes “ Rants Gm arto GRSUCH CC RE CT SOO PAUSE SER < Vin emittance ithe Gee HUD DIES SERGE Aa tA ERE Sat FIGURE 3.4 The basic voltage divider. ater tatat ate ttatete e Ped el el ee er vt ee ae ee Vin Vout 2B — a = ae = FIGURE 3.5 An alternate way to draw a voltage divider, atte . any ae ee CREE at an eee oA ob AOL as wate 1 eee ate CeCe et ee oat eee aD a at fate ta ee fs 3.1 Elements of Circuit Theory 99 ey gy 3. 1, 2. Capacitors and AC Circuits Lek capacitor stores charge, but docs not allow the charge carriers (i.¢., es electrons) to pass through it. It is simplest to visualize a capacitor as a = of arnt: plates, paralle] to cach other and separated only by a ert ae gonstant value independent of the voltage. In general, it is wossible, but ni easy, to calculate C from the geometry of the conducting surfaces. The SI ae je “unit of capacitance is Coulombs/Volts, also known as the Farad (F). As it ee ee tumns out, one Farad is an enormous capacitance, and laboratory capaci- — om are connected in series and in parallel, just using the above definitions and _: rule about the total voltage drop. The answers are : a RS een ea pea —-=—+— apacitors in series G& tt > & Z ZO C=C),+Cz Capacitors in parallel, Bs = carriers to pass through it SO the current / = 0. Therefore the voltage 7 oo across the resistor R is zero, and ere the voltage across the capacitor ee oe ee _ wit time. :, If the voltage changes with time, we refer to the system as an AC circuit. at the voltage is constant, we call it a DC circuit. Now go back to the : = divider with a capacitor, pictured in Fig. 3.6, and let the input = as 'L wu = 1 pF (picofarad). a + ee ne == he 94 863 Electronics and Data Acquisition Vin NIE R 27 HEE Vout oe ar Soe = . FIGURE 3.6 Avoltage divider with a capacitor in it. eat EEL CL tee Se Set ee ea an ett at ha a ea voltage change with time in a very simple way. That 1s, take OE oe Vin(t}=0 fort <0 (3.293 =V fort>@ (3.3) iJ a & a Ha a ma “a and assume that there is no charge g on the capacitor at ¢ = 0, Then for: t > 0, the charge g(t) produces a voltage drop Vou(t) = g(t)/C across the capacitor. The current I(t) = dq/di through the divider string alsa. gives a voltage drop I R across the resistor, and the sum of the two voltage = drops must equal V. In other words LS dV, E V = Voy + PR = Vou + rit = Vou + RC— G.4).2 dt dt os and Voy.(0} = 0. This differential equation has a simple solution. lt is = Vout(t) = VIL — eW/ RC], (3.5): Now it should be clear what is going on. As soon as the input voltage is :: switched on, current flows through the resistor and the charge carricrs pile «2 up on the input side of the capacitor. There is induced charge on the output .: side of the capacitor, and that is what completes the circuit to pround. : However, as the capacitor charges up, it gets harder and harder to put. : more charge on it, and as ¢ — oo, the current does not flow anymore and Vou. ~ ¥. This is just the DC case, where this circuit is not interesting : anymore. : 3.1 Elements of Circuit Theory 95 ee a / Thats a very useful property that we will study some more, and use in lots 2 E of experiments. ee ::- The time dependence of any function can always be expressed in terms of : ze sine and cosine functions using a Fourier transform. It is therefore common B49, work with sinusoidally varying functions for voltage and so forth, just 2 realizing that we can add them up with the right coefficients to get whatever 2 time dependence we want in the end. It is very convenient to use the complex number notation V(r) = me (3.6) = wae watatat Jae ee 2 a a oscillations per amend This expression for V(t) is easy to differentia c gS and integrate when solving equations, It is also a neat way of keeping track we: ‘pf all the phase changes signals undergo when they pass through capacitors ie “and other “reactive” components. You will see and appreciate this better t= a3 we go along. ve Now is a convenient time to define impedance. This is just a general- ization of resistance for AC circuits. Impedance, usually denoted by Z, is : = ‘a (usually) complex quantity and (usually) a function of the angular [re- . eg ‘quency w. It is defined as the ratio of voltage drop across a component to es ‘the current through it, and just as for resistance, the SI unit is the ohm. : fee “For “linear” components (of which resistors and capacitors are common : fe examples), the impedance is not a function of the amplitude of the volt- ee age or current signals. Given this definition of impedance, the rules for ee thie equivalent impedance are the same as those for resistance. That is, for a ro “components in series, add the impedances, while if they are in parallel, add z Abeir reciprocals. The impedance of a resistor is trivial. It is just the resistance R. In : fe © this case, the voltage drop across the resistor is in phase with the current Be = through it since Z = R is a purely real quantity. The impedance is also © independent of frequency in this case. For a capacitor, the voltage drop 5 has J * , SERETS a state ee tte vane 0% 3 Electronics and Data Acquisition Vo Voe! = g/C and the cwrent J = dq/dt =twC x Voei, Therefore, 3 the impedance is : Vio, t) i(w,t) i@c’ Now the behavior of capacitors is clear. At frequencies low compared, *= to 1/RC, ic., the “DC limit,” the impedance of the capacitor goes ta: infinity. (Here, the valuc of R is the equivalent resistance in series with 2% the capacitor.) It does not allow current to pass through it. However, as the. :: frequency gets much larger than 1/RC, the impedance goes to 0 and the.“ capacitor acts like a short, since current passes through it as if it were not, “: there. You can Jearn a fot about the behavior of capacitors in circuits just 2 by keeping these limits in mind. : We can easily generalize our concept of the voltage divider to inciude 4 AC circuits and reactive (i.e., frequency dependent) components like 23 capacitors. We will leam about another reactive component, the induc- “4 tor, shortly. The generalized voltage divider is shown in Fig. 3.7. In this 2 case, we have Bee Z(o) = a7 e saccathnttatasacstiee SSS spetepata ch mite enn ots Vou (@, 2) = Vinlw p22 = Vi(w, tee’? (3.8) one aes Z+Z5 — mn, TEE» ene where we have expressed the impedance ratio Z;/(Zi + Zz), a complex = number, in terms of two real numbers g and @. We refer to g = |Voutl/| Vin] 22 cnn see B, scienasescesh aS tt ‘ a Vin O a = aaa Aaa esis SSSA ietetinante raletete het nati es eee lel Date teat tata a edhe ata TRS SUS S Sannin scent atta FIGURE 3.7 The generalized voltage divider. Z 2 3.1 Elamants of Circult Theory 97 ‘ toe vata relly Z ee as the “gain” of the circuit, and @ is the phase shift of the output signal ass = relative to the input signal. For the simple resistive voltage divider shown ean Figs. 3.4 and 3.5, we have g = Rj/(R, + Ro) and @ = 0. That is, = the output signal is in phase with the input signal, and the amplitude is % ys reduced by the relative resistor values. This holds at all frequencies, % ; including DC, e-- The relative phase is an important quantity, so let’s take a moment to i 7 Mook at it a little more physically. If we write Vin = Voe'™', then according just the real part of these complex expressions, we have iss 7 Vin = Vo cos(wr) ee Z Vout = a%° cos(wt + ¢) fe Eon a ime different than the input voltage, and this time is proportional to ee the phase. To be exact, relative to the time at which V,, is a maximum, ae d oa Time of maximum Voy, = —— x T = a Bee an oa ap . che) Sa ok) b i bchcks Doe a's DF) sa ee es oS Votlage ne ew © eee a ee te ee 98 2 Electronics and Data Acqutsition Now consider the voltage divider in Fig. 3.6. Using Eq. (3.8) we find | Li C | Vout = Vin toe = Vin et l R+- 1+ia@RC iwc The gain g of this voltage divider is just (1 + w7R*C*)—'/? and you can: see that for @ = 0 (i.e., DC operation) the gain is unity. For very large .:: for frequencies in the neighborhood of 1/RC. We have said all this before, = but in a less general language. However, our new language tells us something new and important <2 about Vour, namely the phase relative to Vj,. Any complex number z can. :: es | be written as z=\zle and 2* = |zle%, (3.9) 4 where Q _, [Im{z) OE 1 ae = tan | ——— 3.10} , Fal 610) Be is called the “phase” of z. Therefore, we find that l 1-—r0RC i ie EE ES 2 ltiwoRC i+w*R?c (1 + w2R2C2)!/2 In other words, the output voltage is phase shifted relative to the imput = voltage by an amount @ = —tan—!(wRC). For w = O there is no phase % shift, as you should expect, but at very high frequencies the phase is shifted 2 by —90°. 3.1.3. Inductors Just as a capacitor stores energy in an electric field, an inductor stores “2 energy in a magnetic field. An inductor is essentially a wire wound into °; the shape of a solenoid. The symbol for an inductor is . The key is in, : cue RH CSTE AT) inte ate Par * PP rr the magnetic field that is set up inside the coil, and what happens when the current changes. So, just as with a capacitor, inductors are important when = the voltage and current change with time, and the response depends on the frequency. 3.1 Elements of Circuit Theory 94 ee moe The inductance L of a circuit element is defined to be NO L=—_, I _ aN) _ 7a! dt) dt = JZ, where Z is the impedance of the inductor, and ee A= — Ipe, then | V —lwLI or ae fa eg Be - Z=iwl, (3.11) s Z We can use this impedance to calculate, for example, Vour for the gener- Se “alized voltage divider of Fig. 3.7 if one or more of the components is an seoynductor. : a You can now see that the inductor is, to a large cxtent, the opposite of 7 a capacitor. The inductor behaves as a short (that is, just the wire it is) at te low frequencies, whereas a capacitor is open in the DC limit. On the other ee Nand, an inductor behaves as if the wire were cut (an open circuit) at high ee"? frequencies, but the capacitor is a short in this Limit. s ae One particularly interesting combination ts the series LC R circuit, com- ‘ 2 bining on¢ of each In series. The impedance of such a string displays the as “phenomenon of “resonance.” That is, in complete analogy with mechanical ES ‘resonance, the valtage drop across one of the elements is a maximum for ee . ; ES ca t certain value of w. Also, as the frequency passes een this value, the on _ O14. Diodes and Transistors Sa Se " th = ie _ Resistor, capacitors, and inductors are “Linear” devices. That is, we write ES x S ae Ree Se Parenter sretnbeatah SMS SESEESSS SE SUSSEE Tats aah gabe SERRA tee hate ceetelaatebaatetenteha sTalecereTereTatate tata * 10 93«=63)- Electronics and Data Acquisition a oes oie ee aa then you increase 7 by the same factor. Diodes and transistors are exam- = ples of “nonlinear” devices. Instead of talking about some impedance Z, = we instead consider the relationship between V and 7 as some nonlinear “= function. What is more, a transistor is an “active” device, unlike resistors, == capacitors, inductors, and diodes, which are “passive.” That is, a transistor takes in power from some vollage or current source, and gives an output’ that combines that input power with the signal input to get a response: It':: used to be that many of these functions were possible with vacuum tubes = of various kinds. These have been almost completely replaced by solid- 22 state devices based on semiconductors. The physics of semiconductors and -: semiconductor devices was discussed in Sections 2.1 and 2.4. 3 The symbol for a diode is Pb where the arrow shows the nominal direc- “= tion of current flow. An ideal diode conducts in one direction only. That : is, its V—7 curve would give zero current / for V < © and infinite J for: V > O. (Of course, in practice, the current J is limited by some resistor 2 in series with the diode.} This is shown in Fig. 3.9a. A real diode, how- 22 ever, has a more complicated curve, as shown in Fig. 3.9b. The current. 2% ! changes approximately exponentially with V, and becomes very large.::: for voltages above some forward voltage drop Vp. For most cases, a good:: approximation is that the current is zero for V < Vp and unlimited for’: V > Vp. Typical values of Ve are between 0.5 and 0.8 V. : OC Senet aceioerearmnmentort eae Mitte SHER BaD Hs TrtataBSN, Be Diodes are pn junctions. These are the simplest solid-state devices, made: 4 of a semiconductor, usually silicon. The electrons in a semiconductor fill: ~anrentergy> dia Loormally.¢ annot. move, throueh the bulk material, so {24 the semiconductor is really an insulator, If electrons make it into the next S {a) j (b) I Y= Ve Y=0 Vv V FIGURE 3.9 Current 7 versus valtage V for (a) the ideal diode and (b) a real diode. a ee ta Ce ele ee sg ‘a we i . 3.1 Elements of Circuit Theory 101 : 2 energy band, which is normally empty, then they can conduct electricity. =<"Ehis cap happen if, for example, electrons are thermally excited across the eenergy gap between the bands. For silicon, the band gap is 1.1 eV, but the ="nean thermal energy of electrons at room temperature is ~kT = 1/40 eV. e--'Pherefore, silicon is essentially an insulator under normal conditions, and Snot particularly useful. : Soe That is where the pandn come in. By adding a small amount (around ae _ - eae Pete Ze srsonic, give electrons as carriers, and the doped semiconductor j 1s called a n-type, since the carriers are negative. Other dopants, like boron, bind ats ee 2 gee wa a) al a. cr =] jj = mj ¥ i=s (b tA a i—7 G —_— fa a po a = HE TH oO , ime =< > oO ape oO i fa d tA OQ & tt 3 wT he semiconductor p-type. In either case, the conductivity j increases by a Z factor of ~1000 at room temperature, and this makes some nifty things == “possible. ge So now back to the diode, or pn junction. This is a piece of silicon, 2doped p-type on one side and n-type on the other. Electrons can only flow © from p ton. That is, 2 current is carried only in one direction. A detailed Lk aN! or 2° citing i in Section 3. 10) for more details. If you put voltage across the diode == in the direction opposite to the direction of possible current flow, that is : called a “reverse bias.” A small “leakage’’ current flows as shown in Fig. Z°3.9. If you put too much of a reverse bias on the diode, i.e., V < —V}™, "it will break down and start to conduct. Typical values of Vi are 100 V waa aaa ‘= ge Or less. ize:: "Transistors are considerably more complicated than diodes,’ and we will “= introduction to ransistors in The Art of Electronics (full citing in Section ie. 3.10). For details on the underlying theory, see Dunlap (1988). A transis- Dw main types of transistors, namely npr and pnp, and their symbols are f<-Shown in Fig. 3.10, The names are based on the dopants used in the semi- z sonductor materials, The properties of a transistor may be summarized in f=: *Dhe invention af the transistor was worth a Nobel Prize in Physics in 1956. 1020 «3s Electronics and Data Acquisition Gollactor G cae ae a ‘ is wa eae wa See ths st i ee i a ue a Sete anetateb testa ses case Emitter = npn pnp FIGURE 3.10 Symbols for aga and pnp transistors. ve ae Raat ahah ea the following simple rules for npn transistors. (For pvp transistors, just, =: reverse all the polarities.) E 1. The collector must be more positive than the emitter. 2. The base-cmitter and base—collector circuits behave like diodes. Nomnally the base-emitter diode is conducting and the base—collector diode is reverse-biased, 3. Any given transistor has maximum values of Ic, fg, and Vcr that cannot be exceeded without nuning the transistor. If you are using a transistor in the design of some circuit, check the 58 specifications to see what these limiting values are. Se 4. When rules 1—3 are obeyed, Jc is roughly proportional to 7g and = 23% can be written as Ic = Ayelg. The parameter hx, also called Bees B, is typically around 100, but it varies a lot among a sample of See nominally identical transistors. Bs Obviously, rule 4 is what gives a transistor its punch. It means that a 22 transistor can “amplify” some mput signal. It can also do a lot of other 2:4 things, and we will see them in action later on. eg Pa ntutatet 3.L5. Frequency Filters Simple combinations of passive elements can be used to remove “noise” = from a voltage signal. If the noise that is bothering you is in some specific 2% range of frequencies, and you can make your measurement in some other 3 Tange, then a frequency filter can do a lot for you. Frequency filters are a8 usually simpic circuits (or perhaps their mechanical analogs) that allow = only a specific frequency range to pass from the input to the output. Youthen 24 3.1 Elements of Gircuit Theory 103 =: make your measurement with the output. Of course, you need to be carefu! = of amy noise introduced by the filter itself. The circuit shown in Fig. 3.6 is =a “low-pass” filter. It exploits the frequency dependence of the capacitor == impedance Zc = 1/iwC to short frequencies much larger than 1/RC to = ground, and to allow much smaller frequencies to pass. As we showed =. earlier, the ratio of the output to input voltage as a function of frequency =: yp = w/2n is (1+a7R2C*)—'/, You can also use inductors in these simple ee circuits. Remember that whereas a capacitor is open at low frequencies == and a short at high frequencies, an inductor behaves just the opposite. = Figure 3.11 shows al] permutations of resistors, capacitors, and inductors, es cand whether they are high- or low-pass filters. pe Suppose you only want to deal with frequencies in a specific range. #- Then, you want a “bandpass” filter, which cuts off at both low and high ee frequencies, but lets some intermediate bandwidth pass through. Consider Res m PLEATS Circuil Type Circult Typa “De Pome "fF 7 . te anlar ar heh Se aera aaah Low pass High pass ~T . T Low pass High pass FIGURE 3.11 Simple passive frequency filters. Vin R Voun Oo heed FIGURE 3.12 A simple bandpass filter. 106 3 Electronics and Data Acquisition through either a capacitor or an inductor. Therefore, the output will be zero | Be at both low and high frequencies. Analyzing this filter circuit is simple Vout _ ZLC Vin Zre+Zic’ where Ze = Rand Zic = (Z;' + Zz')™ with Z, = Liok and Zc = iwC. (Note that L and C are connected in parallel.) The result is Vout 1 Vin R2 1f2 2 2 + wipe —w LC) — and as advertised, g > Oforbothw < R/Landforw > 1/RC. However, “2 frequencies near v = w/2m = 1/(22+/LC) are passed through with little <2 attenuation. At w = 1/./EC, 2 = 1 and there is no attenuation at all. Can 2 you see how to build a “notch” filter, or “band reject” filter, that allows ail ; 28 frequencies to pass except those in the neighborhood of w = 1/./LC? 3.2. BASIC ELECTRONIC EQUIPMENT 3.2.1. Wire and Cable Connections between components are made with wires. We tend to neglect = the importance of choosing the nght wire for the job, but in some cases ee ee pe ke a watt ee eae are ae tl beh eed Piha tar iat AL, So ON ed ee a tt BEASASEMSLERELESSLSLS RNS natant nnletels! 7 sonata ate a hal i a NASA thy it can make a big difference. The simplest wire is just a strand of some oe conductor, most often a metal such as copper or aluminum. Usually the wire = is coated with an insulator so that it will not short out to its surroundings, S or to another part of the wire itself. If the wire is supposed to carry some “= small signal, then it will likely need to be “shielded,” that is, covered with another conductor (outside the insulator) so that the external environment = does not add noise somehow. One popular type of shielded wire is the “ “coaxial cable,” which is also used to propagate “pulses.” Do not forget about Ohm’s law when choosing the proper wire. That & is, the voltage drop across a section of wire is still V = IR, and % you want this voltage drop to be small compared to the “real” voltages involved. The resistance R = p x L/A, where L is the length of the = wire, A is its cross-sectional area, and 9 1s the resistivity of the metal. 8 Therefore, to get the smallest possible R, you keep the length L as short -: Bass ERS ewe ia sk Sah et ptenatanatavetatens Nts ee ee ahe ohta ae xv i yo inte ne a*e ay a" be ates . “ Ma tah a el ay eh es 3.2 Basic Elactronic Equipment 105 = as practical, get a wire with the largest practical A, and choose a con- - ductor with small resistivity. Copper is the usual choice because it has = Jow resistivity (0 = 1.69 x 1078 Q-cm) and is easy to form into wire ~ of vanous thicknesses and shapes. Other common choices are aluminum (pe 2.159% 10-*® Q-cm), which can be significantly cheaper in large | quantities, or silver (9 = 1,62 x 10-* Q-cm), which is a slightly better =: conductor, although not usually worth the increased expense. : The resistivity increases with temperature, and this can lead to a partic- ? wlarly insidious failure if the wire must carry a large current. The power * dissipated in the wire is P = /*R, and this tends to heat it up. Jf there is not enough cooling by convection or other means, then FR will increase and = the wire will get botter and hotter until it does serious damage. This is most = common in wires used to wind magnets, but can show up in other high- = power applicahons. A common solution is to use very-low-gage (i.e., very = thick) wire which has a hollow channel in the middle through which water = flows, The water acts as a coolant to keep the wire from getting too hot. A coaxial cable is a shielded wire. The name comes from the fact that the =) wire sits inside an tnsulator, another conductor, and another insulator, all =, jn circular cross section sharing the sarne axis. A cutaway view is shown in SO CN ~ Fig. 3.13, Coaxial cable is used in place of simple wire when the signals are - yery smal! and are likely to be obscured by some sort of electronic noise =. in the room. The outside conductor (called the “shield”’) makes it difficult iss for external electromagnciic fields to penetrate to the wire, and minimizes =. the noise. This outside conductor ts usually connected to ground. A second, and very important, use of coaxial cable is for “pulse trans- mission.” The wire and shield, separated by the dielectric insulator, act as =: a waveguide and allow short pulses of current to be transmitted with little distortion from dispersion. Short pulses can be very common in the labo- < Tatory, in such applications as digital signal transmission and in radiation Saas Sy ss a * SS toate eS a Sex s ay) : detectors. You must be aware of the “characteristic impedance” of the cable when you use it in this way. Coaxial cable has a characteristic impedance because it transmits the signal as a train of electric and magnetic fluctuations, and the cable itself has characteristic capacitance and inductance. The capacitance and inductance of acylindrical geometry like this are typically solved in elementary physics +Wire diumeter is usuully specified by the “gage number.” The smaller the wire gage, the thicker the wire, and the larger the cross-sectional area. 106 3 Electronics and Data Acquisition SEE toa ae FIGURE 3.13 Cutaway view of coaxial cable. texts on electricity and magnetism. The solutions are 2 € lt b = —-_——__— f 4 d = — | — £, Cc in(b/a) x an L an n(2) ye where a and b are the radii of the wire and shield respectively, € and jz are 2: the permittivity and permeability of the dielectric, and 2 is the length of the 3 cable. Itis very interesting to derive and solve the equations that determine 24 pulse propagation in a coaxial cable, but we will not do that here. One = thing you learn, however, is that the impedance seen by the pulse (which 3 is dominated by high frequencies) is very nearly real and independent of ae frequency, and equal to iL 1 /p b 8 = — = -— J— — |. AD) e C we in (2) 6-12) Se This “characteristic impedance” is always in a limited range, typically =: 50 < Z. < 20022, owing to natural values of € and sz, and to the slow 2 variation of the logarithm. You must be careful when making connections with coaxial cable, so 2 that the characteristic impedance Z,. of the cable is “matched” to the “2 load impedance Z;. The transmission equations are used to show that <4 the “reflection coefficient” I’, defined as the ratio of the current reflected <2 from the end of the cable to the current incident on the end, is given by _ Zi — Le 7 ZL + Le That is, if a pulse is transmitted along a cable and the end of the cable isnot :2 connected to anything (Z;, = oo), then! = 1 and the pulse is immediately.“ reflected back. On the other hand, if the end shorts the conductor to the “2 shield (Z;, = 9), then F = —1 and the pulse is inverted and then sent “3 back. The ideal case is when the load has the same impedance as the cable. cS In this case, there ix no loss at the end of the cable and the full signal = is transmitted through. You should take care in the lab to use cable and = i ] ee! an te . ee Cia i NL bg SSS seetetatgetats eee ewes) aaa ers] ad ae re a oe Nee Een SS ttre Pat) oh, at COPESES ESS a ot tae tal tae mth far 3.2 Basic Electronic Equipment 107 : electronics that have matched impedances. Common impedance standards = are 50 and 90. =: Of course, you will need to connect your wire to the apparatus somehow, = and this is done in a wide variety of ways. For permanent connections, = especially inside electronic devices, solder is usually the preferred solution. It is harder than you might think to make a good solder joint, and if you t= are poing to do some of this, you should have someone show you who : Bs has a decent amount of experience. Another type of permanent’connection, =: called “crimping,” squeezes the conductors together using a special tool S that ensures a 200d contact that does not release. This is particularly useful =< jf you cannot apply the type of heat necessary to make a good solder fe: Joint Less permanent connections can be made using terminal screws or bind- eS ing posts. These work by taking a piece of wire and tmserting it between two ==: surfaces that are then forced together by tightening a screw. You may need f=: to twist the end of the wire into a hook or loop to do this best, or you may Be use wire with some sort of attachment that has been soldered or crimped me on the end. If you keep tightening or untightening screws, especially onto aes wires with handmade hooks or loops, then the wire is likely to break at es some point. Therefore, for temporary connections, it is best to use alliga- =< tor clips or banana plugs, or something similar. Again, you will usually “ se wires with this kind of connector previously soldered or crimped on =: the end. Coaxial cable connections are made with one of several special types of = connectors. Probably most common is the “bayonet N-connector,” or BNC, | standard, including male cable end connectors, female device connectors, =! and union and T-connectors for joining cables. In this system, a pin is =. soldered or crimped to the inner conductor of the cable, and the shield is = connected to an outer metal holder. Connections are made by twisting the » holder over the mating connector, with the pin inserting itself on the inner part. Another common connector standard, called “safe high voltage” or =. SHV, works similarly to BNC, but is destgned for use with high DC voltages “. by making tt difficult to contact the central pin unless you attach it to the = Correct mate. For low-level racasurement you must be aware of the thermal elec- =. De potential difference between two dissimilar conductors at different = temperatures. These “thermoelectric coefficients” are typically around 1 upV/°C, but between copper and copper-oxide (which can easily happen = if a wire or terminal is oxidized) it is around 1 mV/°C, 1080s 3s Electronics and Data Acquisition 3.2.2. DC Power Supplies Laboratory equipment needs to be “powered” in one way or another. Unlike the typical 100-¥V, 60-Hz AC line you get out of the wall socket, though, this o equipment usually requires some constant DC level to operate. One way to provide this constant DC level is to use a battery, but if the equipment B draws much current the battery will quickly run down. Instead we use DC “power supplies”, the power supply in turn gets its power from the wall socket. Power supplies come in lois of shapes, sizcs, and varieties, but there are two general classes. These are “voltage” supplies or “current” supplies, = and the difference is based on how the output is regulated. Since the inner workings of the power supply have some effective resistance, when the <4 power supply must givc some current, there will be a voltage drop across that internal resistance, which will affect how the power supply works. In 23 a “voltage-regulated” supply, the circustry is designed to keep the output voltage constant (to within some tolerance), regardless of how much current is drawn. (Typically, there will be some maximum current at which the r regulation starts to fail, That is, there is a maximum power that can be = supplied.) Most electronic devices and detector systems prefer to have a specific voltage they can count on, so they are usually connected to = voltage-regulated supplies. A “current-regulated” supply is completely analogous, but here the cir- cuitry is designed to give a constant output current in the face of some load on the supply. Such supplies are most often used to power magnets, 2 since the magnetic field only cares about how much current flows through the coils. This is in fact quite important for establishing precise magnetic fields, since the coils tend to get hot and change their resistance. In this 2 TST IIE Ste SST Soo oS Saat agabeceeLogetege SOLON log aLac gana le lage Dosa UUM eregeacedanetahe snc hg a ale alee fe SL a acaalelaladaberetay STALELETHLEEAISTAEAEACGREESHLELPasGbByonoeesnanscbrteteSABOMGCONBNBEAAMRERES SAEASAESS SLOP OLN NOLOEASESRADSGHGASSR ACSC ASSRSS#SLASOSGSGISSASAPUSENESLBDOCUCFEREDSSTCESSESOSUC DCSE HLALORECOSOIMRNNSESGOMPaSaa! sole oer Ph) Pare case, V = IR and R is changing with time, so the power supply must & know to keep / constant by varying V accordingly. [n many cases, a sim- ple modification (usually done without opening up the box) can convert a 2 power supply from voltage regulation to current regulation. The output terminals on most power supplies are “floating.” Thatis, they | e are not tied to any extemal potential, in particular not to ground. One output “2 (sometimes colored in red} is positive with respect to the other (black). You will usually connect one of the outputs to some extemal point at known = potential, like a common ground. You should be aware of some numbers. The size and price of a power E supply depends Jargely on how much power it can supply. If it provides a = 3.2 Basic Electranic Equipment 1089 == voltage V while sourcing a current /, then the power output is P = JV. 24 very common supply you will find around the lab will put out several Sn a a = fA = o. fa eo s) = =< ra OQ = Z cP @ # of oO Aa =) 5 = sts) q = = o = tl o =) Ct no) to mI e = a Yeon things like contro! knobs and settings to computer interfacing, they can #2: cost anywhere from $50 up to a few hundred. So-called “high-voltage” ce “power supplies will give several hundred up to several thousand volts, and “ean source anywhere from a few microamperes up to 100 mA, and keep the #25" voltage constant to a level of better than 100 mV. Still, the power output bo such devices is not enormously high, typically under a few hundred Hee watts. The cost will run into thousands of dollars. Magnet power supplies, BE shou gh, may be asked to run something like 50 A through a coil that has a | wsiane of, say, 2 2. In this case, the output power is 5 kW. co ee Dd. Waveform Generators Be. ar =cones, and how to connect these voltages using wire and cable, you must Wink about how to measure the voltage. The simplest way to do this is with Be with AC capability, but we will not go into the details here.) An excellent Pore : 10 =«3)s Elactronics and Data Acquisition eel, ear) reference on the subject of meters is given in the Low Level Measurements 2 Handbook, published by Keithley Instruments, Inc. This handbook, as well “2 as other materials, are available from Keithley at htip:/fwww.keithlay.com/. =: At one time, people would use either voltmeters, ammeters, or ohm- = meters to measure voltage, current, or resistance, respectively. These days, = although you still might want to buy one of these specialized instruments. to get down to very low levels, most measurements are done with “digital 24 mulmeters,” or DMMs for short. (In fact, some DMMs are available now 2 that can effectively take the place of the most sensitive specialized meters.) SS Voltage and resistance measurements are made by connecting the meter in <= parallel ta the portion of the circuit you are interested in. To measure current, “2 the meter must be in series. Realize that DMMs work by averaging the voltage measurement over some period of time, and then displaying the result. This means that if “= the voltage is fluctuating on some time scale, these fluctuations will not = be observed if the averaging time is greater than the typical period of the = fluctuations. Of course the shorter the averaging time a meter has (the “3 higher the “bandwidth” it has), the fancier it is and the more it costs. Meters have some effective input impedance, so they will (at some level) E change the voltage you are trying to measure. For this reason, voltmeters 2 and ohmmeters are designed to have very large input impedances (many “2 megaohms to as high as several gigaohms), while ammeters “shunt” the = current through a very low resistance and tum the job into measuring the = (perhaps very low) voltage drop across that resistor. 3.3. OSCILLOSCOPES AND DIGITIZERS 3.3.1. Oscilloscopes An oscilloscope measures and displays voltage as a function of time. That ‘ is, it plots for you the quantity V(t) on a cathode ray tube (CRT) screen : as it comes in. This is a very useful thing, and you will use oscilloscopes :; in nearly all the experiments you do. A good reference is The XYZ’s of - Oscilloscopes, published by Tektronix, Inc. You can download a copy - from http://www.tek.com/ under “Application Notes for Oscilloscopes.” The simple block diagram shown in Fig. 3.14 explains how an oscillo~ . scope works. The voltage you want to measure serves two purposes. First, after being amplified, it is applied to the vertical defection plates of the ae a a x <_<. , aD AS Sa hehe bce ta dn oles —* “ oe. +" at * tate * eatatatate _ en stere hieP one fate a” a! a4 4 heey SSI eG Ses * sere, _s Nueaso aces TT cree a 3.3 Oscilloscopes and Digitizers Vertical System Ss Trigger Ramp Time Base FIGURE 3.14 Block diagram of an oscilloscope, =" CRT. This means that the vertical position of the trace on the CRT cor- = responds linearly to the input voltage, which is just what you want. The == vertical scale on the CRT has a grid pattern that lets you know what the = input voltage is. The horizontal position of the trace is controlled by a “sweep generator" = whose speed you can control. However, for repetitive signal shapes, you =. want the signal to “start” at the same time for every sweep, and this is e=:» determined by the “trigger” system. The place on the screen where the *. trace starts is controlled by a “horizontal position” knob on the front panel, =: One kind of trigger is to just have the scope sweep at the line (i.e., 60 Hz) £3 frequency, but this will not be useful if the signals you are interested in = do not come at that frequency. Another kind of simple trigger is to have =: the trace sweep once whenever the voltage nses or falls past some level, = i.¢., a “leading edge” trigger. There is usually a light on the front panel that = flashes when the scope is tiggered. Oscilloscopes almost always have at least two input channels, and it is #: possible to tigger on one channel and look at the other. This can be very “ useful for studying coincident signals or for measuring the relative phase 2; Of two waveforms. In any case, the tigger “mode” can either be “normal,” : in which case there is a sweep only if the tigger condition is met, or “auto” wae * 112) 83 Electronics and Data Acquisition SUL where the scope will trigger itself if the trigger condition is not met in: some period of time. Auto mode is particularly useful if you are search- ‘: ing for some weak signal and do not want the trace to keep disappearing :: on you. : ‘You have several controls on how the input voltage is handled. A “ver- : tical position” knob on the front panel controls where the trace appears on : the screen. You will find one of these for each input channel. The input ‘: “coupling” can be set to either AC, DC, or ground. In AC mode, there is ‘22 at a ial a a capacitor between the mput connector and the vertical system circuit. 3:4 This keeps any constant DC level from entering the scope, and all you see 3: is the time-varying (i.e, AC) part. If you put the scope on DC, then the : constant voltage level also shows up. If the input coupling is grounded, °:224% then you force the input level to 0, and this shows you where 0 is on the Pe screen. (Make sure that the scope is on “auto” trigger if you ground the /2:2% input; otherwise, you will not sec a trace!) Se : E Sometimes, you also get to choose the input impedance for each channel. |: 27% Choosing the “high” input impedance (usually 1 MQ) is best if you want ae to measure voltage levels and not have the oscilloscope interact with the «328 circuit. However, the oscilloscope will get alot of use looking at fast pulsed «72% signals transmitted down coaxial cable, and you de not want an “impedance ©2224 mismatch” to cause the signal to be reflected back. (Sce Section 3.2.1.) Cables with 50-92 characteristic impedances are very common in this work, cs so you may find a 50-& input impedance option on the scope. If not, you .: should use a “tee” connector on the input to put a 50-@ load in parallel with the input. “ By flipping switches on the front, you can look at either input channel’s trace separately, or both at the same time. There is obviously a problem, though, with viewing both simultaneously since the vertical trace can only be in one place at a tume. There are two ways to get around this. One 1s to alternate the trace from channel one to channel two and back again. This gives complete traces of each, but does not really show them to you at the same time. If the signals.are very repetitive and you are not interested in fine detail, this is okay. However, if you really want to see the traces at the same time, select the chop option. Here, the trace jumps back and. forth between the channels at some high frequency, and you let your eye © - interpolate between the jumps. If the sweep speed is relatively slow, the interpolation is no problem and you probably cannot tell the difference between alternate and chop. However, at high sweep speed, the effect of the chopping action will be obvious. ae Perey -. "ae 3.3 Oscilloscopes and Digitizers 113 =: You should realize by now that high-frequency operation gets hard, and =>. the oscilloscope gets more complicated and expensive, Probably the single =* most important specification for an oscilloscope is its “bandwidth,” and eS you Wil] see that number printed on the front face nght near the screen. =- The number tells you the frequency at which a sine wave would appear = only 71% as large as it should be. You cannot trust the scope at frequencies = approaching or exceeding the bandwidth. Most of the scopes in the lab have = 20- or 60-MHz bandwidths. A “fast” oscilloscope will have a bandwidth “ of a few hundred megahertz or more. You will find that you can vary the sweep speed over a large range, but never much more than (bandwidth)~!. The “vertical sensitivity” can be set independently of the sweep speed, but scopes in general cannot go below around 2 mV/division. On most oscilloscopes, if you turn the sweep speed down to the lowest value, one more notch puts the scope in the XY display mode. Now, the trace displays channel one (X) on the horizontal axis and channel two (Y) on the vertical. For periodic signals, the trace is a Lissajous pattem from which you can determine the relative phase of the two inputs. Oscilloscopes are also used in this way as displays for various pieces of equipment which have X ¥ output options. Thus, the oscilloscope can be used as a plotting device in some cases, 3.3.2. Digitizers In order to measure a voltage and deal with the result in a computer, the voltage must be digitized. The generic device that does this Is the analog-to- digital converter or ADC. ADCs come in approximately an infinite number of varieties and connect to computers in lots of different ways. We will cover the particulars when we discuss the individual experiments, but for now we will review same of the basics. Probably the most important specification for an ADC is its resolution. We specify the resolution in terms of the number of binary digits (“bits”) that the ADC spreads out over its measuring range. The actual measuring range can be varied externally by some circuit, so the number of bits tells you how finely you can chop that range up. Obviously, the larger the number of bits, the closer you can get to knowing exactly what the input voltage was before it was digitized, A “low-resolution” ADC will have 8 bits or Jess. That 1s, it divides the input voltage up into 256 pieces and gives the computer a number between 0 and 255, which represents the voltage. A “high-resolution” ADC has 16 bits or more. vate ve * Seren ae 1144 = 3) Electronics and Data Acquisition High resolution does not come for free. In the first place, it can mean a: lot more data to handle. For example, if you want to histogram the voltage”: being measured with an 8-bit ADC, then you need 256 channels for each: histogram. However, if you want to make full use of a 16-bit ADC, every: histogram would have to consume 65,536 channels. Resolution also affects: the speed at which a voltage can be digitized. Generally speaking, it takes: rouch less time to digitize a voltage into a smaller number of bits than it aS does for a large number of bits. There are three general classes of ADCs, referred to as flash, peak. 2 voltage sensing, aud charge integrating ADCs. A flash ADC, or “wave- form recorder,” simply reads the voltage level at its input and converts °2 that voltage level into a number. They are typically low resolution, “% but run very fast. Today you can easily get an 8-bit flash ADC that-= digitizes at 100 MHz (i.c., one measurement every 10 ns). This is fast =: enough so that just about any time-varying signal can be converted to BS numbers so that a true representation of the signal can be stored in a, computer. To get better resolution, you need to decide what it is about the signal 2 you are really interested in. lor example, if you only care about the maxi- “: mutn voltage value, you can use a peak-sensing ADC, which digitizes the .:; . : wie whine seh a st i Bre Cea ee : 2 may Ce es maximum voltage observed during some specified time. Sometimes, you 4 are interested instead in the area underneath some voltage signal. This is pee the case, for example, in elementary particle detectors where the net charge “24 delivered is a measure of the particle’s energy. For applications like this, :::2 you can use an integrating ADC, which digitizes the net charge absorbed “22 over some time period, i.c., (1/R) f,? V(0) dt, where R is the resistance at {4 the input. For either of these types, you can buy commercial ADCs that digitize into 12 or 13 bits in 5 \ss or longer, but remember that faster and 224 more bits costs more money. The opposite of an ADC is a DAC, or digital-to-analog Converter. Here ‘2 the computer feeds the DAC a number depending on the number of bits, and the DAC puts out an analog voltage proportional to that number. The —2 simplest DAC has just one bit, and its output is either “on” or “off.” In this 23 case, we refer to the device as an “output register.” These devices are a way == of controling extemal equipment in an essentially computer-independent = fashion. In mary cases, you want to digitize a time interval instead of a voltage “= level. This can be done with a “tame-to-analog converter” (TAC), followed by an ADC. However, both of these functions are now available packaged 22 RST aC TTT NSE Ratner ti ens Maa eae eee eRe eta ete so eatatat ee eT ale ee BRON MN tn on he bt os 3.3 Oscilloscopes and Digitizers 115 in-a single device called a TDC. The rules and ranges are very similar as = for ADCs. Be Devices known as “latches” or “input registers” will take an external see: fogic level, and digitize the result into a single bit. These are useful for > alling whether some device is on or off, or perhaps if something has 7 appened that the computer should know about. ae When a device is busy digitizing, it cannot deal with more input. We ee tefer to the cumulative time a device is busy as “dead time.” ” Suppose T Beg the time needed to digitize an input pulse, and Ro is the (presumably Be ' random) rate at which pulses are delivered to the digitizer. If Rm is the ee measured rate, then in a time 7 the number of digitized pulses is RmT ge The dead time incurred in time T is therefore (R,,T)t, so the number of ee pulses lost is [(Rm7)t]Ro. The total number of pulses delivered (RoT) Ee must equal the number digitized plus the number lost, so ee Bee RoT = RunT + Rn Tt Ro, fe ee and therefore es Ro pe R= 3.13 Be mp4 tRg ) For ae R ae Ro = ——_. 3.14 se 8 T= tRe _ es The “normal” way to operate a digitizer ts so that itcan keep up with the se rate at which pulses come in. In other words, the rate at which it digitizes =: (1/r) should be much greater than the rate at which pulses are delivered, =: that is, TRo < 1. Equation (3.13) shows that in this case, Rm * Rg; that z:" is, the measured rate is very close to the true rate, which is just what you = want. Futhermore, an accurate correction to the measured rate is given by =. Eq. (3.14), which can be written as Ro = Rm(1 + tp) under normal = operation. z On the other hand, if 7Ro >> 1, then Ry © 1/t. That is, the digitizier =: Ddeasures a pulse and betore it can catch its breath, another pulse comes = along. The device is “always dead,” and the measured rate is just one per =. digitizing time unit Essentially all information on the true rate is lost, =" because the denominator of Eq. (3.14) is close to 0. You would have to =: know the value of rt very precisely in order to make a correction that gives =: you the true rate. | a . 4b ee Sh GT LR tii 116 3 Electronics and Data Acquisition 3.3.3. Digital Oscilloscopes The digital oscilloscope is a wonderful device. Instead of taking the: input voliage and feeding it directly onto the deflection plates of a CRE : (Fig. 3.14), a digital oscilloscope first digitizes the input signal using ‘4°22: flash ADC, stores the waveform in some internal memory, and then ha¥: other circuitry to read that memory and display the output on the CRT 32 We then have the voltage stored as numbers, and the internal computer in 2 a aren = a eee | have controls that make them look as much like analog scopes as possible! de aaa The same terminology is used, and just about any function found on. Latte! analog scope will also be found on a digital one. 2 SUSE 3.4. SIMPLE MEASUREMENTS We now outline some simple measurements of elementary circuits. Circuits oe are most easily put together on a “breadboard.” This is a flat, multilayered: surface with holes in which you stick the leads of wires, resistors, capaci-!224 tors, and so on. The holes are connected intermally across on the component: p pads, and downward on the power pads. AS Connect two 1-k@ resistors in series on the breadboard, and then connect 2; the terminals of the power supply to each end of this two-resistor string,233 Measure the voltage across the output of the terminals. Also, measure the current through the string. Now connect two more 1-k{&2 resistors in series: with the others. Move the connections from the power supply so that once: again it is connected to each end of the string. Repeat your voltage andi: current measurements. Now measure the voltage drop across each of the: four resistors. Compare the result to what you expect based on the voltage?:= divider relation. Use your data and Ohm’s law to measure the resistance of: each of the resistors. Compare the resistance values you measure with the nomiual value. 3 Set the waveform to a sine wave. Use an oscilloscope to compare the: 23 voltage (as a function of time) across the resistor string from the waveform: :-# generator with the voltage across one of the resistors. Put each of these™: into the two channels of the oscilloscope, and trigger the scope on the! channel corresponding to the waveform generator output. Look at both’ =: 2 RUMOR RTH MRR Reaierccrearerer eae TC 3.4 Simple Measurements 117 se = of the “input” sine wave across the string, and the “output” sine 7 Z wave across the single resistor. ee ;, Now connect a resistor and ie in series. Choose a resistance R opie across the front and back of the pair. (You should take care to set gene DC offset of the waveform generator to 0 using the oscilloscape to Z measure the offset relative to ere Do this as a function of HEqOEnSY, vat eS sine wave, relative to the input sine wave. Figure 3.15 shows how to make ee “these measurements on the oscilloscope CRT, using the circuit shown, ee ZR efer to Fig. 3.8 for interpreting the input and output waveforms in terms ey “of gain and phase. It would be a good idea to select your frequency values Z logarithmically instead of linearly, That is, use vg, 2019, 49, ..-. Vmax Where B2-yy is your Starting low frequency. Make a clear table of your measurements ie e: “and plot the gain (i.e., the relative amplitudes) and the relative phase as a Ee ee = function of frequency. Do not forget that you measure frequency v, but most Fe eee tam aar™ SOE COLL COLO E IES FILO PETE TT ee eaten ROR OOO OO anni tiaienn canna ch ei a nen eaeae s ¥ wae vii Santas satctatalattstatstalcinnen nen manera aan aceite ene nines * * : SSIS <* a ots ae ee ee ee ee ee pS “ee “y ee ee ee ee ed ee ee es ee ee a te FIGURE 3.15 Measuring gain and relative phase on an ascilloscape eee ' ye , ans ee) * fee Maes ~— ne Ar | os? ay ns Pols en 118 8603 «Electronics and Data Acquisition 10° tor Gain i 1907 10" 10 1o* 105 10° 10° Angular Frequency (Hz) Phase (Degrees) 10 0 10° 10° 10* 16° 10° 10° Angular Frequency y (He) curves were calculated using the known values of the resistor qd A53 Ks s and capacitor (0.1 pF). ae z AS OPERATIONAL AMPLIFIERS ee 4.5 Operational Amplifiers 119 a Noise can getin the way of your measurements by causing things to change “syhen you do not want it. These changes can happen as a function of time, oo = frequency, temperature, etc. To fight this, you want your apparatus to be sable against time, frequency, temperature, etc. The most common way to achieve this is using negative feedback. The idea behind negative feedback ig ahat you take a part of the “output” and subéract it away from the “input,” Z, causing it to “feed back” to the output and scourge it from changing. te : E hes the difference voliage between its inputs to give an 1 output voltage. Let Hee the gain of the amplifier be aw. That is, for the circuit in Fig. 3.17 we have CVn = aVi,. We apply negative feedback by taking some of the output #-Noltage and subtracting it from the input. This is shown in Fig. 3.18. A TEN ae =yesistor voltage divider is used to take a fraction 8 = Ro/(R; + R2) of the output voltage Von and subtract it from the input. The amplifier now ; does not mt amphiy Vin directly, but instead amplifies Vair = Vin — B Vou. a Vour = of Vaig = & Vin — 0&8 Vou, and the net gain g Is g=- tr (3.15) FIGURE 3.17) A genene amplifier. = mm a #20 33 «Electranics and Data Acquisition Vin Vout = Vegi FIGURE 3.18 A generic amplifier with negative feedback. Now’s here the key point. The generic amplifier is designed so it has” enornious gain. That is, a is very, very large. So large, in fact, that a8 >> li no matter how small f is. That means that the gain is 8 ee 3 oe eee oe S a oe 2B : Bee ‘ ag gacalt—t — forop > . 6.16) 5 B Ro ah The gain of the system only depends on the ratio of a pair of restsior values, ae and not on the gain of the generic amplifier. ILis hard to get resistor values: ee fo then so this oe tk circuit is very stable. The generic amplifier | ad See shown im Fig. 3.17 are available i in lots of flavors. They are called opera: . tional amplifiers or opamps for short. Instead of a box, they are represented: by a triangle, as shown in Fig 3.19. The two 0 inputs are labeled aoa and: nea Hut Reon a ee $1 althou gh you can pay a lotif you want special properties. All have very Be large gain, i.e., ¢ upward of 10* or more, up to some frequency. (Rememt= ber that capacitance kills circuits at high frequency because it becomes: Hf a short.) An old, popular opamp is the model 741, which is still widely | used today, A version of the 741 in standard use today (the LF411) has ae re gain of at least 88 dB (Le, a > 2.5 x 104) and can be used up to fre= a an SSeS Se sh ae 3.6 Operational Amplifiars 121 +V Out = FIGURE 3.19 Opamp notation. Vour 10 . oo eee ate ee FIGURE 3.20 An amplifier circuit with gain of 100. ae = Oe Rp so that the gain given by Eq. (3.16) is g © R,/R2. For example, z 0 build a stable amplifier with a gain of ~100 up to a kilohertz or so, you 2 a - Another application of opamps connects to our discussion of passive © fers. (See Section 3.1.5.) The effective input Impedance of an opamp in — —- , a4 122, «3 «Electronics and Data Acquisition 42, seen ‘sree Eee eee rarer) apenas! ete TEES a Pa a ee | Pe eee | Per Pore) oe FIGURE 3.21 A high-pass filter with input load buffering. SR Ke so it draws no current, This makes the opamp ideal for “load buffering.” vt That is, you can use it to make the input to some device (like a filter or! perhaps a meter) large enough so that you can ignore its effect on the circuit 2 that feeds it. For instance, you might build a high pass filter as shown itt: Fig. 3.2]. Ali the output of the opamp is fed back to the input, thus § = 1 and gy = 1. However, Zi, = 00 (effectively) because of the opamp, so‘? all this circuit does is cut off the output of the source for @ < 1/RC hike? a good high-pass filter should. If the opamp were not there, you would: ce a SSK COG 5, need lo add in the filter input impedance Zgrery = R + 1/iwC to the: 2 source circuit. See Dunlap (1988) for further clever variations on active eB filters. ES ee 3.6. MEASUREMENTS OF JOHNSON NOISE Ss In this experiment, we will measure a very fundamental source of noise. te e has to do with the motion of electrons in a conductor and the heat energy. (random motion) associated with them. This is called “Johnson noise?! because it was originally measured by J, B. Johnson. Some people calf: it “Nyquist noise,” because the phenomenon Johnson measured was first 2 correctly explained by H. Nyquist. A more generic term is “thermal noise,” Some journal articles on similar experiments are listed at the end of this. chapter. You might also want to go back and look at the original work: of Johnson and Nyquist, published in J, B. Johnson, “Thermal Agitation of Electricity in Conductors,” Phys. Rev. 32, 97 (1928), and HL. Nyquist “Thermal Agitation of Electric Charge in Conductors,” Phys. Rev. 32, 0: (1928). etn a ae att ete tat wt apetert ee ee atta what wate Paes 3.6 Measurements of Johnson Noise 1273 e 361. Thermal Motion of Electrons : 2 We will outline a simple mode! of thermal noise as presented by W. Henry ‘(see references). The model is based on random thermal fluctuations of be: electrons in a one-dimensional resistor of length L and cross-sectional area ee A. The resistor has resistance R, and a voltage drop V = /R across the % - ends. The current J, and therefore the voltage V, arises from the thermal ee . “- es p On average no current flows through the resistor, and the average value é- of V is zero, That is, (V)=0 “a Ce re g: On the other band, the thermal fluctuations still give rise to a finite voltage 67: as a function of rime; in other words V(r) A 0. Therefore, the variance* = of V is not zero; namely, of = ((V —(V))*) = (V2) — (Vv)? = (V4) 40, = This quantity (V2) = a¢ is called the thermal or Johnson noise voltage, = and it is what we will measure in this experiment. From Ohm’s law and the definitions of current and charge, we can write POONA HLS hoeknienn hon i tae oy =o;R teeta seh ee C. ee —ft ane fp ee fg I! - ane a * eh nee ne ‘ 2 - where L is the length of the resistor, and oy is the net x motion of all the f=: electrons in the measuring time fo. If we can reduce this to the motion of an Be individual electron, then we can use a microscopic description of current t= and resistance. If there is a total of N independent and random electron #2 motions (i.c., “random walks") in time fg, then Bor Bs i *The student may want ta review varions definitions in the theory of statistics, given in a Chapter 10. 124 §=83 «Electronics and Data Acquisition where og is Lhe average distance that any single electron moves. Therefore: Now AW is the total number of conduction electrons in the resistor times the number of walks in time fg, so = (AL) x fo _ nALiy -* where 7 is the number density of conduction electrons and t is the time between collisions of a single electron. The fluctuation in the motion of ; a single electron is ; of = (d*) = (vic?) = (w2}r’, and this is what we connect to temperature by {EF} = Fm(v v2?) = LT 2 where m is the mass of an electron and we note that motion is ‘only Ee the fundamental relationship between temperature and internal energy, | Therefore = a2 kT c2 a om We note that (see Eq. (2.14)) Le 2m iL =—o = R, Anec A° where p is the resistivity.° Finally, put this all into Eq. (3.17) to get 2 2 mp Od. p2 ~ 72 2 2 =7 I mig _A kT ne*t t * R?, ~ | im to "The definition of r used here differs from that used in Section 2.2 by a factor of 2. That oe is because we are dealing with a single electron. cS ree res whet ae cad Deh one inennetnna st 3.6 Measurements of Johnson Noise 125 os or (V7) = oRTR (3.18) io “Jt is customary, however, to express the noise using the equivalent =: bandwidth Av = |/2i9. Therefore, we have a av? = Mer RW) dv. alt te ob tain the expression ae eso (V2} = A4kTRAp. ~ GA9d In order (0 measure the voltage V, we will need to amplify or at least _ Process the signal in some way. Let g(v) be the gain of this processing seerreult at frequency v. Then the output voltage fluctuation d (v2) integrated over some small frequency range dv is given by ead ‘see (v2) = 4kTRG*Ap, (3.20) = Swhere G and Av are constants defined by = Pave | g2(v) dv. (3.21) 0 wees 36.2. Measurements =-We will measure the Johnson noise in a series of resistors, and use the result 2° determine a value for Boltzmann's constant k. . The setup is shown schematically in Fig. 3. 22. The soe ACTOSS the am a ae Woe sas 7 3 :ipasure v2), given by Eq. (3.20). By chanping the value of R (simply by = Changing resistors), you measure {V2} as a function of R, and the result 2 should be a straight line. The slope of the line is just 4kT G* Av, $0 once -e = ae saree” aleyete eee oe te aa” a as 126 04«=63-s Electronics and Data Acquisition Digital Amplifier | Oscilloscope gain we need, take a bandwidth Av = 10 kHz. The digital oscilloscope 2 cannot make measurements much smaller than around 0. 5 mV, so Eq. (3 ane B like the right solution. a8 The bandwidth of the amplifier also needs to be considered. In fact; if we are going to do the job right, we want to make sure that all the: g bandwidth limitations are given by the amplifier, and not by the oscil: g loscope, for example. Thal way, we can measure the function g(v) ae the ene stage a The oscilloscope bandwidth will depend on the ee ein than the amplifier’ s, you will be OK. You ensure this by putting. ae bandwidth filter on the output of the amplifier. In the beginning, you: : Bree: Imits, ES ae The first “amplifier” you will usc, therefore, is shown in Fig. 3. 23:33 2 For now the bandwidth filter is just a box with an input and output, and with knobs you can torn. T he gamn- producing patt ‘Orne! annpifiren, wi i a a Brrr al ‘ eh) ‘ 3.6 Measurements of Johnson Noise 127 ~ SON ne hobe . rake Sere PPG Oe ek “* eee SOS RM HC OCM DLN NDC BL PM PLO ber etic ae ; lel er ia era ve ere e ae Oe) Mrererate rere ere tere letetsi ee ont nnanetatae re nt a ata alate E a FIGURE 3.23 Amplifier stage for measurement of Jobnson noise. SERRE LS NC +V Ow Bal . wee * “Lie ee Bal -In +in -V eZ FIGURE 3.24 Pinout diagram for the opamp chips used in this experiment, We are nol 3 “ee using the “Bal” connections. The notation “NC” means “no connection.” : yen and uses a HAS147.° Good starting values to use are Ry = 102, B Ry = 100 Q, and R3 = 2.2kQ. This gives the first stage a gain of 11 and = the second stage a gain of 221 times the bandwidth function imposed by aS the opamps and the bandwidth filter. - All of these components, including your input resistor R (but not the G ccanmercial bandwidth filter), are mounted on a breadboard so you can F Be change things easily. The pinout diagram for the HAS!70 and HA5147 is “Shown in Fig, 3,24. The opamps are powered by +12-V levels applied in Bee parallel with 0.|-\.F capacitors to ground, to filter off noise in the power Doane Connections to the breadboard are made using wires soldered to pe PNC connectors, a See ee : he credit for figuring out the right opamps and amplifier circuit in general goes to Jeff Reson RPI Class of "94, More details on this circuit design are available. A sors SOO eee Cee ew ey PINAR ANANSI Serena thee a te a ete ee a ene SN SS Ss Se . 1728 3 Electronics and Data Acquisition Set up the circuit shown in Fig. 3.23. Check things carefully, especially if you are not used to working with breadboards. In particular, make sure-the Ee 12-¥ DC levels are connected properly, before you turn the power supply: =: on. The output from the breadboard gets connected to the bandwidth filter::::2 and the output of the bandwidth filter goes into the oscilloscope. The lower and upper limits of the bandwidth filter are not crucial, but 5 and 20 kHz: are a reasonable place to start. 8 First you need to measure the gain of the amplifier/bandwidth filter as ae function of frequency. All you really need to do is put a sine wave input to. oe the circuit and measure the output on an 1 ascilloscope. The output should is just the gain ¢(v). Thereis a problem, though. You have buiit an amplifiek: of very large gain, around 2.4 x 103, and the output amplitude must be less - ee ae eee irre a om ae The waveform generator output passes through a voltage divider, cutting: : the amplitude down by a known factor. This divided voltage is used ag: ‘ ei aa ee Waveform generator FIGURE 3.25 Caltbration scheme for the noise amplifier. Digital oscilloscope SERS ‘ SEL AAA ~~ As, Se SS Pal . re ee Ni What EEE SRNR AROS eal ha | ‘ oF La . . . . os mine ERE ec tet RRR EA SER EASES oS Se a SLSR anc See See DEO ON SUS aCe SaaS i see Sse be Pathe aL Sa Cee Scat atte Por i oe hs bees 1 3.6 Measurements of Johnson Noise 129 2500 Nogatlve feedback gain 2006 1500 1006 ch o o Bandwiaih Irmitts Gain of amplifler and bandwidth filler 0 5 10 15 20 25 30 35 40 Inpul valtage frequency (KHz) “FIGURE 3.26 Sample of data used to determine g(v) for the amplifier followed by the commercial bandwidth filter. The simple negative feedback formula gives a gain of 2431, and be bandwidth filter is set for vig = 5 kHz and yyy = 20 kHz. Make your measurements of g(v) by varying the frequency of the wave- =form generator, and recording the output amplinide. Of course, you must also record the “ (1.e., generator) amplitude, but if you check it every “tment. Measure over a range of frequencies that allows you to clearly see the cutoffs from the bandwidth filter, including the shape as g approaches =: gero. Also make sure you confirm that the gain is relatively flat in between =the limits. An example is shown in Fig. 3.26. The setup used R) = 10Q, =Ry = 1009, and R3 = 2.2kQ, so the total gain should be 2431, and with bandwidth filter limits at 5 and 20 kHz. The main features seem to “Be correct, although the filter has apparently decreased the maximum gain = @ bit. =: Now take measurentents of the actual Johnson noise as a function of R. Remove the waveform generator and voltage divider inputs, and put the fesistor you want to measure across the input to the amplifier. Set the time = per division on the oscilloscope so that its bandwidth limit is much larger than the upper frequency you used on the bandwidth filter. For example, sf there are 10,000 points {i.e., samples) per trace and you set the scope to 139 «63> «Electronics and Data Acquisition 20 Singlo sweep > 10 £ g | © -10 -2) 0 0.6 1 15 2 2 100 sweeps 1 ! S 5 o 9 | oS 5 8 - -2 0 0.5 1 1.5 2 Time (ms) FIGURE 3.27 Oscilloscope traces of the output of the bandwidth filter, and for 100 traces: vas averaged together by the oscilloscope. Note the difference in the vertical scales, ree 0.2 ms/div, then the time per sample is 0.2 5 since there are ten divisions: The bandwidth is the reciprocal of twice this time or 2.5 MHz. If the filter: 242 cuts off at 20 kHz, then this would be fine. Baas Use resistors with K near zero (1082) and up to R © 10kQ. The oscil. a fe loscope trace will look like an oscillatory signal, but that is because you 2222 are (likely) using tight bandwidth limits. What would the trace look like if Se the lower limit was only slightly smaller than the upper limit? ae Figure 3.27 shows a single sweep trace on the scope directly from the. ee : output of the bandwidth filter, and the average (as done by the scope itself.) of 100 traces. The average looks the “same” as the single sweep, but it:)22 fe is 10 times smaller. (Note the difference in the vertical scales.) It is cleary:24= therefore, that the oscillations i in the i input signal are random i in phase, he cae answer using the MATLAB function trapz, which nerforms a sere ea ny te ‘a! rata Pater = a ¥ a! a - . ’ a a 5 a _ att ‘a 2 . ¥ aa dl _ ana! ate “a ” *- at *. Fr, CJ a - oa a! ae Pa = ve "i ry a - a oa eee — oa “a i F vale a Lat, ‘a Sees * a at al a 2 aaa! eS pee a MARATANTATATS Raa Pe al he ke a Ci a ee ia haa bn et at a ot See be rer irra ara atiae ar rr eae ae a 3.6 Measurements of Jahnson Noise 731 Output voltage variance (mv?) 0 2 4 6 8 10 12 14 Input resistance (kQ2) =: FIGURE3.28 Datataken by measuring the standard deviation of the output voltage signal, = asa function of the input resistor value. The slope gives &, while the intercept gives the equivalent input noise voltage, after correcting for the amplifier gain x bandwidth. ee integration given a list of (x, y) values. For the data of Fig. 3.26 one finds =: that G* Av = (7.9+0.5) x 10’ kHz. Next we make a plot of ((V — {V})*} as a function of R. Note that = since (V)} = 0, the above expression reduces to (Vv). The plot is shown in : Fig. 3.28 and a linear fit gives (V44/R = (1.33 £0.08) mV*/kQ and an intercept at 4 mV°. We can now calculate Boltzmann’s constant & from the above data using | Eq. (3.20) and setting T = 298 K (room temperature). Using units of hertz, *; volts, and ohms, we write — (¥4y/R —_— (1.3340.08) x 10~? = Sm __- = (1,420.13) x 1077 TK. 4TG2Av 4x298x (7.90.5) x 10!9 ( )x 107" J/K 3 This result ts in excellent agreement with the accepted value k = 1.38 x © 10-? J/K. 132 «3 «Electronics and Data Acquisition The intercept of the line in Fig. 3.28 is the noise at R = 0. You would - expect this to be zero if Johnson noise in your input resistor were the only: | thing going on. The input opamp, however, has some noise of its own, due : to internal Johnson noise, shot noise, and so on. The specification sheet for .: the HA5170 gives an equivalent input noisc of around 10 nV/./Hz. How | does this compare to your measurement? | There are a number of variations and extensions to this experiment. For ° example, instead of simply using the oscilloscope to determine the standard « deviation, use MATLAB and the trace data (as in Fig. 3.27) to get the values :: and examine their distribution. You can get the data into an array trace, -: and you can use mean(trace} and std{trace) to get the mean and standard :: deviation. The series of MATLAB commands used to plot the distribution 24% might look like ee bins = linspace(min(trace), max(trace), 50); [n, x] = hist(trace, bins); stairs(x, n); The single sweep trace in Fig. 3.27 is plotted this way in Fig. 3.29. The: 2 distribution is rather Gaussian-like, as you expect, but you could test to 2322 30 20 Number of measurements 16 -i0 5 0 5 10 Output vollage (mv) FIGURE 3.29 Histogram of the individual voltage values from a single sweep trace. The line is a Gaussian distribution, with the mean and standard deviation determined from the trace data, and normalized to the nurober of measurements. 3.7 Chaos 133 see whether this is really the case by comparing it to the Gaussian with (he same mean and standard deviation, and considering the x*. (See Chap- ter 10 for definitions and discussions of these quantities,) Some digital oscilloscopes have the capability of performing a real-time Fourier analy- sis of the input. That means that you can actually demonstrate that the noise spectrum d(V") /dv is indeed “white,” that is, independent of frequency. This is straightforward data to take, but will require that you lea more about Fourier analysis to interpret it. One nontrivial circuit modification would be to make your own band- width filter. For example, consider the circuit shown in Fig. 3.12.’ Try assembling components that give you reasonable parameters for the gain integral in Eq. (3.21). A simpler kind of filter might simply be two RC filters, one high pass and one low pass, cascaded in series. If you want to do active buffering, though, be careful to use an opamp that works at these frequencies. Another interesting variation is to use a few-kiloohm resistor as input, but something mechanically large and strong enough to take some rea] temperature change. Lf you immerse the resistor in liquid nitrogen, for example, it should make a large (and predictable) change in the Johnson noise. 3.7. CHAOS We now discuss a measurement that uses nonlinear electronic components to explore phenomena characteristic of complex physical systems. 3.7.1. The Logistic Map and Frequency Bifurcations We are used to the nation that physical systems are described by differential equations that can be exactly solved for all times, given an appropriate set of initial conditions. This is not true in complex systems governed by non- linear equations. A typical example is the flow of fiuids, At low velocity one can identify individual “streamlines” and predict their evolution. However, when a particular combination of velocity, viscosity, and boundary dimen- sions 1s reached, turbulence sets in and eddies and vortices are formed. The motion becomes chaotic, Many chaotic systems exhibit self-similarity: that 7 This, in fact, is what Johnson used in his 1928 paper. You might want to look it up, and compare your results to his. 134 «63> «Electronics and Date Acquisition is when the flow breaks into eddies, the eddies break into smaller eddies and so on. Such scaling is universal; it is observed in all chaotic systems.. A particularly simple case is that of systems that ohey the fogistic map introduced in connection with population growth. Designate by x; the ~ number of members of a group at the time j. Here the group may be the — g eS population on an island, the bacteria in a colony, etc. The index j labelsa_ . finite time interval (such as a day or a year) or the successive “generations” ° of the population. If the reproduction rate in one generation is A, then it = would hold that Xjq] FAX}. However, the population will also decrease due to deaths. In particular if a Bs the food supply on the island is finite the death rate will be proportional to: ee x} . Thus the evolution® is governed by ihe map 2 Xjpi = AX — SX}. oe oe We use the term map, because given x; we can find x;4) uniquely. Both - : i and s are assumed nonnegative. We sec immediately that if A > 1 and = s = Othe population will grow exponentially, whileif A < | the population * 2 Zs will tend to 0. The map of Eq. (3.22) can be rescaled by introducing yyp= ax for all j. Then y; obeys the logistic map yr = Ayj(1 ~ yj). (3.23) © The above map has the interesting property that if the reproduction rate for : one generation is restricted in the range : Q 3 the system behaves in a very different manner. As soon asi > 3 but A < 3.4495... the population alternates between 2 stable values. When A > 3.4495... the population alternates between 4 stable values until A > 3.54,.., where it altemates between & stable values; for \ > 3.56... the population alternates between 16 stable values, and this continues at ever more closely spaced intervals of 4. We say that there is a bifurcation? at these specific values of A. These results can be easily checked with a pocket calculator or a simple program. Table 3.1 gives some typical results for A = 2.8, A = 3.2, and A = 3.5, and the stable points are shown in the graphical construction of Fig. 3.30. What is plotted in Fig. 3.30 is ysna) VS initials DHE Continuous curve is the equation of the logistic map yr = Ayj(1 — yj). In Fig. 3.30a the curves TABLE 3.1 Example of Stable Points of the Logistic Map r=28 y* = 0.6429... A=32 y* = 0.5310.,, = 1.7995... Ave chal y* = 0.3828... = 0.5009... = 0.8269... = 0.8750... *Henri Poincare in 1900 had noticed such bebavior in mechunical systems and named it the “exchange of stability.” (a) tb) y final y final 0 0.2 0.4 0.6 0.8 1.0 y initial 0.6 0.8 1.0 y initial FIGURE 3.30 Plots of the logistic map: (a) for A = 1.0 and A = 2,8; for 4 = 2.8 there is one stable point at y* = 0.6429.... (a) For = 3.2; there are now two stable points at y* = 0.7995... and y* = 0.5130, See the text for details of the path leading tn the stable 0 0.2 0.4 points. a bo en ia . at) iat SOP Ca he a al atta tateterat et tet atatatatan ba hat abet ebenens, 4 2, 3.7 Chaos 137 for A = 2.8 and A = 1.0 are shown, while in Fig. 3.30b the curve for A = 3.2. The lines for yy = y are also drawn. We can follow the path from some initial value yp = 0.1 in Fig. 3.30a to the stable point (indicated by a circle). Given yo we find y; = yr at the intersection with the curve. However, y; must now be used as an input, 4, so we use the ys = y; line 1o locate y; and proceed to find y) and so on. The process converges to the circled point at y* = 0.6429..., It is also evident that the same construction for the A = J*curve will lead to y* = 0.0. In Fig. 3.30b we start (for more rapid convergence) from yo = 0.2. We now find the two stable points at y* = 0.7995 and y* = 0.5130. The map requires that one stable point leads to the next and yice versa, When A > 3.5699... the population no longer reaches a stable point but takes on an infinity of values in the range 0 < yoo < 1. We say that the system behaves chaotically. This persists in the remainder of the range 3.5699.-- 1 but also from 1 —> 2 with: oe the same probability. It is equally reasonable that the photons arising from ©: ae stimulated emission will have oxecry the same frequency and same direc Q a 4.1 The Principle of Laser Operation 153 ae WIV ee 1 —— So (a) (b) (c) =" FIGURE 4.1 Emission and absorption of radiation between twa atomic levels: (a) Spanta- Eo NeOOs emission with transition from state 2 to state |, (b) stimulated emission with transition si 2 — 1, and (c) absorption with transition | > 2. SSS me a by A. Einstein, and the coefficients A, B are related and can be calculated =: from a knowledge of the structure (wave function) of the atomic states aan mene bia an alt ke 3 3 re egrr ic? | (Fp + |i3]?. (4.4) aca Pei re Mi Sal hal eh a, ohhh => Flere { f |p -€ |i) is the matrix element of the electric dipole moment opera- = tor ) between the initial and final states and € the polarization of the EM = field. Loosely speaking, the matmx element ts a measure of the average “] pp. co ES © 5 = Ec gm — & ad = = D or fo =F —s a o an i eo wi 3 Er. o =) —- wi a | Et. 0 Pu mJ o Es Aa] _ = for stimulated emission for the same external field, a consequence of the => seversibility in time of elementary physical processes. So far we have talked =: about single atoms, whereas in reality the lasing medium consists of a col- =: fection of N atoms, It is then important to know how many of the atoms se are in the (excited) state 2 and how many in the (lower) state 1; we will “: designate these numbers by V2, V1, where No + Ny = NV. The number of . transitions per unit of tirne from 2 —» 1 is then given by ah RSenh} SSN we rN Ro) = No(A+ Bu(a)), (4.5) as wee = whereas from 1 — 2 R43 = Ny, Bulw). (4.6) = Here u(m) = du/dmw isthe energy density of the EM field per unit frequency einterval. Normally the relative population N2/N; is governed by the SSS SSS 154 4 Lasers (a) (b) eer have a fast spontariecus decay along the indicated auTOws. Boltzmann distnibution —Ko{kT Roar Na Ne a ele evar aay N, Nef i/k? , ier We can see from Eas. (4.5) and (4.6) that for stimulated emission to eh in preference over absorption, we must have Nz > Ny. Usually the opposite: Z is true because AE for atomic levels is of order of a few electronvolts; whereas at room temperature kT = 0.025 eV, and from Eq. (4.7) No «K Nios It is therefore necessary to create a population inversion, namely to increase N2 while maintaining N; small. This can be achieved by involving three: or four atomic levels as shown in Fig. 4.2. In the three-level laser, atoms:22 are pumped from the ground state ] to the excited state 2 and quickl; 3 decay to state 3 by spontaneous emission. If Ny exceeds Ny lasing can tak place in the 3 — 1] transition. It is, however, easier to use a fore If the lasing medium is placed in an optical cavity (Fig. 4, 3) we cant assume that photons emitted nan the cavity axis are trapped t in the cavity a ie 4.1 The Principle of Laser Operstian 155 ny +73 =A, oe 7 being the atomic density, and dn3 ng a i Won) — BNyns —_ rs (4.8) dN, N. = ates dt BNyn3 Te (4 0) : ‘Here W, 1s the probability per unit ime for pumping 1 — 2 — 3 (transfer- ting atoms from state 1 to state 3) and Bi is the probability that one photon 3 : to spontaneous transitions is t and due to cavity lasses Te: The (mode) vol- 2°" wme in which the photons interact inside the lasing medium is designated es oy V. In all cases the spontaneous transition rate 1/r « B Ny, SO we can £2. neglect this term. With this assumption, the steady-state solution of the rate He equations (i.e., dn3/dt = dN, /dt = 0) is : ee Ny = Wont Vt. (4.10) , ~* ~ feo hb the steady state, the cavity losses per pass equal the gain per pass; the = ‘laser output depends linearly on the pump power, lasing medium density, = and mode volume. Note that VB = co where a is the cross section for the oe 2 absorption of photons in the lasing medium. ed 2 dN, The (logarithmic) gain per unit length of the lasing medium is found fe from Eg. (4.9) if we neglect the cavity losses, Then dé = VBn3dt = V Bn —=onj3at Ny c 156 4 Lasers and Thus in a finite length £, a number of incident photons N, (0) will grow to Ny (2) = Ny Oe*. Often ¢&* = G is designated as the gain per pass through the lasing’ medium, 4.2. PROPERTIES OF LASER BEAMS Lasers emit a “beam” of light, the properties of the beam being determined 2 primarily by the optical cavity. In the cavity shown in Fig. 4.3 the radiation “2 travels in hoth directions and the electric and magnetic fields of the wave ~:: Sy renebtitg stereo aa i my must satisfy boundary conditions at the two mirrors. Standing waves will -23 exist in the cavity as shown in Fig, 4.4, and only frequencies such that the : cavity tength is an integral number of half-wavelengths are allowed. H the 23 cavity length is é, then 20 where g is an integer. The frequency difference between two such adjacent longitudinal modes is Vet — Vg = = = FSR (4.12) = FIGURE 4.4 A laser cavity must support standing waves. =-g and v-=q--, (4.11) = re a Pee ry ce sea eee) rere ay ora wey icinsisatsacennibag prostrate tn eee * ee a ee SSM AN TNS wht ete 2a nEEL! Blatetet cabot satis - . wale ae nae chtalue tlt, STAI hnatend: +14 ota ates =" 57,5, Recher A Se teh eis rk bo 6k ee ee ee ee ee rete peye *,° bh ia hl Sa On ha ne atte ety thatata tate tate tate SAA Sot hy AAO SO oe Schr e bs ae be SSS SAAS SSS SS eevee ase Pa be ee be) vee heh ‘) SHS ee eeeeeae vere . see Cm} 4.2 Properties of Laser Beams 157 G Threshold FIGURE 4.5 The gain curve of a typical lasing material as a function of frequency. Only :~ Vines with gain larger than the threshold will lase. :. and is referred ta as the free spectral range (FSR) of the cavity. As an =: example if we take £ = 0.5 m, we find that FSR = 3 x 10° Hz. This = spacing is very narrow as compared to the frequency of optical lines, i.e., = forA = 600nm, v =5 x 10! Hz, and 2£ Vv 6 =—v= — ~ 1, 10”, q c ‘ FSR ae Only a limited number of longitudinal modes are present in the emitted radiation. This is so because the lasing levels have finite energy width; this width determines the range of possible frequencies as shown in Fig. 4.5 and = js referred to as the gain curve. The width of the individual longitudinal modes is determined by the number of round trips the light makes in the cavity before being attenuated; this is referred to as the finesse F of the cavity. The finesse depends on the losses in the cavity. If we consider only the losses at the mirrors that have a reflectivity R < 1, we find (see Section 4.6) FSR oc (1—R) MS = YT, ree (4.13) or to a good approximation Cc ~ One For R = 0.99 and £ = 0.5 m, we find Av = 10° Hz = 1 MHz. In conwast, the gain curve has a width of several gigahertz. Av (1—R). (4,14) Reece Se ee Besant Rteeaenctines 158 4 Lasers In the transverse direction the optical cavity is not bounded but is open However, the bearn is confined near the axis and its transverse structure is determined by the focal properties of the mirrors and the length of the cavity. A simple example is the confocal resonator, where both (spherical) mirrors have equal radii of curvature,’ R, and R equals the distance, £ between them; oole that f = R/2, $0 that the focus is in the center of the cavity. The transverse beam distribution can assume any of the transverse modes characterized by the indices m, n. The electric field at a longitudinal: distance z from the center of the cavity and at the transverse coordinates x, y is given by ie _ xe4y? . AS E(x, y= Foro Hn, (2 H, (< ") Cz) (4.15). 23 wz) wz) Hm, H, are the m,n Hermite polynomials, and Eo is the peak field value, i For simplicity we have omitted the phase of the field. Reeeee Of particular interest is the lowest mode where m = n = Q, the TEMop eee mode. Since Hy = 1, the field distribution is a Gaussian Beer: w -(S2 ) ae E(x, y)=A ml ys, (4.16) i wz) Bee ee The field falls to 1/e of its peak value, and the intensity to 1 /e*, at a radius Ee r = w(z). We refer to w(z) as the “beam radius” at the distance z. The ee smallest beam radius is at z = 0, where the wavefront is plane and normal 334 ate iS oe £2 a wq (confocal) = a (4.17) 28 The beam radius at the distance z is given by w(z) = woy | + (2/20)*, GIB where zg is the confocal parameter, or Rayleigh range. It is related to the.“ beam Waist through - JEW es 0 as £0 = - >. (4. 19) Rees A 2 It is unfortunate that the same syifibol 4 18 useth ror dae tenecavd she.divs nf curvature of a spherical mirror or lens. ee TS a hoo a aoa aS aa 43 The HeNeLaser 159 28 Zz “| FIGURE 4.6 Focal properties of a TEMgg Gaussian beam propagating along z. Aft the = waist the amplitude falls to 1/e of its on axis value at a distance wy from the axis. Note the =: wavefronts (surfaces of constant phase). The Rayleigh length zp and the di?ergence angie = Ay are also indicated. :: Thus for the confocal resonator, where Eq. (4.17) is applicable, we find = that zo (confocal) = £/2. = In this case the beam radius at the mirrors has grown by /2 over the value at = the waist. At large distances z >> zg the beam divergence is given by d. WG) ok (4.20) = and for the confocal cavity @ (confocal) = ./2A/7r£, = which is typically of order 107? or smaller. Figure 4.6 shows the rays, a wavefronts, and beam waist in a confocal cavity. The fact that the beam = cannot be focused to a point but instead forms a waist is due to the wave = nature of the EM field. Not all mirror combinations lead to stable cavities. The confocal res- “=: Onator in particular is at the limit of the stable range and is not used in = practice. Instead, most laser cavities consist of one perfectly reflecting flat =) mirror and of a curved mirror with radius R > /. Usually the curved mirror has a finite reflectivity, for instance 95%, and thus serves as the output cou- — pler, by transmitting some fraction, say 5%, of the beam stored in the cavity. = 4.3, THE HeNe LASER zs The heltum-—neon gas laser is the most commonly used laser for simple << laboratory work, alignment, and other low-power applications. The first 160 4 Lasers HeNe was built by A. Javan at Bell Labs in 1961 and now HeNe lasers. are available at low cost from many manufacturers, A thin tube is filled: with helium at a pressure of a few Torr and approximately 10% of. neon: gas is added. An electric discharge is established in the rarefied gas by the. application of few kilovolts between the two electrodes. The electrons in the discharge excite the helium atoms to the 2S levels, which lie about; 20 cV above the ground state. By a formitous coincidence these levels: coincide with the 4§ and 5S levels of neon. Through collisional exchange: #4 the neon atoms are excited to these levels, resulting in population inver=: 24 sion. Lasing takes place as indicated in Fig. 4.7, corresponding to the. 3 a al wavelengths | ey 8 55 —+ 3P A = 632.8 and 543 nm 45 —> 3P A = 1523 om 5§8>4P 4 =3391 om. 8 The 3 P level de-excites quickly to the 3.5 state from where the atoms retum = to the ground state by colliding with the walls. By coating the mirrors for = Collisions 20.8 eV 35 rt? ev Collisions with walls Hei's Ne (18? 25% 2p} FIGURE 4.7 Energy levels of helium and neon. The principal lasing transitions are 4 indicated by dowble arrows, Note that the ground state is at a much lower energy. eee Fs es ua . : ul . Saal ae SE Shee SE ad ae ee ee ee nat 2a als hate & an es ah 1 “aS ae SEM elepenon SSSA - '- e A = - . '. = - ve a = . . * ~ ‘. ' + ee! yn) x at ' + - a a * = 5 a ~* on * * _ = we ' en) . On) ae eee oe Ml . . ' . . “ “) eli ate ot an ttt ‘ oh mi -\ s at ts ee oa ae ene ares! eee ai al ey Oe tat? tein, = ey =e cute enet = “) ~* oe ~e +," hao Se ‘e"s ot hee wa bee Sve ee i" a ‘". *. o. = oe 5S _ a fas 4 =~ a. -- o— as oes —= Fa ‘at = or os w- —. ii *« woke ¢.*. 5s oe sale cA >" a -—-- -"- r<- =" = ie =. ~' = oe 2% or ee. ss ws. vs. = ~ < wan = “. UD MAN ASSESSOR E A Te Suits ly OLAPELILLELA LETT LLA EAA E LL DY Z Z = FIGURE4.8 Schematicof a HeNe iaser showing the discharge tubc and the cavity mirrors. (b) FIGURE 4.9 (a) Definition of Brewster's angle &. (b) Transmission of a p-polarized ray at Brewster angle without attenuation. reflectivity at a given wavelength, a particular laser line, most often the red line at 632.8 nm, can be selected. A sketch of a HeNe laser is shown in Fig. 4.8. The tube diameter is chosen so as to maximize the population inversion of the neon atoms, an empirical formula relating the pressure (in Torr) to the tube diameter (in mim) being pD ~ 4Tort-mm, usually D ~ 2 mm. The length of the optical cavity ranges from 20 to 50 cm. As shown in the sketch the electrodes are recessed. The gain in the law-pressure gas is relatively low, resulting in amplification g ~ 0,10 m7. As a result the power level is also low, in the range of a few milliwatts. The width of the gain curve is dominated by Doppler broadening and is of order of 1.5 GHz. A special feature in the sketch of Fig. 4.8 is the exit windows of the tube, which are set at the “Brewster” angle 6. As shown in Fig, 4,9, light polarized in the plane of incidence ( p-light) and incident at , is not reflected. If the refractive index of the window is m, the Brewster condition is Tt 5 A= R= hi 1 864 Lasers but from Snell’s law sinh = Mi sin @:. Ay Therefore we must satisfy sin & _ mt cosh my For n; = 1.0 and m = 1.5, & = 56.3° and the Brewster angle, which is’: B the complement of 6;, is & = 33.7°. Light polarized normal to the plane of “ ee incidence (s-light) is partially reflected from the windows and the higher’, 2B losses prevent s-light from lasing. In Eq. (4.12) of the previous section we showed that the spacing between. aS the longitudinal modes is FSR = c/2£. One can demonstrate the presence of 223% these modes by a simple experiment using a HeNe laser. Since £ = 0.3 m, 4 the FSR = 500 MHz, whereas the width of the gain curve is of order 224 1.5 GHz. Thus we can expect that three to four longitudinal modes could ee be lasing simultaneously. One way of observing these modes is to use a fast 222% diode to record the intensity of the laser light. Because the diode detects the 2322 intensity, 1.¢., the square of the amplitude of the laser field, its signal will ee contain frequency components at the difference between the frequencies “2 of the modes present in the light. To explain this let us consider just two modes at Frequencies @) and wy. : = Then the amplitude (the electric field} is A= Aj cosayt + Az2coswot, (4.21) : te and the intensity (assuming A;, A2 real) i= |A| = A? cos’ wif + 2A, Az COS wf cos wot + A} cos” ant. and |A2|*/2. However, the cross term can be expanded to give 2A,A2 COS w;f COS wt = Ai A2{cos[(a + wo)t) + cos[(@, — an )t}}. respond to the term in the difference frequency Ay Az cosl(w — 0)t]. (4.24) = (4.22). The terms in cos? wt = A + cos 21) andcos” at = 5 + Cos 2w2t):: z oscillate so fast that the diode will respond only to the constant part |Aj |? {2 : = 4.23) As before the term in cos|(@; + @2)¢] will average to 0, but the diode can oe 7 ee par SAREE ORS nS cera ene Shhh oh SN telat tet a hn Pe ee Pe I re ee Oe ete 9 es araretatatatate NAT ae aaa’ a Ss oteletete pe het Shhh ~ — kes), i where €;, 22, and ¢; are the distances from the beam splitter to M1, M2, : oh and the screen, respectively. The resultant amplitude at the screen is As(z5,0) = = [cos(wt — 2ké; — k£,) + cos(wr — 2klo — k£,)] = Ey cosl[at — k(£, + 22 + &,)] cos[k(4; — &)], (4.36) =< and the resultant intensity I, = (|As|*) = (E3/2) cos?[k(é; — é2)]. (4,37) In Eqs. (4.35) and (4.37) we used the fact that (cos*(wt)) = }. Note that SS aS * the light reflected toward the source also forms an interference pattern of - intensity I, = (E5/2) sin?[k(Z) — €2)), (4.38) - go that Is + Ib = Io- = tis much more difficult to observe J, than /x. ?In this section we use trigonometric rather than exponential notation. 170 44 Lasers From the above analysis we conclude that the intensity at the screen will : y vary as cos*[k(€, — £2)|. Since k = 2 /A, it follows that when Seeks 1 wh Ad = | — £5 = Fi 5 n= 0, 12..., (4,39): z the screen will be bright (bright field), and when 1 eee Ag = |£; — 2) = a + ya - n=0,1,2,..., eG cae Pant ae ay at the level of a fraction of a wavelength distort the wavefront and modify _ the interference pattern. ee Nonparallelism between the murors M1 and M2 gives rise to “inter-' = ference fringes” at the screen. We assume that the two mirrors are set so: !3 that their normals are in the plane of incidence (the plane of the paper in. #44 Fig. 4.17), but 12 is misaligned by an angle a with respect to the axis of “224 the beam as shown. Because the rays retuming from M1 are reflected by 55 90° at B, we can think of M1 as located at M1’, and that the reflected rays ee propagate in exact parallelism with the z axis. The z axis is defined from: (24 the screen toward M2 and the x axis is in the direction of the screen as 2:23 indicated in the figure. For a small misalignment angle w, a well-collimated 3:23 beam, and for £2, £5 sufficiently large we need consider only rays from M2 223 that propagate parallel to the z axis. Then the rays reaching the point x on’ 4 the screen have traversed path lengths E zy = 2£) + Ls zz = 2(£2 + x tang) + £z, (4.41) ag nts es EEE ESDE ES on Sata a tale et Reeoaioe reine and their path difference is Be SS (2) — 22) = 2(£1 — £2) — 2x tana. (4.42) “2 cee rer] Sore a] Bright fringes perpendicular to the plane of incidence will appear on the 34 a aT aw screen when (z1 — z2) = mA. Consequently, the fringes are separated on |: cu ee ~ =. os Vv eS een ee ae an eee SS hh Sn Assess fe es 45 The Michelson Interferometer 171 Mi 2% 2 bs SSS BA EE” —E>=E———————— 0 x FIGURE 4.17 Schematic of the Michelson interferometer with one mirror slightly mis- aligned. To calculate the interference pattern M1 can be relocated at the dotted line M\', Vertical (to the plane of the paper) fringes appear on the screen separated by Ax = A/(2 tana). the screen by a distance (4.43) AS OSS cate For example, for the HeNe, A = 633 nm and if we take a = 107“, we find Ax ~ 3mm. As the angle a@ is increased the fringes crowd together and eventually the interference pattern is Jost. In the previous discussion we have implicitly assumed that the expanded HeNe was collimated; for a noncollimated beam the fringes form a circular pattern. Some residual curvature is observed even with a collimated beam Za when the interferometer is not perfectly aligned or when the optics have aberrations. In the laboratory we set up the murror M1 on a translation stage (the same as used for the beam profile measurements), The mirrors are carefully aligned until an interference pattern is achieved. When the translation stage 172. «34 «Lasers have sessed by, for a given amount of motion. It is convenient to meas = Az for ~25 fringes at a time; this corresponds to motion of ~8 jm, which: a can be adequately resolved by the counter on the translation stage. Ie wavelength is immediately obtained from 4 =2Az/N), aa Bee a where Az is the motion of the stage and N the number of fringes that a passed by. 6 ee 4.6. THE FABRY-PEROT INTERFEROMETER In the Michelson interferometer, two coherent waves were made to inter: fere. In the arrangement introduced by Fabry and Perot a very large (in: a eae participation of many waves, very sharp contrast between bright and sak fringes can be obtained and this results in excellent wavelength resolution: og The Fabry-Perot consists of two mirrors, often parallel plates coated oi their inner surface to have good reflectivity at the wavelength of interest The spacing, t, between the plates is maintained by precision spacers. forming an assembly referred to sometimes as an étalon. This is shown schematically in Fig. 4.18, where for simplicity we have shown the plate: as infinitely thin. A ray incoming at an angle 6 with respect to the normal after traversing plate 1 will undergo repeated reflections. We label the raysi22 emerging from plate 2 by A&, CD, EF, etc. The path difference between: He two adjacent rays, say AB and CD, is Af€=BC+CK _ a with BK nomnal to CD. The finite thickness of the plate does not modify 2 ithe above relation. It follows that oe = ee = ae Af = 2tcosé. CO Note that CK = BC cos2¢ and BC cos@ = ¢; thus, A€ = BC(I 4: He cos28) = 2BC cos*# = 2teos0. Therefore, constructive interference 2 er es a! ee ‘= a mt ae Se “ a ScacRh Tea Shania Pe t . aaa aa Se nggnannnogsnnnnre Re LS aoe * Sohhaas oa a tata A Be ERE ere teteberer ean sgREEN EET aaa ean ganan omeragnamaan mma Re Aoki Nh TTC TEESE Sa 4.6 The Fabry-Perot interferometer 178 Coated surfaces —— jf Be FIGURE 4.18 The Fabry—Perot interferometer. A ray incident at an angie @ is shown. =<: For simplicity the mirrors are indicated as infinitely thin, Note that an infinite number of reflections contribute to the transmitted intensity at angle @. * : ‘will occur when the path difference 1s a multiple of a wavelength 2¢ cos @, = ma. (4.46) a Since 8, is a small angle, n is a large number of order n ~ 21/2. The above constructive interference condition holds provided the dis- =. tance form the étalon to the point of observation is the same for all rays, “<< pamely when the observation point is at infinity. To achieve this we use a =. lens to focus the rays emerging from the étalon onto a screen, For a slightly =. diverging incident beam one observes a set of rings of radius r, = f tan6, ~ fn, (4.47) «where @, is determined by Eq. (4.46) and f is the focal length of the lens. = Note that the incident beam sbould not be perfectly collimated but should © contain enough angular divergence to support the angles 6,. To obtain the spacing between consecutive maxima (fringes) we first = note that for @ = O, the path difference between adjacent beams, :, measured in wavelengths, is ng = 2t/h, (4.48) 174 4 Lasers which in general is not an integer. The first observable ring is formed at an angle 6; where n, is the integer closest to (smaller than) ng. Thus myp=np-e€ Ose <] and At Q e= = (1 —cos@,) = = St 2 (). C9) As we move out from the center, the pth ring corresponds to np = (no —€)—(p—). (4.50) ee that the angle of the pth ring is poem anata wpe ed h Oe Op ~ J Cp—1) - (4.51) 2 ee : applicable for moderately large values of p, p = 5. As an example, if 22:2 ee t = 1 cmand 4 = 633 nm we have A/t ~ 6.3 x 10-5 and the p= lirmg. 2 will appear at 9 = 25 x 107? rads; for a lens with focal length f = 40 em: oe the radius is | cm. ne es Next we calculate the intensity of the rings (fringes) and the contrast |: 2 between bright and dark fringes. We designate by T the power transmission: ss coefficient of the inner surfaces of the étalon, For simplicity we also assume: 2224 (hat both surfaces have the same transmission and reflection coefficients. * : The power reflection coefficient is &, so that in the absence of absorption is R+7T =1. The amplitude transmission and reflection coefficients are designated by : Ee t= /T and ru VR, z o a | hae a Cr lear a a ef = : = eu & by S. or 2 4) E =r 2 oO. ob Sh Ap = Aote’®, (4.52) 2 = totale chatyetete Parte ot oe ees Phat a hea aes RS waters SSM SSS See AAA AA SCANS aR RARE 46 The Fabry-Perot Interferometer 178 where ¢ is a phase acquired in traversing both plates and the space between = them, Ray D will have amplitude Ap = Agr’e’™, (4.53) ° ray F Ar = Apr’e!™ (4.53") if and so on. Here the phase angle 26 is due to the path difference of adjacent - rays as they travel between the plates, It follows from Eq. (4.45) that Se atten 2! =. (4.54) From Eqs. (4.53) we sce that the amplitude of successive rays decreases :. by r? = R, but there is an infinite number of such rays. The amplitude of : the transmitted light is LoS] Ar =Agt*e'® }* [1 +74e!@], (4.55) q=! This geometric series can be easily summed = Se Ay = Aot*e a pielas > and the transmitted intensity T? iy =- A z7_y} eT r=g lar!’ = 10 eye aRain’ 8 (4.56) a Maxima occur when 4 is an integra) multiple of 2, whereas minima occur © when 4 is a half-integra] multiple of 7, At the maxima IgT? — —— es . 4. Ir 1 — Ry (4.57) - We see that im the absence of absorption /7 (max) = /p. At the minima IpT? (1—R)? lr = ——_—,; = 1b ——_ ¥, : Tr d+Re. 9 A+R £4.58) ~ showing that very good contrast can be achieved if R is close to 1, = Bquation (4.56) is plotted in Fig. 4.19 for different values of R. 176 4 Lasers hy a int \ i } Wy fr yy > / 1 2 f ‘ 2 f \ = f 4 i ‘ R=0.3 7 / . ; / ‘ ; \ fa06 >" ‘Sy Pid ~L = oe se ee ee A=0.9 0 i.2 o.4 0.6 8 7 Fraction of an order FIGURE 4.19 The width of the Fabry-Perot fringes as a function of mirror reflectivity, The two peaks are separated in frequency by 1 FSR = ¢/2r. The bright fringe will reach half its peak intensity when 4R sin?(8i2) = (1 — R)? or when 5 (1 — R) 2= —_— fe 2/R where the smaii angle approximation was used. The full-width at half- = maximum (FWHM) of the fringe is 25,/2. The spacing between adjacent = fringes corresponds to a phase angle difference of 27, and we define the =: finesse of the Fabry—Perot interferometer as the ratio between fringe spacing ES and the HWHM of the fringe 2m _ xR by LAR For a typical reflectivity R = 0.98, the finesse is F = 155. The spacing between bright fringes defines the free spectral range of | the interferometer. Let the wavelength 4, form its pth ring at angle @, and : wavelength Az form its ( — 1) ring at the same angie. Since these two = rings overlap, (un — Ag = FA] Or hg — Aq =Ag/n. te ape i oe seed ent nhs hee sstnieen th bathe a a nates ee (4.60) = a ee ee ay ve seo ches atta el Ma Teens WUT ISNMN En ngagenen i he 4.6 The Fabry-Perot Interferometer 177 ee However, nA; ~ nAo ~ 2t, so we obtain that Ag — Ay =A}/2t. (4.61) eS If we express Eq. (4.61) in terms of frequency, v = ¢/A we find that AZ => Namely overlapping rings correspond to the free spectral rahge already =: introduced in Eg, (4.12). For instance for A = 633 nm and ¢ = 1 cm the =. wavelength spacing is dA = \*/2t = 0.02 nm. However, lines between = fringes can be resolved if they do not exactly overlap and this depends on He the line width, i.e,, the finesse of the instrument. Thus, the wavelength : gesolution is given by a aoe, Ay) = Lz (4.62) ~ FF 2 i+ Fo ‘ For the above example and for F = 155, dA/A ~ 2 x 107’, showing =. that extremely high resolution can be achieved with a relatively simple apparatus. Fabry-Perot etalons used in conjunction with lasers are frequently made ‘. with two focusing mirrors rather than flat plates. This facilitates the align- == ment but fixes the free spectral range. They serve as high-resolution filters =. to select specific wavelengths and as optical “spectrum analyzers,” which Ih tynayeteinannnnnannnanannnnannndnan nnn aneyeqeyrae syentarneeytaciieetatetatetateratadat tar vnctat acacaacatetidt olptet tletesate Bras Realt Spusetnnnnieio SSE ~ eer tes eee ee he ee ba tal tal el ed ee eu ® v's’ aa oe ae” We SS en ae wh a i ha ht hel) one ee = are in essence high-resolution scanning spectrometers. ee CHAPTER 5 Optics Experiments eee © $1. INTRODUCTION t=: The wide use of lasers in so many applications has increased the need t=: for high-quality optics and for good optical designs. We address some of #2" these questions in this chapter where we discuss the diffraction of light and =: rotation of the optical polarization, as well as propagation in optical fibers. “= When a collimated beam of light passes through an aperture, or if it eo=: encounters an obstacle, it spreads out and the resulting pattern contains z= bright and dark regions. This effect is called diffraction, and is charac- : p imerference between different parts of the wavefront, which was altered in fe Passing through the aperture. The angle of diffraction is of order A/d with ee -) the wavelength and d the dimension of the aperture. Thus, for visible ee = light, apertures in the range 10-100 1m produce easily resolved diffraction ee 2 pattems. 2B 2 ye Bs 2 Cea . tt eee oer: be “ ens Pou FAN . . ey ae ar és , 179 180 )0«=6} sSC(Optics Experiments Very different patterns are formed near and far from the aperture. in : tee Tea ae it is convenient to form an image of it on a screen. In the far field we = ea viatele should be used and the pattern observed in the focal nlane. In the following = three sections we discuss Fraunhofer diffraction from a slit and a circular: aperture. The results shown were obtained with a CCD camera. ee The diffraction grating was already introduced in Chapter 1. o : vo ee ee aa | ee hee i a en in this experiment. The last section is a demonstration and measurement of “Berry’s phase.” This is the rotation of polarization due to a more | change in the direction of propagation of the light. It is demonstrated ES injecting the light in an optical fiber that is wound as a helix. ae a ng ae 5.2, DIFFRACTION FROM A SLIT _ B oe al ae ey ae Bo ae sketches shown in Fig. 5.1. Consider a plane wave of visible light incident - boty teal contains the slit; i.e., the wavefront is parallel to the screen. We can “divide"? the slit in half (ie, AB = BC = d/2) and consider the rays 1 and fe 3 emerging at an angle 9 with respect to the direction of incidence (Fig. 5. lay: The path difference between these rays is BD = ABsin@ = (d/2) sind; If the path difference is 4/2, then at large distances from the slit, rays. 42 and 2 will interfere destructively. However, this will also happen for rays: : 1 and 2’, 1” and 2”, and so on, so that there will be no light in the direction: e ee] rhe) wee d r ie ate 5 S1n Gy > ( ee 5.2 Diffraction from a Slit 181 z 1 me 3 - : 2 = |, cheer 4 Ze — ee Lee om @ os eae c is ae os {a} (b) Bee ees 2 FIGURE 5.1] Finding the mmima of a diffraction pattern (a) the slit of width d is “divided” Ey half and (b) into quarters. The rays are focused at infinity and the path difference is bc indicated. wove b ete eens I ae b ee tn contrast, at @ = Q the path length (out to a large distance) of all rays is panel and the resultant amplitude is maximal. co "To find the next zero, let us “divide” the slit into quarters as shown in © destntively with ray 4 and similarly for all intermediate rays. [bus there = will be no light in the direction 62, where a 2 ind => 5.2 Za ni sin 6, = 5" (5.2) This argument can be continued by subdividing the slit into an (integral) ce jmumber of smaller and smaller segments. By analogy with Eqs. (5.1) and a 6. 2) we find the generalized expression for the minima dsin@, = nh, n=1,2,3,.... (5.3) ee _ For small angles sin@ ~ @ and A 6, =n dq’ n= 1,2,3,.... (5.4) "The complete expression for the intensity distribution of the diffracted . = tent is derived in the next section. It is ee an cane) 1(@) = Io | —-—— | . (3.5) = sin 6 wee ate Po ae 182 #5 Optics Experiments where Jp is the intensity (into a small angular interval d?) in the for: ward direction (@ = 0). Intensity is the energy traversing unit area ity unt time ce I = |S| = |E x Bl = ceol EI, (5.6) where E, H are the electric and magnetic fields of the light wave, and: 2 we usually take the time average, which introduces a further factor of £ Equation (5.5) has zeros in agreement with Kq. (5.3); maxima occur (to ; good approximation} when : dsingy = (m+ 5) A. m= 1,2,3,.... (5:7) 3 The intensity at the secondary maxima decreases as m increasés;% Equation (5.5) is of the general form ee sin” x fQ@)= xe which is graphed in Fig, 5.2. Note that as x — 0, f(x) > i. (6) 3a/d —2ifd —-Md 0 Ad 2hid 3ud FIGURE 5.2 Plot of the Fraunhofer diffraction pattem sin? x/x?. 4.2 Diffraction from a Stit 188 k— 1 onanennnnsenensncntsty parte mee ee so LASER T S L FIGURE 5.3 Schematic of a simple layout to observe Fraunhofer diffraction, The experimental setup ts shown in Fig. 5.3. The laser beam is expanded “in a 4:1 telescope T, to better approximate a plane wave and is then incident Eon the slit D. The diffraction pattern is observed on the screen S, which is =in the focal plane of the lens L. Thus, we observe the image of the pattern = formed at infinity. The slit width was d@ = 200 ym and the focal length =f = 50 cm, so that the first minimum appears at a distance x from the =: principal maximum sinh ae x= fo ~ fsing = f(A/d) = 1.68 mm, =owhere we used A = 633 nm. A picture of the diffraction pattern taken == with a CCD camera is shown in Fig. 5.4. The central spot saturates the = camera. = Instead of using a slit, we can observe the same diffraction pattem by g. placing a thin wire of width d in the path of the beam. Since it is easier to rates c FIGURE 5.4 Diffraction pattern from a thin slit observed in the focal plane obtained with ie ¥-8 OCD camera. 184 «=©6§ «Optics Experiments obtain thin wires (or nas) than to manufacture thin slits, the former ae S B is given by B(x) =A —-df2 d/2. obstacle, is given by C(x) =0 —df2 d/2. Combining Eqs. (5.9) and (5.10) we can write C(x} = A — B(x) (valid for ail x). We know that when the amplitude is constant for all x, the wave mopar only in the @ = 0 direction. Thus for angles 8 + 0, the constant amplitude: = does not contribute, and Eq. (5.11) becomes oe C(x) =—B(x) (valid for 9 # 0). ee is It follows that the diffraction pattern, which is proportional to the square Be of the amplitude ee 5 a ICP = (BO, 2 vate ee is the same in both cases, Equation (5.5) remains valid but with the direction, | nS ue 6 = 0excluded. Another case of interest arises when instead of a slit a square aperture: is used. The result is shown in Fig. 5.5 and consists, primarily, of twoee single-slit diffraction patterns along the x and y directions. The intensity:2 of the maxima in directions diffcring from the x or y axes decreases verges rapidly. ce sie cS & wn sega CSR SSC SENN 5.3 Calculation of the Diffraction Pattern 185 nets wi) ite fet ays > = r es ne vu Qo ae FIGURE 5.5 Diffraction from a square aperture. woe « - Ee §.3, CALCULATION OF THE DIFFRACTION PATTERN To obtain an expression for the diffraction pattern formed by an aperture, =.we will make use of the Huygens—Fresnel principle. The principle states #-° that every point on the aperture D is a source of spherical wavelets with P= amplitude and phase determined by the incident wave. These “secondary” Be wavelets propagate at all angles and interfere at every point in the observa- Be tion plane to determine the diffracted wave amplitude. We take the incident : 2=-wavefront parallel to the aperture plane, whereas the observation plane is =located at infinity (Fraunhofer diffraction). This is approximated in the Be---sketch of Fig. 5.6 where we show both the aperture and observation planes =~ and two typical rays to the observation point P’. We need be concerned g- only with the transverse coordinates, In the aperture plane, the point M z As specified by he coordinates ¢, 7, whereas in the observation planc, the Ze 2 Becankes we observe at Safinsti R, the distance from O to the observation : ae “point is very large (and equals OS’) as compared to the dimensions of the fe Aperture; therefore, rays OP’ and MP’ are to be considered as parallel. py Then the path difference between the ray OP’ (from the coordinate origin Bee tO P’) and the ray MP’ (from the source point to P’) is the length OB oe oe vonamios 185 865 Optics Experiments diffraction. are, e a ay a along the ray OP’ we obtain for OB oO eB ~oMeqer (2 MY = ex! be ny’ OB =OM'q (5) +n(2) [gx +ny ]/R. The direction cosines of the vector OP’ are (x’/R) =u and {y'/R) =v and are well defined whether & is finite or tends to infinity. The phase 2 see difference between the two rays is DIC Ag = — x LEH + Hv]. amnplitude at the observation point P’ dA‘ (x’, y’) —s ef Put ar dy, differential element of the aperture at the point M. 5.3 Calculation of the Diffraction Pattern 187 To obtain the amplitude at the point P’ we must integrate the contribution = from all source points, If the amplitude and phase of the incident wave are = constant over the aperture, we directly integrate Eq, (5,15), For the case of = a square aperture with dimensions 2) and 2ny the integral is elementary, Al(x', y= / : | : ef tburmlaedy —np 4 —%9 } ES sin (22 cou) sin (2 nov) pS =4ton0 | —~—— | | —+——+ (5.16) Be tou 2 ou : za The intensity is given by the square of the amplitude Be 2 2 Be ; 22 as (cox) =D (nov) ae Nx, 9 = 1693. |——— ee (5,17) ee z 50u [Nov d a and is proportional to the square of the illuminated (aperture) area. This is : typical of diffraction phenomena, as compared to incoherent illumination, : which 1s simply proportional to the area. m:. In the case of a long vertical slit, no >> {o, the intensity vanishes very ss rapidly for vu # 0. (Note that nv becomes large and the exponential in e. Bq. (5,15) oscillates rapidly, its average value tending to zero.) Thus we ee observe a horizontal diffraction pattern confined to the x’ axis, as shown = in Fig. 5.4. Exactly on the x’ axis, v = 0 and Eq. (5-17) reduces to ee 2 oe sin { [-Sou sin (2 sin @ eI (x', y' =0) = 16g5n5 sn (ios) =I an (3 vino) | dy Be : 2 rou a sin 6 Z (5.18) Zz In the last step we made use of the relations u = x’/R = sin@ (valid for Bey! = 0), where @ is the angle from the z axis and tg = d/2; we also set 7 fe : eK = Igto os. the amd at 0 = 0. Note that the above result a . oe ee ore = en Ao ee oo a ~ “ ae = “See ——* = - —— 188 8=65 «(Optics Experimenis modulated in magnitude and/or phase. In this case we express the amplitude on the aperture by re Fen), contained in the pulse 4.00 m A(w) = | F(the' dt. (5 20) z —oO os eee Similarly, in Eq. (5.19) F(¢, 7) describes the spatial dependence in the: aperture plane and A’{x’, y’) can be thought of as describing the spectrum: op of “spatial frequencies” 27u/i and 2rv/X. We will make use of these one concepts in Section 5.4. ee 5.4. DIFFRACTION FROM A CIRCULAR APERTURE Instead of a slit we shall now use a circular aperture. Some skill is required” a in manufacturing such small apertures, but they can also be purchased commercially, The smaller the diameter, the larger the diffraction angles _ rare present experiment the aperture » diameter was d = 150 um, which is 2: | good compromise for the HeNe wavelength. a To obtain the diffraction pattern, we integrate Eq. (5.15) over the circular | cele ae okt, Ss a viratat reba DPC wit SNM ia o 1 nares ana wae ee i SS we . ae 5.4 Diffraction from a Circular Aperture 188 FIGURE 5.7 Coordinate systems in polar coordinates for calculating diffraction. Es shown in Fig. 5.7. In the source plane we use the coordinates a, @ so that f=acos¢ n=asndg, (5,21) "whereas in the observation plane we use p, ¢’ so that = where a = p/R ts the sine of the radial diffraction angle. Expressed == in terms of these new coordinates, the argument of the exponential in ss Eq. (5.15) becomes 7) i = [cCu+nyv] =1 = aa cos(@ — ¢’). (5.23) ? __ a i i aw cos 71) A'(a) = ey ada d@. (5.24) 0 0 “Here ag is the radius of the circular aperture, and we have assumed uniform illumination. 196 5 Optics Experiments The intepral in Eq. (5.24) cannot be performed in terms of gonometric:: functions but is well known. One finds that ae A’(a) = R 5 ; i ee) a where J; is the Bessel function of order 1. The intensity is given by we square of the amplitude : 2 2 244 (2400) I(@) = (a) aga nated area. Since a = sin @ is the diffraction angle, Eq. (5.24) is similar ae Eq. (5.5) with the replacement of the sine by the J, Bessel function, ©2232 Equation (5.26) is plotted as a function of its argument, x = (23 (aoe in Fig. 5.8. The zeros occur at the following values of x, : ee a 2 xy = 3,83, 2x. = 7.02, = 10.17, etc, — cee 223 whereas the maxima fall in between. The pattern is that of an intense central disk surrounded by altemating dark and bright rings, as shown in Fig. 5. ee The first dark ring occurs at an angle : 3. | 4; ~ sin é, = Zz —— = 1.227 — (5.27 oe ee where D is the diameter of the aperture. If the lens uscd has a focal lengthy 2 f, the radius of the first dark ring on the screen occurs at pee ae ay ae eae ee a ie a ont ee Coe ia ae eae central disk contains 76% of the total intensity. . The experimental setup is the same as shown i ip re 3. 3, except tt ae! © Cr far eal 5.4 Diffraction froma Circular Aperture 191 ee: a o 1 2 3 4 5 6 7 B6 3 a a : FIGURE 5.8 The intensity distribution for Fraunhofer diffraction from a circular aperture 2 _ aga function of x = (2m /A)ag sin 4: @ is the diffraction angle and ag the aperture radius. PA ee ge Folntes ween ares F, r measured at the angles ay = (5.25+ 1) x 107 radians ay = (10.542) x 107° radians a ma Tata eee a3 = (14.542) x 107? radians. at a af as eta ta te 1h, . tt, rn B | Using the values for the zeros of J) as given previously, we obtain the ee ee gorresponding values for A/D 192 «35 Optics Experiments a te FIGURE 5.9 Observed diffraction patie of a HeNe beam from a pinhole of diameter 150 pm. | 5.5, THE DIFFRACTION GRATING We have already made use of the diffraction grating and discussed the: : physical principles in Chapter 1. Here we will carry out a more detailed: analysis and demonstrate a compact spectrometer using a digital readout. 222 If instead of a single slit, two shits are illuminated by a plane wavefront, a series of interference tamiges-pardial'a.beclits will anpear on a far 2 screen. This is the classical experument of Thomas Young (1800) shown m Fig. 5,10a. If the spacing between the slits is d, the intensity distribution on the screen is! I(@) = 4ig cos” (= sin 8) . (5.29} s The angle @ is measured, as usual, with respect to the normal to the plane: containing the slits. If one of the slits is blocked, the fringes disappear and S the transmitted intensity is Jp. H3e We take the wavefront parallel to the plane in which lie the slits. §.5 The Diffraction Grating 193 i : | a “FE a EEN hahansateaonrehaiehmmien ete maee anime {a) {b} =< FIGURE 5.10 (a) Young’s two-slit experiment. (b) Multiple-stit interference, the diffrac- = tion grating. - In Eq. (5.29) we have not included the effects of diffraction due to the =. width of the slits. Let the slit width be a. Then Eq. (5.29) is modulated by the =< diffraction pattern of Eq. (5.5), and we obtain for the intensity distribution 2 sin (22 sin 8 1(8) = 41g cos? (= sin 6) ee | . (5.30) i : If instead of two shits, several equidistantly spaccd slits are illuminated =: by the wavefront, the interference maxima become much sharper, and the =. interference pattern is given by 1(@) = Ip (3.31) 2 sin (N44 sin 8) sin (32 sin@) | fe Here @ is the spacing between the slits, and N the total number of slits; we = have not included the effects of diffraction because in practical applications the slits are so nartow that the modulation is not important. Note that = Eq. (5.31) reduces to Bq. (5.29) for N = 2, as it must. ce; Whats of particular interest is that the pattern contains principal maxima =when the denominator of Eq. (5.31) becomes zero, namely when sind = tnA/d, n=0,1,2,.... (5.32) SSE Sncoannannuas ai Ce ead al Da ath a aL Siesta 19 865: «Optics Experiments The intensity at the principal maxima can be found as follows. Near a princi- pal maximum (7d@/i) sin @ = nw + and therefore sin((#d/%) sin 6] ~ € so thai Eq. (5.31) can be wnitten as sinf N (nx tony. Ion? pace = pn? (22s) | € Ne x RSC ARTCC Ed I (9) max = Ip (5 33 Be : et ie ce at ; oe oH e where x = Ne = Nor(d/d)AQ@, and A@ is the departure of @ from the’ condition of Eq. (5.32). Since the function (sin x/x}* > 1 as x — 0, thes intensity at the principal maxima [A@ = 0] is ee Imax = N7 Ip. al é This pattern is shown in Fig. 5.11, SS 5 The width of the pnncipar maxi s'gertuthe.Grst minimum of the 244 function (sin x /x), which occurs when x = -bz, namely _ Ag =4—~., 6.35) Note that (Ad) is the total extent of the region covered by the slits. Thus = the principal maxima are as narrow as if the wavefront diffracted froma slit of width Vd@. By combining Eq. (5.35) with Eq. (5.32) wecanexpress = 70 Intensity (a) b [a] ar -1.5 al -0.5 0 0.6 1 1.6 2 {afAjsing FIGURE 5.11 Different orders of monochromatic light scattered from a grating. Note that the puincipal maxima are Very narrow peaks, whereas the stcondary maxima ale suppressed. Plotted for N = 5, 5.6 Tha Diffraction Grating 1% the resolution of the system of N slits by AA I 9 ~ ] 5 %6 i) Nn ~ Nn 0.36) A diffraction grating is equivalent to such a system of many slits and can be used either in transmission or in reflection. The angle of incidence 6; can be different from the normal to the grating, in which case Eq. (5.32) must be modified to read ° A sin & — sin @, = kn a n=0O,1,2,.... (5.37) The diffraction angle is 6, and is taken positive if it is oppasite from 6; with respect to the normal, These definitions are shown in Fig. 5.12. Fora reflection prating n = 0 corresponds to specularreflection (sin 4, = sin 4). Reflection gratings are often manufactured so as to enhance reflection at particular angles. Recall that Eq. (5.37) was already used in Chapter 1 (see Eg. (1.16)). The arrangement used in the laboratory is shown in Fig. 5.13. The light source is focused on the slit and the emerging beam is made parallel by lens L1, which has focal length f; = 20 cm. The parallel beam is incident on the 4 x 4 cm” grating, which has 1200 lines/mm. The angle of incidence was chosen to be & = 55.7°. The beam diffracted in first order was focused with lens L2, identical to Ll, onto the “reticon” where it formed an image of the slit, The reticon is a linear array of pixels, which can be read out on an oscilloscope. In the present case the array contained 128 pixels; the clock speed was 80 kHz so that a pixel is read oul every At = 12.5 ws. The pixel I 6<0 | 6>0 Grating ee FIGURE 5.12 The convention used for labeling the incidence and reflection angles for a ete mt Ry ah te RyRy Seba rari ieee che 4 oe Se ee 2: reflection grating. iat Chea nn in a speeetcy RHODESIA ut Scena nS as i eee. hee 196 5 Optics Expariments Light source pnd order | Display/Scope Raticon oO” order Contre! ckt FIGURE 5.13 Layout of a simple grating spectrometer read out by a reticon (a one: dimensional solid-state detector array}. os size was Axo = 100 wm fora total array length of 1.28 cm. Thus we have e the conversion factor ss Ax = 8 um/s, (5.38) and since Ax = fA6, f = 20cm 5 Ad = 0.04 mrad /us. (5.39) The spectrum of a Hg arc lamp is shown in Fig. 5.144, The horizontat scale (sweep speed) corresponds to 200 .s/cm. The spectrum was observed in first order, and from Eq. (5.37) with @ = 55.7°,d = 1073/1210?) m we find that for the green line of Hg (Ay = 346.) nm) sin é, = sin& — - = 0.170, namely @, = 9.8° in the quadrant opposite to the incident beam. The second: order appears in the same quadrant as the incident beam at @, = 29° a8. shown in Fig. 5.13. The green line corresponds to the peak on the right-hand side of the eraph ree (Fig. 5.144}, whereas the doublet on the left corresponds to the yellow lines =: (Ay = 577.7 nm and Az = 579.1 nm). Knowledge of these wavelengths, : allows us to make a more precise calibration of the spectrometer, including. ee at ah 4*a*n*nte) c%ete tats talela™ ae atta! eee a Pe 1? yt NLC MMP PC CRI he CT eee meson nbesetaladatabatalatatatateca®atcseeytetasetetyteterary ‘ Pe ne le a a eA Rains As r se ed a athe be ae 5.5 The Diffraction Grating 197 {a} fremom iat 720s. SOG ee TE... ae ee ee | a etre ataeeea a. +. /-_ a I eee ee pwn == af , etre eieantians p Ld i ot or e8 We ae “hes Frelervesdesastosanl ta-a7e due” B21 sams ~ AS Bloog ns d= Boole 7 : | 7 (b) - - . - - - >- - Ct ed ie hee ee ere CMe Nettie —— — a i 3 3 =f arret artmeg ere —re > i- ciysenipent es tad wierd pod t teehee =763.0)5 12=239.0ps aera” tie 17.86 KHz FIGURE 5.14 The observed spectnim (a) of the green line and yellow doublet of the Hg spectrum obtained with the spectrometer of Fig, 5.13b, (b) The yellow doublet on an expanded scale. musalignment and other instrumental effects. Differentiating Eq. (5.37) with 6 fixed, we obtain nAd/d = —cos 6, A@,. (5.40) In our case n = 1, cos6, = 0.99 and AA, from the first yellow line A> to the green line A,, is AA = 33 nm, or A@ = 40.0 mrad. The time interval between these lines as measured off Fig. 5.j4a is At = 1030 ws, and thus the calibration A@ = 39 x 1077 mrad/ps (5.41) in close agreement with the direct calculation. To measure the fine structure of the yellow doublet the sweep speed is increased so that the scale factor is 50 ys/em as shown in Fig. 5.14b. 19 «=6©—5 «Optics Experiments One can now recognize the response of individual pixels. The separation of the two lines is 56 j1s; using Eq. (5.41) we find A@ = 2.18 mrad and from Eg. (5.40) : AA = 1.8 mn. Our result is only in modest agreement with the accepted value of AA == 1.4 nm. This is not surprising because one pixel, the ultimate resolution of our detector in this configuration, contributes an uncertainty of 6A =" 0.42 nm. Thus one must be cautious when using digital techniques, which=:2 often do not have the advantages of the high resolution of photographic? film or of visual observation. : 5.6. FOURIER OPTICS In Eq. (5.19), we showed that the amplitude of the electric field in the focal plane of 2 lens is the Fourier transform of the near-field amplitude incident a lr by E. Abbe in Jena, Germany, but found much wider use as lasers became . : a available. Ee A transmission grating is arepetition of regions in space that alternatively: : transmit/absorb the incident wavefront; we can represent the transmission. * of the grating by the “square-wave” function shown in Fig. 5.15a. We are =: immediately reminded of the analogous square-wave function of time that. 22 has period 7’, and thus frequency v = 1/7. Therefore we can assign tothe “2 ory a ae i eT ee | ! | t kod [yg= to] r hd} [v= 1/9} x x ss (a) (b) x FIGURES.15 (a) Representation of the transmission of a grating: the spatial spectrum con- tains the fundamental frequency v, = 1/ed and its higher harmonic¢s. (b) If the transmission Se is sinusoidal, only the frequency v, = 1/d is present in the scattered wave, : 5.6 Fourier Optics 199 grating a spatial period d and a spanal frequency 1/d. Spatial frequency is measured in cycles per unit length and has dimensions of inverse length. For instance, the grating used in the experiment described in the previous section has a spatial frequency of 1200 lines/mm. From circuit theory we know that a square pulse in time contains the fundamental) frequency as well as higher harmonics, Similarly the square grating contains not only the fundamental spatial frequency 1/d, but also its harmonics n/d. This is seen when light incident on the grating is diffracted at the angles @, with A iné, =" —. sin 6, na If the grating profile was sinusoidal, as in Fig. 5,1Sb diffraction would occur only forn = Qandn = 1}. We can place a lens after the grating to relocate the far field into the (back) focal plane of the lens as shown in Fig. 5.16. We will then see the diffraction maxima, namely the Fourier transform of the grating: we refer to this plane as the transform plane. If the distance 5; from the grating to the fens exceeds the focal length /, an image of the grating will be formed in the image plane located at 52, where ae the Jens), and the image plane. 2000S 5s«Optics Experiments Laser Se ee Expand | Se ] | | 1 1 | Masks h ccD Mes image plane Transform plane FIGURE 5.17 Experimental layout for demonstrating Fourier optics. This image is the Fourier transform of the amplitude in the transform plang oe Therefore, by altering the pattern in the transform plane, we can modify. z the image being formed. There are several applications of this principle; as for instance in smoothing out images that contain noise or in pattern recognition. | A simple demonstration of Fourier optics can be carried out in the labor ratory with the setup shown in Fig. 5.17. The laser beam is expanded and. allowed to illuminate a mesh with 270 lines/in. and transmission factor ~50%,. Lens L3 is used to image the grating onto a CCD camera. Various: masks are then inserted in the focal plane of the lens, the transform plane, to modify the image. The results are shown in Fig. 5.18. In Fig. 5.18a, no obstacle is in the transform plane, and the pattern represents the image of the mesh. Next, a vertical slit 1.5 mm wide is placed in the transform plane, and the pat- tern in the image plane contains horizontal stripes as shown in Fig. 5.18b.248 The effect of the mask is to allow passage only of components of the 224 wavefront that were dispersed vertically in the transform plane. These com- + ponents carry the information about the horizontal structure of the object SS (the mesh) and thus show horizontal lines in the image plane. Figure 5. 18¢ 22 was obtaimed with a horizontal slit as the mask in the transform plane. Se wee a 2 ae ne a a] rar a | ore] ASSEN ST Mn oe te RNAS os hie AA atitvNate te AAAI ALLA thet eet be eee ha we ee ee Pb we SEES 4.4, CP PLP ae) anata R DS SNS CATS SNEN ESOS ‘ . te oe SATAN ‘ 0,7, 50h, FPO bias wes} phi bb) anes "ers SPL dere 'ats . RRESNN SERENE EROS tated eRe noes i he hb ont pe haa ke bl inthnxks nhs Sieh . ' . - ’ P + ot) Ut) A SASSANAS NNN SE 5.7 The Faraday Effect 201 FIGURE 5.18 Results from placing masks in the transform plane: (a) Image of a square mesh in the absence of a mask, (b) placing a vertical slit in the transform plane, (c) plac- ing a horizontal slit in the transform planc, and (@) placing a pinhole in the transform plane. Finally Fig. 5.18d shows the result of placing a 1-mm_-diameter pinhole in the transform plane. Now all high spatial frequencies are filtered, and the pattern in the image plane is significantly smoothed out. Spatial filtering by using a pinhole is often used to “clean” laser beams that have acquired structure due to imperfect optics, dust on components, and other aberrations. This is analogous to Using a capacitor to filter out high-frequency noise in an electric circuit. 5.7. THE FARADAY EFFECT 5.7.1. Discussion As already mentioned, the Faraday effect refers to the rotation of the plane %: of polarization when light propagates through certain media subject to an axial magnetic field. It was discovered in 1845 by Faraday long before 202 § Optics Experiments the nature of light or mattcr was understood. We now know that the elec: tric field of light is transversely polarized with respect to its direction of propagation, z, and we can express it, in exponential notation, as B(z, 1) = Re{ Ege fe}, (5.42) Here Re means to take the real part of the expression; for simplicity of notation we will omit this designation in what follows but it is always: implied. As usual @ = 27 and k = 27/2. e is the polarization vector, which can be expressed in terms of fvo unit vectors (since e is restricted: “22 to the x, y plane). We can choose linearly polarized unit vectors 7 e| = uy, a Uy (5.43) ee qi or circularly polarized unit vectors Qp = Uy + iuy, e, = Ux — iUy. (5.44) os If we now examine the electric field at a fixed position z, in the case of circular polarization we will have the two components ER = Eg|cos wtu, + sinewruy| Ey, = Ep[cos wtu, — sin wfuy]. (5.45) = These were obtained by introducing Eqs. (5.44) into Ey. (5.42), The fields rotate in the transverse plane, in the first case according to the right-hand — rule (with the thumb along the direction of propagation), in the second © case according to the left hand. This is shown in Fig. 5.19 where we use a | atta Saas nr tay ey oor Pa iow rl "an a p aa ope aa a ae ae on ? a . a caer ‘ete ‘4 aa "a are ae aaa es oad cores ae oa va neat we an a aa "a < FIGURE 5.19 The right-handed coordinate system used to define right- and left-circular polarization. sf fete ES oe) SN SEN aS . s “ ‘ - . Sree Bn un oes vey ‘an St) hh) '" . AR mie NDA ay Wh re TUTeTy Tay ty Etat ta Delarare ree trees eerie yt atat Seas x le ») hs araeatatalataa ete niet ar tet Kaa hes at Mb ie iB el ha A - x he eb arate) ‘ . - d 7S Ve dade wee es « pt EL Sana * *** stats Tete, “ 7! rehire tate tate te ty Me eeed rar ab Ya by hs Kp) 5.7 The Faraday Effect 203 3 Be right-handed coordinate system. Note that we can write Eqs. (5.45) as Fp = E, +1By By = Ex ~ ik, (5.46) 2 and by solving i 1 The Faraday effect arises because in certain materials the application of a magnetic field results in different refractive indices for the nght and left circularly polarized light propagating along the direction of the field. Materials that have a different refractive index for two given polarization orientations are called birefringent. The birefringence is natural in certain crystals or can be induced by the application of an electric field (Pockels = effect). The physical interpretation of the Faraday effect is related to the shift of * the atomic energy levels when an external magnetic field is applied. This is the Zeeman effect, which is discussed in some detail in the following chapter. When the light propagates along the axis of the field the nght polarized light can excite only a particular set of sublevels (Am = +1, where m is the magnetic quantum number) and conversely for the left polarized light (Am = —1). These levels have different excitation energy and this results in different refractive indices, np and my. For more details the reader should consult the references cited at the end of the chapter. We know that the velocity of propagation of the wave, the phase velocity, is given by c’ = c/n; thus the phase advance in a length L of material is 2 2 5 i gs ee By (5.48) xX c’ c where the frequency v of the light is fixed and 7 is the refractive index of the material. Thus the nght and left polarized light will acquire different phases. If the incident light was linearly polarized when entering the material, say along x, Ep and £; would have the same phase (see Eq, (5.47)), However, upon exiting the material their relative phase would be shifted and the light, while still linearly polarized, would also contain a small £ component. Namely it will have rotated by an angle 1 b= 5(6t — 61) = = Ling — ny). (5.49) rr Cr a es a 2044 85 Optics Experiments FIGURE 5.20 The rotation of the plane of linear polarization after propagation in a region where the nght- and left-handed circular components have different phase velocity. Because Ep and E, rotate by different amounts, the plane of linear polarization for E = Ep + Ey Totates away from the x axis by an angle @ = (Gp — 6,,)/2. —— TABLE 5.1 Verdet Constant for Distilled Water A (nm) Cy (rad/T-m) 590 3.8! (Na D-linés) 600 3.66 800 2.04 1000 1.28 1250 0.84 This is shown in Fig. 5.20. The change in the refractive index is proportional © to the external magnetic field, B, so that we can write = CyBL, (5.50) where Cy is called the Verder constant. We expect Cy to be a function of 2 zs wavelength, as well as of the medium. Values for distilled water at various | 25344 wavelengths are listed in Table 5.1. : 5.7.2. Procedure and Analysis It is difficult to generate axial magnetic fields in the kilogauss range. 2 Instead, a small but oscillating magnetic field will be used. The size of -:2: ray oo 4, q 2 a a ie ae fe oe wad ‘a caine 2Pata from E. U. Condon and H. Odishaw (Eds.}, Handbook of Paysies, second ed., oe McGraw-Hill, New York, 1967. oi ouese ‘ ws pss % k Ata che hee ae a at white! 1 4 at ara hh eer CORT REDE NNUENTAD Liste ht Shite . “4 hate a hs ta tae al ay oe Her SHS Se NSE NSSN SHE rete 5.7 The Faraday Effect 205 Linearly polarized Analyzer HeNe laser solenoid Photodiode [| _ Sample Output vallage an Solencid coaxial cable to driving clrcuit -DMNM * - Scope - Lock-in =. PIGURE 5.21 Experimental setup used for the Faraday effect. The photodiode output = goes to the DMM for the polarization calibration, and to the oscilloscope or lock-in to measure the Verdet constant. =: the effect will be smal, but the oscillations make it possible to pick it out =. of the noise, by using lock-in detection. The experimental setup is shown im Fig. 5.21, The source of polarized light is a HeNe laser. The magnetic field is supplied by a 1026-turn solenoid =: driven by the amplified signal of a waveform generator, in series with a monitor resistor. After passing through the sample and polarization ana- = lyzing filter, the light is detected in a photodiode. The signal is measured =" by the output voltage of the photodiode, and is given by 1(@) = Igcos* @, (5,51) where @ is the angle of the linear polartzation with respect to the analyzer axis. We are interested ind¢ /d? and in this case the sensitivity is maximized by “biasing” the polarizer at dp = 45°, where @ = do + (ft). Note that di(g) _d¢ al do do =— — = —— Io sin 26 ~ —— f 2 5.52 “a ae dé de OSAP = Gy fosin2eo. = O52) We can calibrate the polarization analyzer by recording the photodiode voltage as a function of the analyzer angle. The result is shown in Fig. 5.22 and exhibits the cos? @ dependence of Eq. (5.51). The maximum sensitivity dVp/d@ ts found near @ = 180° and as predicted by Eq. (5.52) equals Vax namely dVp/d@ * 0.4 Virad. The magnetic field is provided by the 1026-turn solenoid coil around the sample, driven by a sinusoidally varying current. The current is pro- = vided by an HP33L1A waveform generator (sine wave, 600 Q output) : amplified by a Bogen MU10 monaural audio amplifier. The driver setup 206 43865 Optics Experiments Photodioda yofttage (mV) a 150 290 250 300 Analyzer angle (degrees} FIGURE 5.22 Sample polarization calibration data. The plot shows the full range of <2 anples. wd ory Re oy hia a se atest a a a ee ae eat anal wed at, al ae ee ng trelelaylatitininiys ltt ea te el he is shown in Fig. 5.23. The wave generator provides the Input to the audio a amplifier, and the output loops through the solenoid coil with a high-power resistor Regi in series. The current and thus the mapnetic field are deter- mined by measuring the voltage drop across this resistor. Do not ground either side of the amplifier output signai. Using clip ieads on a coaxial cable measure the voltage Voy across Roi on an oscilloscope. The shape should be a good sine wave with no DC offset and amplitude on the order of 10 V peak to peak. This is achieved by adjusting the amplitude of the HP331L1A and the amplification (i.c., “volume”} of the audio amplifier appropriately. li may be necessary to adjust the distortion on the amplifier so that the shape is alight. The photodiode output is now connected to the other channel of the oscilloscope. The scope trigger is set to fire on coil voltage, and both chan- nels are viewed simultaneously. If the channel on which Vp is measured is DC-coupled, one sees a large DC level, corresponding to the mean light intensity on the photodiode. (This DC levei should agree with what was measured with the DMM.) The Faraday effect, on the other hand, shows up as a small oscillation on top of this DC ievel, in time with the V,g. One is just able to sce this small oscillation if the channel sensitivity is set to . Soe ee Be el Pe ee D LP Berateteta! ars steel wale ew STAN MIE rented SRA LE earseteta te To oscilloscope 5.7 The Faraday Effect 207 {should be 10V sine wave) Solencid HT Bogen MU10 aucio amplifier Sins wave 600 Of out HPS311A WF Gan (connections at rear panel) Tee off to lock-in relarence a FIGURE 5.23 The driver circuil used to generate the oscillating magnetic field for °° measurement of the Faraday effect. $ its lowest scale and AC-coupled to the input so that the large DC level is = gemoved. Confirm that the amplitude of these small oscillations move up :* or down with the amplitude of Voy, which is best adjusted by changing =::. the amplifier gain. Confirm also that the oscillations disappear if the pho- = todiode is blocked from the laser. In fact, the amplitude of the oscillations should change (and the phase reverse) as the analyzer is rotated. We can now check that we are getting about the nght Verdet constant, = although it is hard to do a careful job with the small signal on the oscil- loscope. From Eq. (5.50), we know that the small changes in polarization angle A@ are related to the changes in magnetic field AB through Ad = Cv AB: L sample, (5.53) and from the calibration, we can convert A@ to a change in photodiode voltage A Vp through Ad ee = AVp. (5.54) The magnetic field in a solenoid of length Z cojenoig and N = 1026 turns is given by B= pwolcouN/L sotenoid (5.55) 208 5 Optics Expariments when a current foo; passes through the coil. By combining Eqs. (5.532288 (5.55}, one obtains an expression for the Verdet constant Cy in terms of 22238 Vp, Veo, and other quantitics that you know or can measure separately, 238 Consistent definitions should be used for V,,;, and for Vp. That is, if Veoit Br is the amplitude of the sinc wave, we make sure to do the same for Vp. 2 5.7.3. Results Using the Lock-In ee | See Pate) roe ae an explanation of lock-in detection. ee The lock-in is a PARC Model 120 with a fixed reference frequency te oh ~\00 Hz. It is best used by defining the reference wave extemally, but: it us that we are using a reference signal with orecisely the same Fomenayag the Faraday effect signal in Vp. The photodiode output should be connected, ae to the Jock-in input. 3 One stil] needs to tune the phase of the lock-in amplifier so as to have maximum sensitivity to the oscillating Vp signal. There are a few ways io do this, but the most instructive is to use the oscilloscope. om {. With the oscilloscope still triggered on the V,.4) signal, use the other channel to view the “monitor out” port of the lock-in, with the switch set to “OUT x 1,” which is the basic output signal of the lock-in. If the time: constant is sel to a yalue much smaller than (100 Hz)~! (1 ms will do);' then you should just get the sine wave folded with the reference signal f oscillating between +1. That is, it should look pretty much like Fig. 3.372224 or Fig. 3.38, or something in between, depending on the phase setting. .© te 2. Adjust the phase knob so that itlooks like Fig. 3.37, thatis, oe relative phase quadrant knob so that the phase is 90° lesser or greater, the trace should look like Fig. 3.38. On the other hand, it should change sign if you flip by 180°. 3. With the phase adjusted so the output looks like Fig. 3,37, aa “y a sng a eat ate a a MAEM MAMMA ARMA AHR AIC ee en eer ee ao ke SO A he er ae ha . Srhroch hee CAL Panache n Ta rte ate mate nureea tt en a i et eM eS a SURAT CRC MCRL RTC RCC TOL a ak ae ae 8 . . ee a ee ee ek ay ae ae ee . 5.7 The Faraday Effect 209 ee DMM, or use the meter on the lock-in. It is probably a good idea to block “> the Itght to the photodiode, and adjust the zero-trim so that the lock-in a output is 0. Vary Veoy by adjusting the audio amplifier gain. (You should not touch =~ the wavcform generator settings anymore, since it is now serving a dual =: role as both amplifier input and lock-in reference.) Make a table of Vp as = qeasured with the lock-in and V,o. Realize that the value of Vp provided ==: by the lock-in is the RMS value, ie., 1/2 times the amplitude. Plot Vp = versus V.gi and make sure you get a straight line through 0, Either fit to <. find the slope or average your values of Vp/ V.o4 to determine the Verdet =. constant with an uncertainty estimate. Results obtained by a student are shown in Fig. 5.24 for a water sample. -- The parameters used to obtain these data were Reoit = 5.3 2 N = 1026 L solencid = 0.265 m L sample = 0.265 m. 0.8 0.7 o oh 1 Lack-in output voltage (mv} a bh 0 2 4 6 A 10 #2 Voltage across resistor (V) 7 FIGURE 5.24 Results on the Faraday rotation angle as a function of magnetic field, “. obtained by a student. 21a. § «Optics Experiments . Ae We first calculate the magnetic field as a function of Vooiy Vooil N B= pio Kooi EL solenoid = Voy < (9.18 x 1074) T. @ = Vp/(0.098) rads. : The measured values (sec Fig. 5.24) are 4 Vp/ Veoit = (6.7 + 0,52) x 107°, . Thus we find for the Verdet constant 8 ~ BLsampie Lsample \ Veo) 9.8 x 1072)(9.18 x 10-4) = 2.80 + 0,2 rad/T-m From Table 5.1, extrapolating to 4 = 633 nm, we would expect Cy ~ e 3.2 rad/T-m. The difference could be accounted for in part by the short length of the solenoid, which results in a weaker field than what we © calculate. 2S coat etetetr tn ateleg aint! ae Ce eel a AGUS toe NC 5.8. BERRY’S PHASE We will demonstrate this effect by the rotation of the polarization vector of a beam of light, as in the Faraday effect, but in the present case the ight = propagates in a vacuum. The reason for the rotation of the polarization : is that the propagation vector of the light, the & vector, performs a closed = circuit around its direction of propagation. This is shown in Fig. 5.25 where : light propagates from point A to point 4. In part (a) of the figure the & vector - describes a helix on its way, namely a closed loop in the transverse plane; — therefore the polarization rotatcs. In examples (b) and (c) the initial and final values of k are the same as in example (a) but there is no looping | around the direction of propagation; therefore the polarization does not rotate. We speak of a “topological” change in phase because the effect - depends on the path followed while the initial and final points (in phase space) are the same. This effect was first predicted by M. V. Berry in his 1984 paper (see Section 5.9). He analyzed the behavior of a quantum mechanical wave LAP 5, *, *, " WWE Oe SSRN ERRSRNRTR wepeeeetes LMT NNT 5.8 Berry's Phase 211 Kiraias {b) eee pee ee A B a Kin {e) A — A Kenai = FIGURE 5.25 Topology of the optical fiber between A and B with kguat = Kiniiul! = (a) helical winding, {b) direct (straight line path), and (c) citcular path on a flat surtace. = function when a parameter on which the wave function depends is slowly =: yaried over a closed circuit. He showed that the wave function can acquire ": an extra phase factor even though the final state is identical to the initial “= state. It was soon realized that the same results should also hold for the ~ electric field (the wave function) of a beam of light. Thus, the extra phase : appears in classical as well as quantum-mechanical systems. In fact the = precession of the Foucault pendulum or the Bohm-Aharonov effect can be =: interpreted as manifestations of Berry's phase. When the k vector of light is transported through a closed circuit sub- é tending a solid angle AQ at the origin, the right polanzed light acquires a =: phase factor Epy =e!“ Epy, (5.56) i whereas the left polarized light acquires a phase factor E_r = e tAQg, | (5.57) =. Where { and f refer to the initial and final state. This is a consequence * Of Maxwell's equations, which require that the k vector and the two :’ polarization vectors always form an orthogonal triad. To become convinced about this statement we show in Fig. 5.26 the unit sphere on which we can indicate the directions of k, e;, and e2. Suppose * we start from point A on the sphere and parallel transport the triad to point *. B along the equator. We then parallel transport it to point C along a great ;. Circle and return to point A by the corresponding great circle. At each point 212 «#35 Optics Experiments equator and two preat circles. Note that k retums to its initial position but e and ep are 24 rotated by 90°. The solid angle enclosed by the path is 90°. we have shown the orientation of the triad, and it is evident that upon return: “22 to A, the k vector has not changed but the e; and e polarization vectors: :::2 have been rotated by 90°. The solid angle subtended by the path that we 2 followed is 1/8 of 4x or #/2 = 90° equal to the observed rotation of e;, €2. Let us now assume that the incident light is linearly polarized along the :2 x axis. From Eg. (5.47) we can write Bs vena Ein = Ex = ¥(Ep + Ex). After completing the circuit, we will have according to Eqs. (5.56) and (5.57) Ef= 5 (Eel A® + Eye 4%), ners However, this corresponds to linearly polarized light at an angle BE @ =1 [Ag —(-AQ)] = AQ (5.58) = with respect to the x axis. The argument is exactly the same as that used in & To carry out the experiment we must find a way to adiabatically change == the onentation of the & vector. This can be done most conveniently by injecting the light into an optical fiber and then laying out the fiber on the desired path. One must use a single-mode fiber in order to preserve the polarization of the light and the path must be continuous (i.e., po kinks in PONIES uO ARERR ES SATIN ST NESS Pe a a A oa a ate Cn pk 7 ee a ee enna ate tana 5.8 Berry's Phase 213 | FIGURE 5.27 Layout of the fiber winding on a cylinder. Here the fiber length is s and the <> radius of the cylinder r. x :: the fiber). For instance we can wind the fiber on a cardboard tube as shown ” in Fig. 5.27a. Lf the radius of the tube is r and the length for one revolution (the pitch) is £, the winding angle 4 is given by cosé = £/s sos 4f@24 (27r)-. (5.59) ee The solid angle described by the fiber is then AQ = 2x(1 —cos@) = 27 (1 — £/s). (5.60) The experimental setup is relatively simple. A HeNe laser beam ts polar- =" ized and injected through a fiber coupler into the (single-mode) fiber. At the 500 450 | we _ o es g0 400 a4 + ry q a ¢ + 350 A a LK © a a Q > 300 | _ = 4 o o 256) 4 o oO r Co oa . Ss 200 ¢, a A : 150+ g ¢ ° o é 100 o* 6 Oo e ® + 50 Oo | | _ oO + e oo on Ke a 4 1 0 60 100 «1600S 200s 250s ds“ (aési8D.—“‘é‘ésK Rotation (Degrees) a FIGURE 5.28 Results from a measurement of Berry’s phase. The transmitted intensity is shown as a function of the angle of the analyzing polarizer. Open squares are for the flat ao topology, filled squares for helical winding, The polarization has rotated by 245° between the two measurements. 214 «=©§ Optics Experiments end of the fiber the light exits through another fiber coupler and is ae : by a rotatable polarizer and a photodiode. We use two configurations, one 3 ee en ie lel obtained with the flat fiber, the solid squares with the helical winding. Wee see that the polarization has rotated by @ = 245° (or it could be 115° in the g opposite direction!). Ss In this case the radius of the cylinder was r = 14 cm and the pit £ = 28 cm, for one complete tum. Thus s = 92 cm and a AQ = 2n(l — &/s8) = 4.37 sr. More details on the first demonstration of Berry’s phase with an tea fiber are given by Tomiia and Chiao (1986). SE 5.9. REFERENCES M. ¥. Besry, Proc. R. Soc. London Ser, A 392, 45 (1984). 2 M, ¥. Berry, Phys. Today 36 (Dec. 1990). Ones A. Tomita and R. ¥, Chiao, Phys. Rev. Lett. 57, 927 (1986). EE ea er aT i ae Be | CHAPTER 6 High-Resolution Spectroscopy See earmtatare SSS cot tas = 6.1. INTRODUCTION In 1896, P. Zeeman observed that when a sodium source was placed in =<. a strong magnetic field, the yellow D lines were split into several com- “2: ponents. Faraday had performed the same experiment some thirty years =< earlier but had failed to observe an effect because of the low resolution of = his spectrograph. We also know from Chapter | that even in the absence of =: a magnetic field the atomic spectral lines have a fine structure that was eas- zs ily observed with the smal] grating spectrometer, with a high-resolution f=. instrument, however, it becomes possible to observe that each of these _ fine structure lines may again be resolved into closely spaced componients, which form the so-called hyperfine structure (hfs) of atomic lines.! pier nana ROARS 'To set the reader af ease, no further splitting beyond the hyperfine structure has been pe observed, nor can it be expected for free atoms; in the fryperfine structure we include both wi: the splitting due to nuclear spin aud that due to the isotope shift. ee ee a Pa Fen tet ty hae bale rie ta ha or rh Pha ha Baha id re eh a a ee a) oa . ar ar is ae ae ~ . 216 «66 «(High-Resolution Spectroscopy order of e e ny E=p2p-B= L-Bw~ B, JAS em “ : where jt is the magnetic moment of thc state (see Section 2 of this chapter}; The constant B= eh/2m = §.79x 197-8 MeV/T is called the Bohy 2 magneton is Av AE _ 2 Apa ot = B=46.69B m7! (62522 che ore “Se or Av = 14.018 GHz with B in Tesla. energy for the magnetic-dipole terms is of the order of 1 2 AE = py (Bz{0)}) ~ uwiea( 5} = 1837 ( where jin is the nuclear magneton ent LB HN = = Tor? 2mn 1837 = and (6; (0)) is the expectation value for the magnetic ficld of the electrons a at: oe eS the origin; it is equal to jeg / ry (except for configurations with £ = 0)..: a Instead of evaluating (1/r7) we recall that the fine structure splitting is”: ee due to an L - 8 coupling of the electrons, and therefore is of the order of ee LA 2 {1/r?} so that we expect AECE | AE(his) ~ 2209) 64) 1837 Pd AY ae NN NOOO AUN AL a oe Parerohete no de eb Bote oe SENSES hinge eth ay CD a ‘ ote ‘e's : rat hr rh ia babe he ee neve sree id et lal tl tl a es te APPL eh tate te Meet e ~\* SAINTE NWNUR SEEN PEDED DDE CPG Re oe oe oe oe S A 45 Seba ee eee 6.1 Introduction 217 Let us substitute reasonable numbers in Eqs. (6.2) and (6.4); for example, . Pe~e LT AD (Zeeman) ~ 46.0 m7 |. eS and since? AY (fine structure) ~ 10* m~!, we find that Ap(hfs) ~ 5.0 m~! = 1.5 GHz. . Thus the splitting of the lines is very small and can be observed only with a high-resolution instrument, Assuming 2 ~ 500 nm and Av = 5.0 m7! g we find that the required resolving power is Such a resolution may be achieved in two ways: (a) With a large grating used in a high order; the resolving power of a Es grating is given by X —=WNn, Axr where mt, the diffraction order, can be as large as 20, and for a 10-in, grating with 7000 rulings to the inch, the number of rulings is N = 7 x 104, so that A 6 AL 10”. Such gratings, are, however, very difficult to construct, but can now be obtained commercially. (b) With a “multiple-beam” interferometer, the most common one today and easiest to use being the Fabry—Perot, which was discussed in Section 4.6. One can directly observe the “rings” of the interference pattem for = a diverging beam. An optical filter or a dispersive element is needed to - select the line of interest. Alternately one can use the Fabry-Perot in the “scanning mode” by moving one of the end-murrors, through half a wave- length, and observing the transmission of a collimated beam. For instance a Fabry—Perot with 5 cm spacing has an FSR (free spectral range) of 3 GHz; *See Section 1.6.3 and recall that 0 = w/c = 1/2. 2m& 866 6 High-Resolution Spectroscopy even with modest finesse F = 100, the resolution (sce Eq. (4.62))}:is = Av = 30 MHz. Thus for 4 = 500 nm, namely v = 6 x 10/4 Hz : _ 7 Ay 2x 10. : In the following two sections the Zeeman effect and the theory of hypér: fine structure are discussed in some detail. We also discuss the isotope shif€22 and present data on the shift between the spectral lines of hydrogen and 2 deuterium. We then describe a measurement of the Zeeman splitting of the = 546.1-nm green line of Hg, using a Fabry—Perot etalon. The final section oe is devoted to a measurement of the hyperfine structure of rubidium using tie Doppler-free saturation spectroscopy. _ The bibliography on atomic spectroscopy is vast and because of oo refcrences is given at the end of the chapter. 6.2, THE ZEEMAN EFFECT 6.2.1. The Normal Zeeman Effect As already discussed in Section 1.4, the solution of the Schrédinger equa- 2 tion yiclds “stationary states” labeled by three integer indices, n, J, and 324 m, where f < nandm = —/, -i+1,..., 1-1, /. For the soreened 2 Coulomb potential, the energy of (hese states depends on n and / but not on; we therefore say that the (2/ + 1) states with the same 7 and / index. ~ are “degenerate” jn the m quantum number. Classically we can atiribute.” this degeneracy to the fact that the plane of the “orbit” of the electron may. 2 be oricoted in any direction without affecting the energy of the state, since se the potential is spherically symmetric. a Ifa magnetic field B is switched on in the region of the atom, we should « ea expect that the electrons (and the nucleus*) will interact with it. We need Es only consider the electrons cutside closed shells, and assume there is one 3 such electron; indeed the interaction of the magnetic field with this electron = 3“Quantim Mechanics" A. Das and A. Melissinos, Gordon and Breach (1986), = New York. Or any other text on quantum mechanics. ae 4¥or our present discussion this interaction of the nucleus with the external field is so small that we will neglect it. % ory . oe “a ee bear thr hr ke he hele oi Oe oe Bt pee . ee a ta ea te he a a ee Sl BD ee ee a nb erie is ae he] ee eee Ne ee ee ee de A a ae hier ee ee eoa eb ea ee See ae ee he CRNA PATSSESRISTATATESU AgNO ADMD SATAN arS aH 6.2 The Zeeman Effect 215 i 7 dé FIGURE 6.1 Magnetic moment duc to a current circulating in a closed [oop, 2 yields for each state an additional energy AZ, given by AE = mupB. (6.5) = Thus, the total energy of a state depends now on a, /, and m, and the = degeneracy has been removed. To see how this additional energy arises we consider the classical =. analogy. See Fig. 6.1. The orbiting elcctron is equivalent to a current i density? J(x} = —evd(x —1r), where r is the equation of the orbit and x gives the position of the electron; the negative sign arises from the negative charge of the electron. Such a current density gives rise to a magnetic-dipole moment _! at wa=5 [xx sooa’s = 5 er x Y). For a circuJar orbit, the electron is equivalent to a current / = AQ/AT = e/T = ew/2n, where w is the angular frequency w = v/a: a js the radius of the orbit, However, a plane closed loop of current gives rise to a magnetic moment je = 7 A, where A ts the area enclosed by the loop; in our case A = ma’, hence Li) as in Eg. (6.1), 720 «866 High-Resolution Spectroscopy However, the angular momentum. of the orbit is given by L=rxp=m,(r x ¥), so that e eh —— L=— ley, - 2ihe 2ite b where we expressed the angular momentum of the electron in terms of. - quantized value L = /(h/277)uyz, and uz 1s a unit vector along the direction: of L. The energy of a magnetic dipole in a homogeneous field is E=-p-B=——L-B, (63 Sm but the angle between L and the external field B cannot take all possible : values.© We know that it is quantized, so that the projection of L on the 2 axis (which we can take to coincide with the direction of B since no oth 2 preferred direction exists) can only take the values m = —/, —1 + 1,. g 1 — 1,1. Thus the energy of a particular state n, 7, m in the presence of magnetic field will be given by’ En tm = Ent + mB tip, where® In Fig. 6.2 is shown the energy-level diagrarn for the five states with given n and / = 2, before and after the application of a magnetic field B. We note: that all the levels are equidistantly spaced, the energy difference between: them being - AE = ppb. Let us next consider the transition between a state with n;, 4, m,; and one with ny, ls, my. As an example we choose /; = 2 and/; = 1, so that the = ®This was first clearly shown in the Stem—Gerlach expenment. W. Gerlach and 0, Stem, 2 Z. Physik 9, 349 (192). ne "The energy in the field is positive because the electron charge is taken as negative. Srp in this expression is the mass of the electron, not to be confused with the magnetic “2 quantum number m. oe SRT ON retrace RTS Ht ASR R SRSA Ain are ut * a eh rete oF ve DAI > 4 suaanareraaamRaraRaR NNN Renee eo bee es ee eta pate “a7u"a"eMa"e "aha het Matera POO ar a aie te a tatatalg ea pete SRT Late ee ~_*, * *, %"* *, * tH DSS Sn SNR SSNS NY EAR sitet NSTN TN heey gins PERNA ER RR en Ny Deeepercrataracecetetethatytonony ys - Me GH IE . ey ae ” hh A Se SRAM . 6.2 The Zeeman Effect 221 v — 4 s m=+1 “7 a ——" m=0 Ep, i=2 TN a \ m=—1 %& a m=-2 No field With field 2 = FIGURE6.2 Splitting of an energy level under the influcace of an external magnetic field. = The level is assumed to have / = 2 and therefore is split into five equidistant sublevels. (c) mT (a) (b) +2 Ta Cim2 : fe | a: a A ti “E =<. FIGURE 6.3 Splitting of a spectral linc under the influence of an external magnetic field. ne (a) The initial level (/ = 2) and the final leyel (J = 1) with no magnetic field are shown, = A transition between these levels gives rise to the spectral Jines. (b) The two levels after the magnetic field has been applied. (c) The nine allowed transitions between the eijht sublevels of the initial and final states, energy-level diagram is as shown in Fig. 6.3; without a magnetic field in Fig. 6.3a, and when the magnetic field is present in Fig. 6.3b. However, for an e/ectric-dipole transition to take place between two levels, certain selection rules must be fulfilled; in particular, Al =1. (6.9) :. Thus, when the field is turned on, we cannot expect transitions between the = m sublevels with the same /, since they do not satisfy Eq. (6.9). Further, = the transitions between the sublevels with /; = 2 to the sublevels with 222 =6«G.-:s«High-Rasolution Spectrascapy ig = 1 that do satisfy Eq. (6.9) are now governed by the additionai selection rule” me Am =0, +1, (6.10) and thus only the transitions shown in Fig. 6.3c are allowed. : Let the energy splitting in the initial level be a, and in the final level be 5, and jet A be the energy difference between the two levels when rio magnetic field is applied. Then the energy released in a transition i = :£ is given by : Ei - Ef = Aif + mia — m gb. (6.11) are given in matrix form in Table 6,1; x indicates that the transition is forbidden and will not take place. ies At this point the reader must be concerned about the use of a and bt: according to our previous argument (Eq. (6.8), as long as all levels are sub= 3:24 Thus, we see from Eq. (6.11) (or Table 6.1) that only three energy differences are possible ty shtmantheneann a 4 uy a atta Seca E; —Es=A+almy —m;) =A+aAm, TABLE 6.1 Allowed Transitions from !; = 2 tof = 1 and the Corresponding Energies m. of initial state mt of final state +2 +1 i = 9 +1 Ati2a—b A+a-—b A-—b ™ x ° * A +4 A A-a x VOnne -| x x Atb A-a+b A-lat+b <2 4 i a *The selection rules of atomic spectroscopy are a consequence of the addition of angular. ae momenta. In this specific case the selection rules indicate that we consider only eleceric- ces oe dipole radiation. 8 aes * Ss ta 6.2 The Zeeman Effect 223 Source I FIGURE 6.4 The polarization and separation of the components of a norma! Zeeman muloiplet when viewed in a direction normal {o, and in a direction parallel to, the magnetic field. spectral line of frequency v = A/jA Is split mto three components with frequencies v.=(A—peB)f/h, vo=Afh, and vi =(At+pupgB)/h urespective of the values of /; and /,. Furthermore, these spectral lines are polarized, as shown in Fig. 6.4. When the Zeeman effect is viewed in a direction normal to the axis of the magnetic field, the central com- ponent is polarized parallel to the axis, whereas the two outer ones are polanzed normal te the axis of the field. When the Zeeman effect is observed along the axis of the field (by making a hole in the pole face, or using a murror), only the two outer components appear, circularly polar- ized. The lines from Am = +1 transitions appear with nght-hand circular polarization, and from Am = —1 transitions with left-hand circular polar- ization, The central line does not appear, since the electromagnetic field must always have the field vectors (EH and B) normal to the direction of propagation. The splitting of a spectral line into a triplet under the influence of a magnetic field is called the “normal” Zeeman effect, and is occasionally observed experimentally, as, for example, in the 579.0-nm line of mercury arising in a transition!? from ! D2 to ! P;, However, in most cases the lines are split into more components, and even where a triplet appears it does not always show the spacing predicted by Eq. (6.8). This is due to the lONote that both the initial and final states have S$ = 0. ear | | pee 224 Gs «High-Resolution Spectroscopy noaee intrinsic magnetic moment of the electron (associated with its spin) aid will be discussed in the following sections. ee B 6.2.2. The Influence of the Magnetic Moment 2 of the Electron In Scction 1.6 it was discussed how the intrinsic angular momentum (spiny ae of the electrons § couples with the orbital angular momentum of the elec: os trons L to give a resultant J; this coupling gave rise to the “fine structure” of the spectra.'! The projections of J on the z axis are given by my, and we could expect (on the basis of our previous discussion) that the total 8 magnetic moment of the electron will be piven by IaR aa" J. Consequently, the energy-level splitting in a magnetic field B would be i in analogy to Eq. (6.8): AE = —my RB. These conclusions, however, are not correct because the intrinsic mag- netic moment of the electron is related to the intrinsic angular momentum of the electron (the spin) through and not according!” to Eq. (6.6). Consequently, the totai magnetic moment 2 of the eleciron is given by the operator 2 = (2p /h)EL +28}. (6.15) : We will use the following notation: L, 8, J represent angular momentum vectors chat: have magnitude A/I(P 4+ 1), fists +1), RY FG + 1). The symbols J, 7, etc. (s is always: = 5) are the quantion numbers that label a one-electron state and appear in the above square root expressions. The symbols L, 5, J, etc., are quantum numbers that label a state. with more than one elcciron and are then used instead of /, s, 7. ae l2The result of Bq. (6.14) is obtained in a natural way from the solution of the. 222% Dirac equation; it also emerges from the classical relativistic calculation of the “Thomas Se precession.” : 6.2 The Zeeman Effect 225 | We can think of y as a vector onented along J but of magnitude The numerical factor g is called the Landé g factor and a correct quantum- mechanical calculation gives!4 i(F4+)4s64+D-/104+) 2j(j +1) | ; The interesting consequence of Eqs. (6.16) and (6.17) is that now the splitting of a level due to an external field B is g=1+ (6.17) En jdon; = ~Ea gj t+ guaBm; (6.18) and in contrast to Eq. (6.8) 1s nor the same for all levels; it depends on the values of j and / of the level (s = 5 always when one electron is considered). The sublevels are still equidistantly spaced but by an amount AE = guspb. Consider then again the transitions between sublevels belonging to two states with different f (in order to satisfy Eq. (6.9)). However, since we are taking into account the electron spin, / is not a good quantum number, and instead the / values of the mitial and final levels must be specified. If we 13This result can also be obtained from the vector model for the atomic electron. In Fig, 6.5 the three vectors J, 1., and § are shown, and L and § couple into the resultant J, so that J=L+S5. By taking the squares of the vectors, we obtain the following values for the cosines 2 2 2 sz 2 2 I~ — -f cos(L, D= 2 —* cos, Na. adj 25} From Eq. (6.15) we see that fen =fcosthL, J) + ascos (6, J. Thus wp fttP—-s* 272 4252-27 pPts?-? ary ar ar te Finally we must replace j*, s*, and I? by their quantum-mechanical expectation values J(7 +1), etc., and we obtain Ea. (6.17). Se Ss et i Pee hear Ly Lk Lt po 9 CotPUa ate Mt 226 «66 «High-Resolution Spactroscopy FIGURE 6.5 Addition of the orbital angular momentum L and of the spin angula:3 momeéntum § into the total angular momenium J, according to the “vector model.” choose for this example fi = land/y = 0, we have the choice of j; = 5:3 T= 5 whereas jy = 5 Transitions may occur only if they satisfy, ii fe addition to Eq. (6.9), aéso the selection rules for j “ Aj=0,+4! not j=0>j=0. (63) z Furthermore the selection rules for m; must also be satisfied; they are ei 8 same as given by Eq. (6.10) re Amie Of]. (6. 100) In Fig. 6.6 the energy-level diagram is given without and witha magnet : field for the doublet initial state with / = 1, and the singlet final State, f= 0. * Six possible transitions between the initial states with } = : to the inal state with ]= ‘ are shown (as well as the four possible transitions from j= - io j= 4). By using Eq. (6.17) we obtain the following g factors i=1 733 s=% g=% I=0 jst s=h ge? The sublevels in Fig. 6.6 have been spaced accordingly. In Table 6.2 are listed the six transitions from 7 = : toyj= - in anal- ogy with Table 6.1. However, since now a 5, the spectral line is split into a six-component (symmetric) pattem. This structure of the spectral line is indicated in the lower part of Fig. 6.6; following adopted conven- tion, the components with polarization parallel to the field are indicated above the base line, and with polarization normal to the field, below.!* As before the parallel components have Am = 0, the normal ones Ay + 1, It is also conventional to label the parallel coniponents with a, and the normal ones by o (from the German “Senkrecht’”}. yt es . BURL omansaanans hho hac Mh oa bet Mati thorntniennneniner a inate Ran a MNES NNN Ninn ieee nite TN a OL ae SoS Moe canna ene ee Eee RN CUSHING MINN shteasnegrtrtgtenntianrte teat WSEDDRAN AMMAN MASSER ASI A pon neat TMG MLSEM SAIN ASS S SN * 6.2 The Zeeman Effect 227 TABLE 6.2 Allowed Transitions from /; =3 to fr = ; and the Corresponding Energies mj of initial state m; of final state + 42 _! 3 " 5 2 2 2 l 3a a .o ag os _ an ey Cee +5 Atty Atz~3 2 2 “ I a ob a ob 3a)—Stiéi a At—-+-—_ —~iy lt Ae 3 * i ae 272 ho a 8 \ 1 3 | + 4 I | | l \ ~ N | f hy ra i } Polo Aofe+ AP AFG © a cal \ Cc — \ \ | \ 2P ip _ id Mm=+ 5 3 ane PN SS OS HR GO - 2 *Sr2 oa mata ; oe g= are ¥ . 1 2 p+—C—} Am=0 Tt | | Am=+1 49 FIGURE 6.6 Energy levels of a single valence electron atom showing a P state and an £ stale. Due to the fine structnre, the P state is spht into a doublet with f = - and j = . Further, under the influetice of an extemal magnetic field each of the three levels is split into sublevels as shown in the figure where accoont has been taken of the magnetic monient of the electron. The magnetic quantum number m ; for each sublevel is also shown as ts the e factor for each level. The arrows indicate the allowed wansitions between tbe initial and final states, and the structure of the line is shown in the lower part of the figure. The horizontal spacing between the components is proportional to the differences in the energy of the transition, and the vertical height is pro- portional to the intensity of the components; the relative intensity can be predicted exactly since it involves only the comparison of matrix elements between the angular parts of the wave Function. 229 «€©6©—6 «High-Resolution Spectroscopy : we can expect that the “spinning” charge of * the nucleus wll give rise to a magnetic moment (see Eq. (6. 6)) oriented along the spin axis = where M is the mass of the nucleus. In addition, nuclei exhibit an intrinsic magoetization,'> so that in general we have = —gr —— T= gruniuy, im mp =: where uy is a unit vector along the spin direction, and eh. 2M p Ln = = js the nuclear magneton: m p is the proton mass. The numerical factor g; =, includes all the effects of intrinsic and orbita) magnetization of the nucleus “ and can be obtained only from a theory of nuclear structure. The magnetic moment of the nucleus, yz, will interact with the magnetic field B, (0) produced by the atomic electrons (at the nucleus; Fig. 6.7). This :. interaction then results in a shift of the energy levels of the atom by the = amount AE = —yn-B,(0). (6.20) The direction of B, (0) is that given by the total angular momentum of the “: atomic electrons, namely,!® J, so that Le oS) AE = I-J. ; aGall iH J (6.21) 1S This Bives rise to the so-called “anamalous” magnetic moment of the nucleon; for example, the neutron (an uncharged particle} has a magnetic moment of —1.91 zy. l6The direction of B-(0) is really opposite to J because the electron has negative charge. 230 #866 High-Resolution Spectroscopy E,{0) FIGURE6.7 Interaction of the nuclear magnetic moment with the magnetic field produced by the electrons at the nucleus. Thus, we expect the splitting of a level of given J according to the possible values of (I - J), which, as we know, are quantized. The situation is analogous to that of the fine structure, where the interac: tion was proportional to the (L-§) term. In that instance the two angular momenta coupled into a resultant J = (L + §) according to the quantum=:7 mechanical laws of addition of angular momentum, !n the present situation, 2224 J and I couple into a total angular momentum of the atom designated: by F: ¥=(+J). (622) An energy level of given J is then split into sublevels having all possible | values of F’, namely, the integers (or half-integers)} |\J-f)< F< |J+] |. Thus if J = 2 the level is split into two components, with F, = J +4 7 SE and Fp =J—+5 1 (provided J > 4); if f = 1, the level is split into three === ae components with Fi = J-—1,F = J,and fy = J+] (provided J > 1); etc. This situation is shown in Fig. 6.8, and we see that if J is known, the © number of hyperfine structure components of a spectral line provides direct == information on the spin of the nucleus. ES If either § = 0 or J = O, no splitting of the energy levels can 2% occur since the interaction energy specified by Eq. (6.21) vanishes. This “= is to be expected because if / = 0, the nucleus cannot have a dipole = moment, and if J = 0, then by symmetry, the magnetic field at the origin B.(0) = 0. SS Using Eq. (6.22), we can now obtain the expectation value of the operator (1. J) that appears in Eq. (6.21); referring to the vector model = Oa wh ua nN Ee neon hh heh ehhh ehh ahhh hihi shh aahh hhh hhh aie hha TNE AAR RU NN AREOLA TAN NNR M RIN AN HARA L NAL TM NHL AMLHSCINS ig aan ene AsA eens es SS ee i SaaS ae Se eee eee En LEie Sena Sean Ene RRMA hi oh RENDER Re eke ne REDE REEMA ES aR ESP SARS USE REST TT REI praninnntarna ia 6.3 Hypertine Structure 231 Ee FIGURE 6.8 Hyperfine structure splitting of a? Py atomic energy level, and the allowed =. transitions between the hyperfine structure components of this Jeve) and a ! So final state *. when the spin of the nucleus ts (a) f = - atid (b) J = 3. we write “classically” F2— jy? — J? cos I, J) = ay; = and replacing F?, etc., by the quantum-mechanical expectation values F(F + 1) we obtain AE =< (F(R +1)-104)-JU4+01 (6.23) where the constant A is given by _ p (Be(0)) —| lie (6.24) Note that the energy splitting between sublevels, as given by Eq. (6.23) (and shown in Fig. 6.8), is not symmetric. Further, if we succeed in extract- ing from the experimental data the constant A, we can obtain the nuclear ” Thagnetic moment if (8,(0}) is known. The calculation of the average value of the magnetic field of the electrons al the nucleus (B,(0)), however, is not easy to perform, and depends on the orbital angular momentum of the valence electron or electrons. Expressions = oxy 2222S «6s «High-Resolution Spectrascapy for the “constant” A in terms of the atomic wave function can be found: the references (see Kopferman). yt 6.3.2. Isotope Shift Figure 6.9 shows the hyperfine structure of the 253.7-nm line of naturat mercury when examined under high resolutton. When the lines are correctli identified we note that the different isotopes have different energies. Indeed natural mercury consists of several isotopes with the abundances shown in Table 6.3; the nuclear spin, nuclear-dipole magnetic moment, and electric: quadrupole moment are also indicated. oF The isotope shift arises from two effects: (a) The finite mass of the nucleus: The nucleus is much heavier than the electron, but we can con: sider its mass as infinite only to a first approximation. (b) The finite size of the nucleus: The nuclear radius is much smaller than the orbit of thé electron, but we can consider the nucleus as a point only to a first-order approximation. For light elements the isotope shift is mainly due to thé effect of the finite mass, whereas for the heavy elements it is mainly duc té: the finite size effect. It should also be evident that we cannot measure the shift in the energy level of a single isotope, but only the difference in the smift between two or more isotopes. This is shown in Fig. 6.10a A> 253.7 nm _ — -0.507 —0.491/ -G.022 06.230 —0.156 FIGURE6.9 High-resolution spectrogram of the 253.7-nm line of natural mercury. In the’ lower part of the figure the various components are identifted and their separation from the position of the !8Hg component is also indicated. (Note that the !9*Hg component appears in the spectrogram as the longer line.) —0.51 9} Maat). 0 jo? 0 - Cin 6.3 Hyperfine Structure 233 TABLE 6.3 Properties of the Isotapes of Natural Hg (Z = 80) / Abundance N (nuclear le Jsotope (percent) (neutrons) spin) (units of ppv) (cm? x 10-4} 198 10.1 118 0 Q 199 17.0 119 uy 0.876 200 23.2 120 f) 0 > 201 13.2 121 3 ~0,723 0,38 202 29.6 122 0 0 204 6.7 124 0 0 In terms of the solutions of the Schrodinger equation we must consider both the electron and nucleus as revolving about the center of mass of the electron—nucleus system, This leads back to the Schrédinger equation for a stationary attractive center (nucleus) if the mass of the electron is replaced by its reduced mass M M +m," where M is the mass of the nucleus, Then the energy of a hydrogen-like level is given by eet if MN KeRAE f, tey aay n* M+m, n> M m' = nl; (6.25) E, = where Z is the nuclear charge. For instance, the value of the Rydberg as obtained from the spectra of hydrogen and deutenum will differ by Ru Me — ~|{1- ‘ 6.27 Rp ( a) 020) where we set the mass of the deuteron mtg ~ 2m,. This will shift the spectral lines by 3 x 10~*, which we can observe in the laboratory. For the heavier elements the isotope shift due to finite mass becomes very small, Instead it is the finite size of the nucleus that is the dominant reason for a shift of the energy levels. Consider Fig, 6,10b where curve (a) represents the Coulomb potential of a point charge. If it is assumed that the electric charge of the nucleus is distributed on a spherical surface of radius rg, then the potential will not diverge at y = 0, but will be constant for all r < ry. Thus the potential seen by an electron will be of the form shown 2734 = 6, «High-Resolution Spectroscopy {a) AE,(1) AE} | owels for (b) point nucleus AEA) AE (2) (A) {A+1)} fet[AE(1)-AE,(4)] hv-+[AE;(2)--AE,(2)] FIGURE 6,10 The isotope shift of atomic spectral lines. (a) The energy levels of the initiat ee and final states of two different isotopes with mass numbers A and A +- ! are shown. The ae dashed lines show the position the levels would have if the nucleus was an infinitely heavy: aD point; the solid Lines show the actual position of the levels, which are shifted by a different 2 ee amount for cach isotope, and for each level, (b} Modification of the Coulomb potential cee ee the nucleus due to its finite sizc. by the solid curve of Fig. 6.10b. This leads to a significant energy shift as a tunction of ro. Since the nuclear radius can be expressed as =A‘? x12 1074 cm where A is the number of nucleons (protons and neutrons) in the nucleus we see that Ary/ro = AA/3A, which can be significant. 6.3.3, Measurement of the H-D Isotope Shift The hydrogen—deuterium shift is quite large and can be measured with aft instrument of modest resolution. The results presented here were obtained: with a Jarrell-Ash grating spectrometer. A schematic of the spectrometer is shown in Fig. 6.11 and conforms with the generic spectrometer design intro duced in Fig. 5.13. Instead of lenses, focusing mirrors are used to image the: entrance slit onto the photomultiplier tube (PMT). The advantage of using a PMT is that very low levels of light can be detected so that the entrance and detector slits can be set to very narrow width. The grating had 630 rulings per millimeter, and the focal length of the lens was f = 0.5 m. The. spectrum was viewed in second order with a resolution AX /A ~ 2 x 1075: The angle of the grating was computer controlled so that the speed at which* 6.3 Hyperfine Structure 235 Top View Viewport FIGURE 6.11 Schematic Jayout of the high-resolution Jarrell—Ash grating spectrometer. 4.5 4 4.5 2 3 Deuterium; A=655.77 nm oa £25 = a Hydrogen: .+655.94 nm 5 at ~—_ O.5 655.7 655.75 655.8 65595 6559 655.95 656 Calibrated wavelenath (nm) FIGURE 6.12 The red Jine of the Balmer series for a source containing hydrogen and deuterium observed in high resolution. The absolute wavelength calibration is not exact but this has insignificant cffect on the wavelength difference between the two lines. ihe spectrum was swept could be adjusted; slaw speed for high resolution and vice versa. Furthermore, the grating angle was calibrated to indicate wavelength in nanometers. For this experiment the source was a discharge tube containing deu- terium and an admixture of hydrogen. The entrance slit was closed to a few hundred jzm, and the first (red) sine of the Balmer series (nj = 3, ny = 2), A = 656,28 nm, was examined. The resulting spectrum is shown in Fig. 6.12 where the hydrogen line (longer wavelength) is well sepa- . rated from the deuterium line. Note that the absolute calibration of the - Wavelength scale is off by almost 0.3 mm; this is not important in the - present case where we are interested in the wavelength difference, 236 2«=66:,—_« High-Resolution Spectrascopy In terms of the calibration we find that Ay = 655.94 nm Ap = 655.77 mm. Convert the wavelength difference into frequency difference AD — AH — Up = c ———— = —i1.85 GHz, cece MH — YD =e ae namely, a fractional frequency change wae epE Ct reee &, as atte Avy. Aa Be aeHeP 6 2 ~2,59 x 1074. ee YD A Hee From Eq. (6.27) we expect that . Avg-D _ Ry — Rp ~ _ ite = —2.72 x 1074 ba Wa aac at in close agreement (within 5%) with the measured value. Ramana ba PSI 6.4. THE LINE WIDTH Since we are trying to resolve very small differences between the compo- : nents of a spectral line, it is evident that the width of these components :: must be narrower than the separation between them. Before the advent of: the laser, this was a very difficult task, but today laser lines can be stabilized. to aremarkably narrow width, and used for spectrescapic studies. ; Spectral lines have a natural width given by AE 1 : Ay = —_ = —_, 6.28):-: h 27 At \ es where At is the lifetime of the state; this is usually negligible, since atomic | lifetimes are on the order of r > 10~* s. Thus | Avs —! 15 MHz. Qn x 10-8 In wave numbers we find Av < 0.05 m—!. However, external influences do broaden spectral lines considerably; the main causes are as follows: -. (a) Doppler Broadening. Due to their thermal energy, the atoms in® the source move in random directions with a velocity given by the: 6.4 The Line Width 23? Maxwell-Boltzmann distribution. Consequently, the wavelength emitted : jp a transition of the atom is Doppler-shifted; this results in a broadening : of the line, which can be shown to have a half-width A [T a =10-° ft. (6,29) p A : where T is the absolute temperature in Kelvins, and A is the atomic number =: of the element. Doppler broadening is most serious for the light elements and in sources that operate at high temperatures. For example, in an arc discharge operating at 7 = 3600K, a hydrogen line of A = 500 nm will have a Doppler width of 36 GHz, which will mask any hyperfine structure, For heavy elements, as in Hg (A ~ 200), Av == 3 GHz, which is still quite broad. (b) Pressure (or Collision) Broadening. When the pressure in the source : vapor is too high, the atoms are subject to frequent collisions, which ina way ~ can be thought of as reducing the time interval Ar entering into Eq. (6.28). : (c) External Fields. Magnetic orelectric fields produce Zeeman or Stark splitting of the components, resulting in effective broadening of the line. Electric fields of 1000 V/cm can cause a broadening of tens of gigahertz. (d) Self-Absorption and Reversal. This phenomenon is most pronoun- ced with resonance lines. As the radiation emitted from the atoms in the middle of the source travels through the vapor, it has a probability of heing absorbed that is proportional to the path length it traverses and to the absorption cross section; this will be strongest in the center of the line and weaker in the wings. The result shown in Fig. 6.13a is that the line becomes “squashed” in the center; that is, it is broadened. :: FIGURE 4.13 Broadening of a spectral line due to self-absorption in the source. The “. solid cvrve is the emilied line, the dashed curve represents the part of the radiation that is absorbed, and the dash—dat curve shows the transmitted line, which is the difference of - the two former curves. (a) Normal absorption, and (b) strong absorption especially in the central region leading to self-reversal. 238 «=sodG§Cs High-Resolution Spectrascony with almost none in the wings. The result is a “self-reversed” line as shows: a in Fig. 6.13b. This effect is very pronounced in the sodinm D lines, and: ie when it is viewed with a high-resolution instrument, the line exhibits. aos excite the source. 6.5. THE ZEEMAN EFFECT OF THE GREEN ee LINE OF "*Hig Es RORRERN SONS [Sar teh aie at nih ‘ abastetatatatatate ta hat 6.5.1. Equipment and Alignment Be ee We now discuss the observation of the Zeeman effect on the A = 546. 1- mh ne line of '*8Hg. The choice of the green fine i is due to its predominance 5 iit: ‘7 detail in Section 6.2.2, In the present observations, a polarizer parallel * te the magnetic field was used, so that only three of the nine components (the: x light) appeared. Furthermore, natural mercury exhibits in the green line a: large number of hyperfine structure components, and each of them forms 2 Zeeman pattern. To avoid a multiplicity of components in one spectral line,’ a separated isotope of mercury was used as the source. }°°H¢ is well suited: for this purpose since 7 = 0, and therefore it exhibits no hyperfine structure. The optical system used for this investigation 1s shown in Fig. 6.14. The Fabry-Perot was crossed in the parallel-beam method with a small constant-deviation spectrograph (see Chapter 1}. The etalon and lenses are all mounted on an optical bench to which the spectrograph is rigidly “22 attached. The pair of lenses ZL; forms the light from the source into a "24 parallel beam, while the pair £2 focuses the Fabry-Perot ring pattern onto *222: the spectrograph slit; the effective focal length of Lz is 8 cm, and a further /225 magnification of 2 takes place in the spectrograph. Be The discharge tube is mounted vertically, as is the spectrograph slit; the :2: slit width was | mim. It is clear that in this arrangement not only is the ting “2 re eh pattern focused onto the spectrometer slit but also the image of the source, |: 24 A sheet of Polaroid film that could be rotated at will was used as a polarizer. #224 vane! 6.5 The Zeeman Effect of the Graan Line of 8Hqy 239 Spectrograph elit in Ly focal plane of etalon ow Source prajaction system Polarizer | Ftalon t ai Constant | +9.6 W +68 -5 I” UE] deviation Lo ; Prism Doublets Stit to admit only 1 i distortion<1% fight produced in - uniform field Flald-current > coniral \ Excitation coll —t— Position af (to r.f. asciilafor) photopiate FIGURE 6.14 [Experimental arrangement used for observing the Zeeman effect with a _ Fabry-Perot etalon, crossed by a constant-deviation prism spectragraph. L, &-P lp Mirror Ly | F | | by, = FIGURE 6.15 Optical arrangement for aligning a Fabry-Perot etalon. Rough adjustment ts made by viewing the image formed by L>. Final adjusiment is made by viewing the etalon from the point F (or F’). The spacing of the Fabry-Perot etalon is ¢ = 0.5002 cm; namely, the free spectral range is FSR = 30 GHz. It is important to adjust the plates carefully for parallelism. This can be done either by viewing through the spectrograph with a frosted glass in the focal plane, and adjusting for the best quality of the pattern, or by a much more sensitive arrangement as shown in Fig. 6.15. A very small aperture (less than 1 mm in diameter} is placed at the position of the source and illuminated with an intense sodjum lamp. The Fabry—Perot plates are adjusted to be normal to the Optical axis by bringing the image of A reflected by the etalon back onto A. Next, 4 is adjusted until a senes of multiple images of A appears when the observer is located at 7; the plates of the etalon can then be roughly adjusted for parallelism by bringing all the images into coincidence. The final adjustment is made by removing Z3 so that the observer locates his eye at F (or a mirror can be used); then fringes of equal width do appear 240 «866 High-Resolution Spectrascapy Current / (Amp) o 2 4 6 B 10 12 14 Magnetic held 2 (kG) FIGURE 6.16 Calibration of the electromagnet used in the Zecman effect experiment; The magnetic field ts plotted against current; note the saturation at high fields, parallel to the base of the wedge formed by the two plates. As the plates aré moved into parallelism, the fringes become broader and finally the wholg image of the aperture A seems to have a uniform illumination (bright or dark depending on the exact value of ng = 2¢/A). It is equally important that the ring patiern be in sharp focus at the plane of the photographic plate, For this experiment Kodak Royai-Pan film was used. The electrodeless discharge tube was placed in a magnetic field. A smalt iron core electromagnei powered by a 220-V DC supp! y, was used to pro- duce the field. The diameter of the pole faces was only 15 in., and a small gap G in.) was used. By tapering the pole faces, higher magnetic fields can be achieved but this reduces the effective area of the field as well as the homogeneity. The magnetic field was measured with a “flip coil” and. the calibration of field against current is given in Fig. 6.16. It is seen that field strengths of 1.2 T could be reached. 6.5.2. Data on the Zeeman Effect The data presented below were obiained by students. Figure 6.17 shows the. Z 546.1-nm Hg line photographed at various magnet settings. As explained: apeereey ate et Swish bebe de ede sete ae re ee de A be ei ie pe ee We oe be be'Se be he Se be Ba be ee be be De eee Be be he an setae SN bei hen ie totn ben heir ine eh hohe hee Dehetre Rete Bt te Be Be et ed a) ate Sanh heh ” Dan ee 65 The Zeeman Effect of the Green Line of Hq 241 = | = a) = | = a i (b) S () e a ee (2) 2,00 A FIGURE 6.17 Fabry—Perot patterns showing the Zeeman effect of the green line of mer- cury. (See the text for additional details.) (a) No magnetic field applied. (b~e) A magnetic field of progressively preater strength is applicd. Note the splitting of the onginal line into a triplet of increasing separation. earlier, the source contains a single isotope, and the polarizer allows only the observation of 7 light. We note that the fnnges are rather broad, but it can clearly be seen that when the field is applied the single—line pattem breaks up into a triplet, the separation between the components of the triplet becoming larger with increasing field. The initial step in the reduction of the data is the measurement of the diameters (or radii) of the rings. To this effect a traveling microscope was used, and readings were taken directly off the plate; care must be taken to ensure that the travel of the microscope is indeed along the diameter of the rings and that the crosshairs are properly onented. When the fnnges in the pattern are as broad as those in Fig. 6.17, it is much more accurate lo measure the two edges and take the average rather than try to set the crosshairs in the center of the fringe. The ring radii squared in the absence of the held provide the calibration of the data, 242 § High-Resolution Spectroscopy 0.4 G.a 0.2 Av fem) 2 4 6 8 10 12 14 16 Magnetic field & (kG) FIGURE 6.18 Results obtained on the Zeeman effect of the green line of mercury, Oe ma text}. The observed displacement of the three components from the zero fleld value (of thei: single line) is plotted against the magnetic field. Beer Next the radii of the rings for the exposures taken at 1.0, 1.3, and 2.0 A were analyzed, and it was found that the central line is not shifted. However;: the fallowing shifts are observed for the outer rings for the 1.0-A data: Avy = 6.81 GHz Av_ = 6.60 GHz. The complete set of data is plotted in Fig. 6.18, and we see that as predicted the spacing varies linearly with the field, yielding AVS GEE EOS (6.30) The green line of Hg (546.1 nm) connects the *5, state to the ?P. 4 Its Zeeman splitting is shown in Fig. 6.19 where the g factors have been 44 calculated according to Eq. (6.17). Since the polarizer was set to select = only components arising in transitions with Am = 0, we expect to observe 22 only the three central components, which will be separated by SBE LB LB aie Av = — (g;—gr)B = =—B. 6.31) 26 vs (si — BB = 55 (6.31) ge 6.6 Saturation Absorption Spectroscopy of Rubidium 243 mat wf 3S, —__<<.. 0 g=2 aA, 4 ma+2 ” for +1 *P; ——_€-— 0 9! ers -4 . a : FIGURE 6.19 The Zeeman multiplet splitting of the 546.1-nm preen line of Hg. It arises :; froma 35) to Py transition. By comparing with the experimental result of Eq. (6.30), we obtain | wR = 5.95 x 107!! MeV/T in good agreement with the accepted value of itp = 5.79 x 107! MeV/T. _ From these data we conclude that indeed spectral lines are split into com- - ponents when the source is placed in a magnetic field. Further, the splitting - observed was in excellent agreement with the theory of the anomalous ~ Zeeman effect; the normal Zeeman effect can be excluded, since the energy _ difference between the components of the line was not jg B but 4 LBB; ~ compare to Eq. (6.1). 6.6. SATURATION ABSORPTION SPECTROSCOPY ; OF RUBIDIUM : 6.6.1. Introduction ’ We mentioned in Section 6.4 that if an intense spectral line is passed through a region of dense atomic vapor of the same element it may become absorbed 244 #6 High-Resolution Spectroscopy from the ground to an excited state. If one monitors the transmitted light; ag a function of frequency, a Bee cl he, absorption spectrum, such: ea probe beam, the experimental arrangement being as shown ii in Fig. 6. 212 3 The signal at D2 will exhibit the same general behavior as D; except thal penne there will be a sharp spike at the center of the profile: see Fig. 6.20b. =2:222555 (a) (b) ¥g CCtt camer Dy Pafisocoe Rb cell mirtor Doppler broadened absorption FIGURE 6.21 Schematic layout of the saturation absorption experiment, a! eg 6.6 Saturation Absorption Spectroscopy of Rubidium 245 Let us examine what happens when the pump beam of frequency v+ (refer to Fig. 6.20a) is incident on the cell; it excites atoms with a particular velocity v4 moving toward the wave vector of the laser beam. When the pump has frequency v_ it excites atoms that move in the same direction as the wave vector kp with velocity v_. At vg the excited atoms have no . velocity component along kp. The probe beam has the same frequency . as the pump at all times but its & vector is opposite to kp. Thus when vy = v4, the atoms excited by the pump cannot absorb photon$ from the probe since they are moving in the v4 direction, namely along the probe wave vector; similarly when v, = v_. However, when vz, = vp the atoms that could absorb the probe beam are already in the excited state due to the presence of the pump beam. As a result there is less absorption and a spike ; appears in the profile when v sweeps through vo. The spike is very narrow *. as compared to the Doppler profile. The situation becomes more complicated when there are several lines ~ (that is, hyperfine structure) under the Doppler profile. For a single line of ~ frequency vo we found that the spike appears at vg. For two lines present “atv; and yz, one will see spikes not only when the laser frequency reaches vy =, V2 but also when!” be = (vy + v2)/2. (6.32) - Such spikes are “crossover” lines and are often stronger than the direct > lines. Saturation spectroscupy can be easily observed in rubidium, cesium, : and sodium and is used to lock lasers to 4 narrow frequency, For a practical '’Note that if for the laser frequency vy the Doppler shift (for the pump beam) by a class = of atoms with velocity Ug iS vg, then the state that is excited has frequency v; where UL + Ve = YY. * For the probe beam the effective frequency (for this same class of atoms) ts VE Ve. g If this frequency happens to correspond to another atomic transition, say at frequency 9, * then the absorption will again be saturated. Therefore the condition is ee ee | i ie hes ee hee he ote ya's Ey. Me = 22 = OF vz, = (vy +v3)/2 as given by Eq. (6,32). 245 experimental details can be found in a classic paper by K. B. Mac athe =e SS A. Steinbach, and C. Wieman, Am. J. Phys. 60, 1098 (19972). ae 6.6.2. The Rubidium hfs Spectrum Rubidium is an alkali (2 = rubidium has two isotopes Rb 87 Rb 8 High-Resolution Spectroscapy 37) with a asl’ Ss valence electron outy ae. with nuclear spin Sinan Fe ee eter” tei athemme a Reeser era In the absence of nuclear spin the ground state is a 1g, /2 State and the: | excited states are * Pj 2 and ? Pyj2. energy level diagram i is as shown i in a 6.22. When the nuclear spin is icine 121 MHz 267 MHz a 2 64 MHz 157 MHz —. 2 1 72 MHz 1 0 B2=7e0.23 nm F=4 De=760.23 nm F=2 "fe sabia Ot=794,76nm 01=794.70 rm ~———-Fr3 Foe ; 3.036 GHz 4 6,835 GHz c ee eee ; 8SAib (72%) ®7Rb (28%) ees Tee FIGURE 6.22 Bnergy level diagram of the low-lying atomic states of rubidium: (a) sk and {b) 2?’ Rb. a ee — m SSNS * eres a SESE wae was “aaah na hin nacniaeaa aa heheh hers nt ate on rere ee en RASC HR th irit 6.6 Saturation Absorption Spectroscopy of Rubidium 247 “has two F levels F=3 and F=2, e whereas the excited state has four F levels F= 4, 3, 2, and 1. % OMS can be seen from Fig. 6.22 the hfs in the ground state 1s quite large, of the order of 3 GHz, so that one can tune the laser to select transitions from “either the F = 2 or F = 3 state. Obviously the Pj, state is too far away to cause confusion, However, the Doppler profile, which is of the order eof 1.0 GHz, covers all four hfs levels of the excited state. Recall that only =< transitions with AF = 0, +1 are allowed for electric dipole. The laser frequency must be at 780.23 nm, which is in the infrared. It 4s conveniently obtainable from a diode laser. The diode laser is mounted ="jn an external cavity, which is used to select the desired wavelength and “gan deliver up to 10 mW of power. Usually it suffices to send 3 mW to the _ pump beam and only a tenth of that to the probe beam. Paar “te grating angle with piezo controls. The diode laser output is a very strong function of laser temperature. Figure 6,23 shows such a calibration curve, and one selects the appropriate =: temperature with the help of a medium resolution spectrometer. Then the =: piezo is set to sweep the frequency, and one adjusts the laser current to = shift the central frequency while the pump beam is going through the cell. “At some point one will observe fluorescence, with an IR viewer or a CCD a hh te =, camera, or by monitoring the transmitted beam. At this point one can reduce the sweep and setup for saturation absorp- “tion measurements. It is convenient to display the probe beam on a scope zewith the sweep on the horizontal axis. A picture of the observed fluores- z-eence and of the saturated absorption of the probe beam are shown in oF g. 6.24. It is always possible to run a second low-intensity beam through 248 866) «High-Resolution Spectroscopy 40 ——— a0 25 Temperature (°C} 20 15 10 792 7a 794 795 796 a7 738 799 i {nm} FIGURE 6.23 Wavelength as a function of temperature for the diode laser used rn i expenment . oe (r) she be FIGURE 6.24 (a) Fluorescence emitted by the pump beam when properly tuned onto ie: Rb resonance fine. (b) The probe beam signal when the frequency is swept over the entire: Doppler peak. The displaced curves are duc to hysteresis in the piezo eleciricdnver — Biever a the nonsaturated part of the cell to obtain the Doppler absorption pofle | and subtract it from the saturated absorption. 8 Data obtained by students on °°Rb pumping from the F = 3 eround: eA state are shown in Fig. 6.25. The two prominent lines are the crossover: lines [v(F’ = 2) + v(F’ = 4)]/2, and [v = (F' = 3}4+ v(F’ = 4)]/2, and the v(F’ = 4) line can also be distinguished. On the assumption that the: swecp is linear, the position of the other expected lines is indicated. Delays ee ee ene aa) he SL cotetenatreoavetsiaiets es ite td 6.6 Saturation Absorption Spectroscopy of Rubidium 249 Nuthayecachseterstentersessnetaeehtepysten ge ty ) Ray +, . veers vee Voltage (Volts) SS A = : Be ¥2 Veg” V4 ae V24 Vag FIGURE 6.25 Saturation absorption spectrum obtained by students for ®5Rb Ee ( F=3— F’), The position of all expected lines is indicated. 5 AAAN NS ty oh Mad, tre se AN . hee ass 4% $$%™“a % Wa Mg "3 “FIGURE 6.26 Subtracted saturation absorption spectrum obtained by students for 87Rb “(F =2—» F’). The position of all expected lines is indicated. ~~. _ SAVIN SHANA 250 «=6© 6. «High-Resolution Spectroscopy Asis evident from the data the saturated absorption lines are very snap Thus instead of sweeping the laser frequency one cali use a scrvo circuit: reaching a stability of - few megahertz, in absolute terms. 6.7. REFERENCES an advanced level, EEE ee Hs HL EL White, Introduction to Atemic Spectra, McGraw-Hill, New York, 1934. This book contig ae cores extensive data on atomic spectra, and the treatment of the theary is based un the semicsiel peat approach of the vector model. ae fs eee ies H. Kuhn, Atomic Specira, Longman's, London, 1962. A gond book on a slightly more advanced level Le ee than White’s boak referred to above. rere S. Tolansky, High Resolution Spectroscopy, Methuen, London, 1947, A very comprehensive and at S : treatise on the instruments and weatniaues of hi she resojuion spectroscopy. a obtained from it. a ee W. Demtrdeder, Laser Spectroscopy, 2nd ed., Springer-Verlag, Berlin, 1996. A very comprehen zB _ and up-to-date coverage of the field. os Th ens Se oh a a eh a a a SSE aE Mea ut nnn tty . eS CHAPTER ? Magnetic Resonance Experiments = 7.1, INTRODUCTION :: We saw in the previous chapter that when an atom (or a nucleus), with =: angular momentum L (or I), different from 0, is placed in a magnetic field a B the states that correspond to different values of the quantum number m = acquire an additional energy AE = - Bm. (7.1) =: Here yt is the “magnetic moment” of the atam or micleus. When electrons “ are involved, is on the order of the Bohr magneton jz while for nuclei = i218 On the order of the nuclear magneton, j4;,. In convenient units p/h = 14.01 GHz/T Bn/h = (e9/h)/1836= 7.62 MHz2/T,. (7.2) 251 i le Ee ' m 2o2 0 7 ~Magnetic Resonance Experiments Mip=+1 FIGURE 7.1 Splitting of an energy level with / = 1 into three components when plas Barc: in a magnetic field. eee aaa ee et oe ee pets See ae between sublevels with different, because they do not satisfy the scleciie 2 i Pore a) rule Ai = +1. Instead the splitting of a level is observed through tHe small difference in the frequency of the radiation emitted in the transitions: between widely distant levels (with A/ = <1). It is clear that if we conte ee splitting would be obtained. eo a The selection rule Al = +1 is applicable to electric-dipole ration He dipole radiation is emitted, but the probability for such a transition: ts: ag reduced by a factor! (v/c)* from the case of an electric dipole transition. We 3 therefore conclude that spontaneous transitions with Al = 0, Am = Ef ae os ae Sphceterstiat will be very rare, especially if the system can preferentially return to: it S Browne state (lowest enerzy state) by a Al= +1 transition. On the othe SE me She oe oe ord aaa “et electromagnetic field (that is, the total number of quanta) so that if a suffi L ciently strong radiofrequency magnetic field (of frequency vg) is available, : - magnetic- cipole transitions should take place, UE a. ee ee aa ee ‘For utamic systems v is on the order of the velocity in a Bobr orbil, namely, (v/ at me Be 5 x 107%, pe 7.1 Introduction 283 © sfectsic-cipole transitions are induced by the external electnc field (at the os = frequency) of the laser beam. ate. = By referring to Eq, (7.2) we see that for a 1-T magnetic field the energy © pitting of either nuclei or electrons falls in the range of frequencies that es be easily generated. It i is also of interest to estimate the magnitude : designate by H, to distinguish it from the static magnetic (induction) field 2B. in vacuum B = oH. An H field of magnitude 103/47r A/m (equivalent p10 a B field of 10~* T = 1 G) corresponds to an energy flow of ee 4n x 10-7 py (S)= (= CR RLY Pal Bs ons (= hd =; Ser * ( ~) x10 panels, (7. » ee P Necuae for inducing transitions. Finally we must be able to detect the fact =: that a transition took place; this may be done in several ways and is one of ‘the distinguishing factors between the various types of magnetic resonance 2 “experiments. a For example, in the first magnetic resonance experiment, performed by eI. 1. Rabi and collaborators in 1939, a beam of atoms having J = + was == passed in succession through two very inhomogeneous magnets A and B = shown in Fig. 7.2. A homogeneous magnetic field existed in the intermedi- = ate region C where a radiofrequency (RF) held was applied. If a transition =. took place in region C from a state m = +3 to m= —3, that particular pee ge atom was deflected in an opposite direction in field B ad thus missed a the detector. Hence, resonance was detected by a decrease in beam current 2: when the frequency of the RF field was the appropriate one for the magnetic #:° field strength in C. : QA Ks LZ = | ate Detector # Giz { m=} aH = m=+} fen EP } Sz a FIGURE 7.2. The atomic beam arrangement of I. I. Rabi and collaborators used to detect = magnetic resonance transitions in atomic enerey levels, 254 #7 Magnetic Resanance Experiments nuclear magnetic resonance (NMR) experiments and in electron magnetig a resonance (called “electron spin resonance,” ESR) experiments. In exper iments with atomic vapors or transparent materials it 38 possible to detect: radiation (Am + 0) or by selective absorption effects. ae Apart from its intrinsic interest as a way of inducing transitions betwee . rere tn =o = the atom to which the nucleus belongs must have J = 3 material), since otherwise the nuclear spin would be coupled to J and thi ee jarge electronic mapnetic moment would mask the effect. By means. - 1 é eae ar ay ee a high accuracy. The NMR signal depends not only on the nucleus under study but also G1 the environment in which the nucleus finds itself. In fact the observation 6 nuclear magnetic resonance in solids and liquids depends on the relaxatioi of the nuclear spins through their interaction with the lattice. Thus, nuclear magnetic resonance studics have yielded a very large amount of informatio: on the properties of many materials in the solid or liguid state. S Soon after the first successful nuciear magnetic resonance experiments it was realized that the width of the observed resonance line for proton was mostly due to inhomogencities in the constant magnetic field used t split the energy sublevels. When a very homogeneous field was applied, the proton resonance line was shown to exhibit a fine structure on the order of: 245 0.01 G (10-° T). This structure “pens on the organic compound to which e ioe aa 2 ee = chemistry. The term electron spin (or paramagnetic) resonance is used for tran sitions between the Zeeman levels of quasi-free electrons in liquids and: solids. In principle, we should always measure a g factor of 2.00 (if we dea: ee 2 Pra ie | oan ; Sion Me Be Wa MOA tata tatanetecaten St beh ee o Pees ; 7th" 7.2 The Rate for Magnetic-Dipole Transitions 255 Swit free electrons); instead a great variety of g factors and stnicture appears in the resonance lines due to the different effective coupling of the electron =: with the crystalline field. These effects depend on the relative orientation =: of the magnetic field Bo and the crystal axis. Thus, electron spin resonance = is a very important tool in the study of crystalline structures as well as in =. the identification of free radicals in chemistry, medicine, and biophysics. This chapter is organized as follows, In Section 7,2 the conditions for = inducing magnetic-dipole transitions are discussed from both the quantum =-~and classical point of view. In Section 7.3 we introduce the mechanisms =: essential for the observation of energy absorption in nuclear magnetic = resonance and electron spin resonance experiments, namely relaxation = and saturation. We also discuss the idea of free induction decay and Se pulsed NMR. The techniques and results of nuclear magnetic resonance =: @ discussion of an electron spin resonance experltarst that operates at = qmicrowaye frequencies. As was the case in the previous chapter the discussion is limited, and Ee the reader may wish to refer to some of the many excellent monographs eS and texts on this subject. A list of suggested references is given at the end = of the chapter. © 72. THE RATE FOR MAGNETIC-DIPOLE TRANSITIONS 7.2.1. Quantum Calculation The experimental signals in NMR involve the participation of many nuclei. In this section, however, we wil] consider the effects associated with a single nucleus: we use the term a single spin. We will return to an ensemble of nuclei in Section 7.3. Let us consider, for example, a nucleus with angular momentum I (mag- nitude fi./7(/ + 1)) and magnetic moment z oriented along the spin axis. For nucle} it is customary to express the proportionality between the spin I and magnetic moment p by a = yh, (7.4) where y is called the gyromagnetic ratio; as can be seen from Eg. (7.6) below, y has dimensions of radians per second-tesla. The gyromagnetic 2% $7 Magnetic Resonance Experiments Vv Fg +1) By m=-3 a 3(§ 4B) m=-i Le ————— 3 (3 1. By) —ae ee [ cea m= —t ————- - 4. uBy) Cees ee oo - FIGURE 7.3 The energy of the four sublevels of a nucleus with spin f = 3 when, bre é se ina magnctic field Bp. Note that the energy “epenws on a “orientation” of the one i os 7 Ae att sr “ re r a, ratio y cannot be calculated from a en expression such as found. te : so that the energy difference between any adjacent sublevels (Am = + is simply : Thus for protons in 4 field of 1 T the resonance frequency will be Vo = 5.586un Bo = 42.58] MHz (Bp = 1 T). Taking the z axis along Bp we write the two components of A, as (A), = H, = Ay cos wt (Hi)y = Ay = A sin et, is Av= UME ff /(2) 2 wg /2m. mae ae ee eee, Lat a i 7.2 The Rate for Magnetic-Dipole Transitions 257 and we assume that : oly < Bo. = the additional energy of the nucleus, due to the ficid Hj, is RH m= = 0+ By = yh(Hele + Hyly) = (pe + Let), Be Cal hoaktily and l=, ily. (7.8) a a ON ee 4 Lely, 79) ce E where i and f stand for the initial and final state. As usual the matrix s element i is evaluated by performing the integral Z M= f ufHud?s at, (7.10) es = where {, is the perturbing energy of Eq. (7.7). We must include the time ee". dependence of the wave functions vy = ul, m’)exp (-i%1) L 7 yr; = uC, m) exp (i £1). (7.11) = Here primes refer to the final state, and u(/,m) stands for the hme- ee “independent part of the wave function. Evaluating Eq. (7.9) with the help ry a a a ee borne eta ay 3We expand the exponentials and obtain : Be (/y coset + ily(—1) sin wt) + (fy cos wt — tfy(4i) sin wt} oo = 2(/, cos wt + Jy sin @t). 4See, for example, E. Fermi, Motes on Quantwn Mechanics, Lecture 23, Univ. of Chicago Press, Chicago, 1961. Cr ee SSIs 258 7 Magnetic Resonance Experiments of Eqs. (7.10) and (7.11) we find that RH E— &’ : M = i (Lm (Te, m) J exp \-7 ; +0) | di oo — ES + i, m'\I_{Z, m) | exp | (= = 0) | ai} Bs The matrix elements of the operators J, and J_ are? m’| Lele) = =/i(it+H-— min + 4) } bn’, m+] (mr [L_|m) = fT +1) — mim — Y Sarm—1, and thus fy. connects only states with mm’ —m = 1 while IL sonnet states m'—m= ~1., Por f= + the above matrix elements reduce tok: 4 functions (but s¢ see e below) expressing the conservation of energy mde showing that the transition probability is different from zero only if Fi’ -~E=fea form’ =m+1 and | E-—E' =i for m’ =m — 1, (7.13) that is, when the angular frequency of the rotating field is equal to the energy . difference between adjacent m sublevels. Using Eq. (7.6), the conditions of Eqs. (7.13) become simply fiw = hy Bo = he. ES To complete the calculation of the transition rate we must integrate (the 2 absolute square of Eg, (7.12)) over the density of final states. This leads to: 22 Fermi’s golden rule® 2 “ee Rip = IM|?o(E), (7.14) 2 See E. Fermi (1961), Lecture 28, 6See FE. Kermi (1961), or L. Shiff, Quantum Mechanics, Chapter 8, McGraw-Fill, es New York, 1968. as heath ganna nna nna SAE TERRE ESSERE SIRS bye BY ae SCC MeN aT Shi th RRA RHERHER ARAN RRR 7.2 The Rate for Magnetic-Dipole Transitions 259 Ee ‘where R; is the transition probability per unit time (or transition rate) from = the initial state / to the final state f. In Eq. (7.14), Mis the time-independent =. part of the matrix element given by Eq. (7.12) (that is, without the integrals). =. p(B) is the “density of fina] states” and gives the number of states f per =<- unit energy interval that have energy close to E’. For example, if the final = state f has an extremely well-defined energy Ep, then p(E) > 8(E— Ep); :. if the final state has a certain width due for instance to a finite lifetime or :.. other broadening effects, then o(£) expresses this fact mathematically. = We require the function (£) to be nonnalized and can also express it in =. terms of frequency ] p(E) = phy) = i e(v) Bo with fowaz = few dv= 1). (7.15) Combining Eqs. (7.12), (7.14), and (7.15) we obtain for the transition rate : in the case J = 7 the elegant result 2 H? 4 R-1 2941/2 = Reij2s-12 = g(v). (7.16) = In the above equation v is the frequency of the perturbing field (RF or = mmacrowave), and g(v) gives the shape of the resonance line; note that g(v) will be significantly different from zero only for v *% vp. Note also that “in Eq. (7.16) and in the equations leading up to it, A, must be expressed in tesla, namely its value in amperes per meter must be muloplied by « the permeability of free space ti5. We have deliberately not included this : factor in the equations to avoid confusion with the symbol for magnetic moments, There are two important comments we want to make at this point. First as can be seen from Eq. (7.12) or (7.16) the rotating field Hy will induce =: transitions from m; = —4 to my = +3 with exactly the same probabi- =“ lity as from my = +5 to m; = —5. As a result, in che presence of the field H; both levels will, on average, be equally populated. This arzument ~ yemains valid for any value of the nuclear spin. Secondly, while we used a perturbative calculation the two-level sysiem can be solved exactly in terms of simple functions as described, for instance, in the Feynman Lectures, rec im aos — 260 #7 Magnetic Resonance Experiments Bis Vol. II, Lecture 30.’ We will make use of the exact solution in Section. 2 a 2 when we discuss pulsed NMR and free induction decay. : a 7.2.2, Classical Interpretation ee ee ee moment, given by t= px Bo = y(J x Bo). This must equal the time derivative of the angular momentum a NGa) =N. Hix: — f seat ha hehe ba in fe Baie a Sat tf ca a af ta ho eb bi Dt he Ban he rf To tnt TS te Cc A am a Battech aha hes a. : Sa Raa a at a ee er ee er ee ee ee on . RENEE Peete 264 7 Magnetic Resonance Experiments and since fim is always much smaller than kT, we may write for the above. equation oe N feo “e Ny — —. 7.26 * 2 kT \ 9 It is only these N, nuclei that can contribute toward a net absorption 0 energy, and the power absorbed from the RF field is given by _ N (Feo) Before proceeding further, we introduce some numerical values: for: protons y = 2.673 x 10° rad/s-T, so that for By = 1 T and T = 300 Ky Wi obtain N,_ ooh — (2.67 x 10°) x (6.6 x 107'®) eV N 2kT 2(1/40) eV : which justifies the approximation used to obtain Eq. (7.26). If we furthe consider a sample of 1 cm? of water, the number of protons contained i in: “itis x (fan) x R. 0.21 4x 1076, N = No X (2/18) = 6 x 10 x (2/18) = (2/3) x 107. If we use for R = 1/s (as can be seen from Ea. (7.16), this is a conservative ° ae value; R, however, can be as large as 10°/s as discussed below), we obtain a from Eq. (7.27) SHE a = (hag) x (1 x ion >) R~5x1l0%ev/s =8x 107 W. 2K Ty OEE (7.28) © This is a very small amount of power, especially since the applied radiofre- a quency field may be on the order of milliwatts. Therefore, a sensitive null =: method greatly facilitates the observation of nuclear resonance absorption. In writing Eq. (7.27), we assumed that the power absorbed is propor- =: tional to the number of excess nuclei which we now designate by ns; = however, as transitions are induced to the upper state, the number n, will * continuously decrease. The decrease will be exponential at the rate R . hy — N,ew ® Soon the populations of the two levels will be practically equalized, N41;2 N_1i/2, and po more absorption will be observed. arnt SRS RSNA ESN! oe wate SURE TENS ee a eal 7.4 Absarption of Energy by the Nuclear Moments 265 However, while the radiofrequency field tends to equalize the popu- lations, the “spin-lattice” interaction tends to restore the Boltzmann distribution at a rate characterized by 1/7). We say that the nuclei are “relaxing” through their interaction with the lattice, and the characteristic time 7; for this process 1s called the spin—lattice relaxation time. Therefore, in the presence of a radiofrequency field tuned to the resonance frequency, the number of excess nuclei at equilibnum mn, depends on T) and on R; if R << 1/7], thenn, > Ng, while if FR > 1/7,, ny > O. The value of n, can be easily obtained '9 ~ 14+2RT;' where N, (Eq, (7.27)) is the equilibrium excess of population in the absence of the radiofrequency field. By using Eq. (7.16) for R, we obtain Ns ie = — “144 p27? Tia) From the above result we see that when too much radiofrequency power is used, the number of excess nuclei ns, decreases, and so does the resonance signal. We say that the sample has been saturated, and the ratio n,/ Nz is frequently referred to as the saturation factor Z: (7.29) Mg (7.30) Ns | “S$ = le (7.51) Ns 14+5 y*H?T e(v) 10D ot == A41/2 — M-1/2 be the mstantaneous excess of nuclei in the presence of both radiofrequeucy and relaxation. The effect of the radiofrequency is to make n > 0 (F) = —2Rn, at RFE (The factor of 2 arses because each transition up decreases n+ /7 by 1, and also increases n—s2 by 1.) The effect of relaxation is to return 1 > Ng d(N, — 7) 1 dn a 7 MO = (F) Equilibnum iy reached when the sum of the two rates 1s zero; that is, which yields Eq. (7.29). 266 7? Magnetic Resonance Experiments The maximum useful value of the radiofrequency power therefapa=4 depends on the relaxation time 7;. For solids, 7; is large (it takes a long Lye time for the spins to reorient themselves in the equilibrium position), ayé=# therefore only weak radiofrequency fields may be applied. For exampl = for protons in ice 7; = 10* s. In contrast, in liquids, especially in solutiog : containing paramapnetic ions, the relaxation time for protons may be as ae short as T; = 1074s. aaa 7.3.2. Line Width and 7 Pat Just as optical spectral tines can be broadened by external factors cae Section 6.4) the NMR signal is not perfectly sharp but has a certain width Excluding inhomogeneitics of the magnetic field Bp over the size of the: ss yy sample, the principal cause for the line width is the interaction betweert:=: neighboring spins. In the classical analogy of Section 7.2.2 we say that the spin-spin interaction is destroying the phase cohcrence between the ae precessing spins and the rotating radiofrequency ficld. Another way of: thinking of the spin-spin interaction is that one nuclear spin produces. local magnetic field Bjypay at the position of another spin, which then find itself in a field ae By = Bo + Biocal and consequently has a resonance frequency w, = y B, slightly differen from wy. To estimate this effect, we calculate the magnetic field produced by a magnetic dipole one nuclear magneton strong, at a typical distance o O.l mm. HO\ HN Lo eh l — —__ = | — x =, ia) r3 (a) * 2M, Pr where jn is the nuclear magneton e/2M, and jig = 47 x 107? V-s/A-m is the permeability of free space. Numerically we find that Brocat ® ( Boca) ~ 5 x 107* T, which is a significant broadening of the line. In liquids and gases, however, : the reorientation of the molecules is so fast that the average local field is : very close to zero, and thercfore very narrow lines can be obtained. : In Egs. (7.15) and (7.16) we introduced the function g(v) to describe. Syne ta AR i Vera se. hide tio Hath sorta Waa to: % 7.3 Absorption of Energy by the Nuciear Moments 267 the spin-spin interaction. Since g(v)} has dimensions of inverse frequency, namely, of time, we define one-half of tts maximum value by 7> I 5 e(vg) = fr, (7.32) where vo is the resonance frequency in the absence of any broaden- we effects. 72 is called the transverse relaxation time. In view of the normalization condition (Eq. (7.15)), > f eo dv=l, (which also fixes the dimensions of g(v)), we see that a short 72 mmplies broad lines, whereas when 7> 1s long, the line is narrow. Using the definition of Eq. (7.32), we can then write for the saturation factor Z (Eq. (7.31)) at resonance L Zo = Z{vg) = —— ° (vo) [1 + y?H? TT] (7.33) It is of interest to estimate T> for protons when Bioca) = 5 x 1074 Tas found previously. From the uncertainty principle AF Ar ~ fA and the line width AE = ¥ Blocat sO that 1 Bocal (5.58 pty /fi) where we used yp = 5.58 and pny /h = 2 x 7.62 MHz/T (see Eq. (7.2)). Finally, as already mentioned, inhomogeneities in the magnetic field introduce spurious broadening effects that not only mask the fine structure of the line but also decrease the signal amplitude: hence the use of very homogeneous magnets and of the “spinning sample” technique. T) ~ At ~ ~7x107's, 7.3.3, The Bloch Magnetic Susceptibilities!! F. Bloch, who shared with E. M. Purcell the Nobel prize for the discovery of NMR, gave a macroscopic description of nuclear magnetic resonance, ‘This section may be omitted without a loss of continuity and the reader can procecd directly to the discussion of the experimental technique and results in Section 7.4. However, the discussion should be quite helpful for understanding the meaning of the “dispersion” curve as Well as the observed line shapes far both absorption and dispersion. 268 7 Magnetic Resonance Experiments where the effect of the RF field is accounted for by the polarization of thé nuclear spins. We know that when an electric (or magnetizing) field E (ar H)is applied in a region containing matter, the material becomes polarized: (or magnetized). We write es = a ad cry P=yE M=x,H, (7.34) where x, and x,, are the electric and magnetic susceptibilities. The polarizas:324 tion is due primarily to the alignment of the permanent electric (magnetic) #24 Be dipole moments of the atoms or molecules in the direction of the applied: field. Materials that have such dipole moments and exhibit large polar: ization should be called paraelectric (or for large magnetization, they are indeed called paramagnetic). a The refractive index of light is related to the electric and magnetic susceptibilities, since SE + 1 astababybababebeecerbeetstetatel, LY betes e= (1+ xe)€0 w= O+ Xp) ieo and _¢ _ YG/eoro) = /TF xi tx) to en) = (1 + xe} + Xp). The refractive index and therefore also the susceptibilities are a function of © the frequency, as is evident from the familiar phenomenon of the dispersion © of light. Thus the susceptibility at optical frequencies differs from the static: one and is a function of the frequency.!* Frequently the transmission of * light through matter is accompanied by absorption that may be strongest:’: at a particular resonant frequency. We may account for the absorption by. attributing an imaginary part to the susceptibility. : The same formalism can he used as well for the description of nuclear . magnetic resonance phenomena. The static susceptibility arising from.: the nuclear moments in an otherwise diamagnetic material differs from ° zero, but is very small and difficult to measure. For the radiofrequency : susceptibility, we write 3 X(w) = x'(w) — ix"), 12For optical frequencies and for almost all materials, x,, is and the variation inn arises 2: entirely from xX¢. 2 7.3 Absorption of Energy by the Nuclear Moments where both ¥‘(w) and x”{w) exhibit a resonant behavior when w reaches wy = y Bo. The real part ¥‘(w) is given by ] (wp — w)T> ‘(w) = ~ 1h. | ———————————_——_———— |, 7.35 while the imaginary part y”(w) is given by "(w) = t xowoT: (7.36) EOS TOON TT ay — oT + AEDT: | Here xo is the static magnetic suscepnibility defined as in Eq. (7.34) Mo = xofh. and 7; and f> are the familiar relaxation times introduced before; the term y?H fT T> appearing in the denominator is 4 measure of the saturation as defined in Eq. (7.31). Equations (7.35) and (7.36) are shown in Fig. 7.5 under the assumption that y?H°TiT2 < 1; they have the typical behavior of a dispersion and a power resonance curve, We also note that Eq. (7.35) is proportional to the derivative, with respect to w, of Eq. (7.36). By adjusting the detection equipment, we may observe experimentally either of those curves, or a combination of both, as a function of wo — w. Experimentally we can vary (a) (b) x’ in units ot 5 Y9mpTo x" in units of 2 xyc9 7 -4 -9 -2 -1 -4 -3 -2 -1 0 41 2 3 4 (aig—aa) Ty FIGURE 7.5 The radiofrequency magnetic susceptibilities near resonance. (a) The real part of the susceptibiliry exhibits a typical dispersion shape (Eq. (7.35)). (b} The imaginary part of the susceptibility exhibits a typical absorption shape (Eq. (7.34)). ane 270 )=6?)~Magnetic Resonance Experiments aan Bo fixed. 7.3.4. Free Induction Decay and Pulsed NMR” It is convenient to consider again the classical interpretation of NMR die ae Be cussed in Section 7.2.2. Refer to Fig. 7.4b and assume that the RF field 12 applied along the x’ axis in the rotating frame, for a short time t, such tage e wot= y Mt = = 2/2. Then the net magnetization vector M will be rotate? into the x’—y’ plane; in fact it will be along the y’ axis. In the laboratary::: frame this situation corresponds t toa magnetization vector rotating i In: the: - = Sy va a the individual spins that ¢ contribute to M become dephased either becausé: ee of held inhomogeneiues of oecanse Of ne spirusspuriiacraiawy Whew shai a spins are completely dephased (.e., when they are pointing uniformly ¢ tH df all directions in the x—y plane) dM/de through the coil vanishes and sa:34 does the induced signal This effect occurs on a time scale 7, which ee ny = the effects of the spin-spin interaction, magnetic field id inhoogeneity. and spin-lattice relaxation x SST RN “s 1 Th suet ie eee ch atric ttc ESS = + yABo. (7.37). 8 T| Me I3This section, too, can be omitted on a first reading without loss of continuity. How-" ever, it provides insight on the interpretation of wansient effects and of the modern. NMR techniques that are based on pulsed excitation rather than continuous wave (CW): 3 mcasurements, oe mater ats oe ised Bice SRERS x fs a on: oe aan is: es a ane ar: ae ho ate aie SERN oe 7.3 Absorption of Energy by the Nuclear Moments 271 Pick-up cail x1 2 TF, FIGURE 7,6 Free induction decay following a /2 RF pulse. (a) The magnetization vector “Mf in the rotating frame of reference before the application of the RF (¢ = 0). (b) After phe nf2 pulse, the M vector will precess in the stationary frame with angular velocity ap. £0) The induced signal in a stationary cou in the x-y plane will have period T = 2/uiy and will decay exponentially with time constant 7,". _ Therefore the free induction decay (FID) signal contains information on =~ poth the resonant frequency wp) (namely on y) and on Ts for the sample : © being investigated. Note that if one performs a Fourier transform on the FID signal, which = js acquired in the time domain, we obtain the spectrum of alf the resonant “frequencies of the sample. This is much more convenient and efficient than = searching for each resonant line separately. We now briefly return to the quantum-mechanical description of these | phenomena. It was mentioned in Section 7.2.1 that the response of a two- : level system to a resonant perturbation can be solved exactly in quantum om mechanics. If at t = O the spin jS in state m = 5 the probability of 7 finding it at time f in state m = +5 Is L P,1/2 = sin?(wt/2), (7.38a) : with @; = y A. The probability for the spin to remain in state m = —4 1S Pip = cos?(a1/2), (7.38b) : as it must be since for a two-level system it must hold that P_1;. + Pups. l4See foomote 7 of this chapter. 272 #7 Magnatic Resonance Experiments First we reconcile the result of Eq. (7.38a) with our perturbative calcu lation for the transition rate ablained in Eq. (7.16). The transition rate ig of course, the time derivative of the probability and we have . AP1/2> 41/2 _ 1 dt The perturbative calculation is valid when wt < 1, and therefore we ca expand sin cof to first order to find that yi. Sin yf = a $in @1f. AP.1j2341/2 _ y?H? ; at 2 over the time interval t. The maximum such time is given by Ty = = : BG Wy se (see Eq. (7.32)). Thus A AP -1j2-41/2 _ y Ay 7 —zT 8 (v9) as expected from Eq. (7.16); we recall that in deriving Eqs. (7.38) it was 3 x assumed that @ = a. Note also that Pee a APs. /2-4—1/2 _ a P_}/2-++41/2 g dt di ee as repeatedly emphasized for the transition rate. | Tt is clear that applying a m-pulse (w;f = mw) to a spin } in the state m= —§ will make P.1;2 = 1 and P_\;2 = 0; namely the spin will flip: states, as we also concluded from the classical analogy in Section 7.2.2. Sas However, what is the result of a 2 /2-pulse (w@;f = 2/2)? Then we find 34 that il ae Pay == = Pliys. Se 2 eo we Namely the spin is in a coherent superposition of the m: = +4 and m = —4 ee states. It is described by a wave function ee “ae — +2\4\m,=— 7.40) 2 = — fil _ = . . ade Jf2t| * 2 ; 2 es Been Ei % 74 Experimental Observation of the NMR of Protons 273 2 This represents a spin oriented in the x—y plane, and in the presence of : Be a magnetic field Bo along the z axis it will precess‘> in this plane with angular frequency #9 = y Bo. The quantum and classical descriptions lead eto precisely the same conclusions. PLN) et aMate etn ete’ rene erat t Se SSAA aa ° aah te tas ct ad AT el al ls dh ON ne eS rte bans = SRE RRR SNM NANA arnnoennnnhintrtema aay he Bea eal tae taba ed eke Nairn od) We conclude with the remark that the same formalism is used for atoms when two states of energy FE, and E> are connected by an clectric-dipole be moment 2. If such an atom is subject to an oscillatory electric field €) at the resonant frequency between the states, wo = (E ¢ — E;)/h, transitions will : e occur, Of course wp is now an optical frequency rather than RF frequency. = If the atom is initially in the state |i) and the optical field is switched on at = 0, the probability'® for finding the atom in the state | f) at time ¢ is Eid = sin? | —— Al P71) = sin (= t), (7.41a) 2. and for finding it in the state |7) : E\d ine fea P(t) = cos ( t). (7.41b) ge These are of course the exact analogues to Eq. (7.38). The precession frequency for the atomic case 2; = €)d/2h is called the Rabj frequency. With the availability of lasers one can achieve strong enough electric fields to generate 1/2, 7, eic., optical pulses. In this way atoms can be placed in specific quantum states. Such manipulation of single atoms has recently found applications in quantum cryptography, and it could eventually lead to quantum computing. 7.4. EXPERIMENTAL OBSERVATION OF THE NUCLEAR MAGNETIC RESONANCE OF PROTONS 7.4.1. General Considerations To observe nuclear magnetic resonance we need a sample, a magnet, a source of electromagnetic radiation of the appropriate frequency, and a detection system. 13See Das and Melissinos (1986) cited in Footnote 7 of this chapter, Shere we gloss over the fact that d is really the matrix element of the electric-dipole Operator berween the initial and final states. 274 7 Magnetic Resonance Experiments results, the inhomogeneities over the volume of the ample should be ese ees than 1/1000. The choice of the field strength is arbitrary, Provided be tes and therefore give weak signals (see Eq. (7.33)). Similarly it is profitable 2 _ to have a narrow line; hence materials with long spin-spin relaxation time oe Tz are chosen. Liquids will meet this condition, and in most instances the width of the line will be determined by the magnet inhomogeneity #2 (T = 3 x 107% 5 will give for protons a line width of 107° T), Plain tap water makes a good sample, or tap water doped with 1 wt% manganese nitrate Mn(NO3)2 or copper sulfate. The size of the sample is limited by the area over which the magnet is homogeneous, but also by practical considerations of the coil used ta couple the radiofrequency to the sample. In usual practice a 1-cm? sample is adequate; it is contained in a small tubular glass container, around which ig: wrapped a radiofrequency coil as shown in Fig. 7.7a. The whole assembly: is then inserted into the magnet gap and should be secured firmly, since vibration is picked up by the coil and appears as noise in the detector. In deriving the probability for a transition between the m sublevels, and in all our previous discussion, we have assumed the existence of a rotating field at the angular frequency w close to wg. In practice, a magnetic field”: Bee oscillating linearly as A sin wt is established in the interior of the radiofre-®: Le quency coil (Fig. 7. a). Linear harmonic motion, however, is ee to ee cats, ‘aaa since a A A Acos wif, = 7 (eos wim, + Sin winy) + 73 keos(— ofin, + sin(—wr)iny) (7 42). : ens ae ings Cr ey Sueuies peaaneaangaaannagnanans ah 1 we rac) wt 7.4 Experimantal Observation of the NMR of Protons 275 (a) Polefaces (b} 2A sh 60~ Helmhoitz cai!s a FIGURE 7.7 (a) Schematic arrangement of a nuclear magnetic resonance apparatus. The c. sample is placed in a homogeneous magnetic field and radiofrequency is coupled to it by Be means of the coil. The Helmholtz coils ace used to modulate the constant magnetic field. “'(b) A linearly oscillating field of frequency @ is equivalent to two fields rotating in opposite “directions with the same frequency w, =: where my and ny are unit vectors in the x and y directions. The component =: rotating in the same direction as the precessing spins wil] be in resonance = amd may cause transitions; the other component is completely out of phase ==. and has no effect on the sarnple. When the radiofrequency reaches the resonance value wo, energy 1s se absorbed from the field in the coil and this fact is sensed by the detector. Because of the low signal l¢vels involved and the difficulty of maintaining a very Stable level of radiofrequency power it is advantageous to traverse =: the whole resonance curve in a relatively short time. This can be achieved =’ either by “sweeping” the frequency of the radiofrequency oscillator while Maintaining the magnetic field constant, or by “sweeping” the magnetic field while the frequency remains fixed. In early NMR experiments as =. well as in this laboratory the choice is to sweep the field with a pair of =, Helmholtz coils,’’ as indicated in Fig. 7.7a, because it is easy and does not = require fancy frequency generators. The sweep coils are fed with a slowly = -yarying current,'® which results in a modulation of the magnetic field B. = If this sweep covers the value of By, which is in resonance with the fixed ‘=. frequency of the oscillator, a resonance signal modulated at the frequency of WA pair of coils of diameter d, spaced a distance 4/2 apart and traversed by current in the same direction, produce a very homogeneous field at the peometrical center of the me configuration. '8In the absence of a sweep generator and audio amplifier the 60-Hz line voltage can be i used through a variac and an isolation transformer. Ce ei deh 276 0«=6©7 )0~Magnetic Resonance Experiments rt Generator Alienuator Receiver Ss Filament transformer ST fat tn etter nate hh at Data Sarr art . . he SS . rj eat on ae Sas ih mane Helmholtz coils FIGURE 7.8 Block diagram of the nuclear resonance measuring apparatus. SS ash 110V ac ne the sweep will appear at the detector. A modulated signal has the advantage: of easier amplification and improvement in the signal-to-noise ratio by!2 using a narrow bandwidth detector. oo The radiofrequency oscillator and detection circuit can be of severak 24 designs. Today, commercial frequency generators “are“usdu au powide lS the RF drive and low-noise amplifiers for the detector. A single coil is: used as both a transmitter and receiver. A block diagram of a CW NMR: apparatus as used in this laboratory is shown in Fig. 7.8. The signal = was detected by a bridge circuit; this arrangement has great sensitivity but can be used without retuning only over a fairly narrow frequency == range, & Commercial magnetometers often use a “marginal oscillator” circuit = where the oscillator and detector are combined in one unit. In this design the RF power is kept low so as to allow the direct observation of the = absorption, as well as to avoid saturation of the sample. To cover a wide =: frequency range the coil containing the sample is changed since it is part : of the resonant circuit that sets the oscillator frequency. A unit suitable for ”: lahotatary demonstrations is available from Klinger Educational Products, ° as well as from other sources. 7 LS ae Le SS me ere eee Te ce Sey Siar eae a ee ee ae ee a ae Rear t ie eete ea he Lee BN LR SCRE b 7.4 Experimantal Observation of the NMR of Protons 277 = 94.2. Detection of Nuclear Magnetic Resonance with a Bridge Circuit =. The coil in which the sample is located is part of a resonant circuit with high QO. The Q value, or quality factor, of a device is defined as 27 times the ratio of the time-averaged energy stored to energy dissipated, in one : cycle. Por a coi! of inductance £ and resistance R, 27a on ae (7.43) R When resonance ts reached, the real part of the magnetic susceptibility (Eq. (7.35)) changes, and thus the inductance of the coil also changes. . Alternatively, an increase in the imaginary part of the susceptibility (Eq. (7.36)) corresponds to the absorption of power from the field and thus to increased dissipation and therefore increased resistivity of the coil. This small change in the Q value can be detected with a bridge circuit, as shown in Fig. 7.9. The radiofrequency voltage is applied between points a and ¢g (see Fig. 7.9a), and therefore radiofrequency current flows through the load L and the dummy branch D; if the bridge is balanced, no voltage should appear at the point d (since b and ¢ were in phase and of the same ampli- tude, and the signal from c and d is shifted by A/2). Any slight unbalance of the bridge produces a small voltage at d. The actual bridge circuit is shown in (b) of the figure. The R’C’ elements are effectively generating (b) & ws FIGURE 7.9 A radiofrequency bridge circuit that can be used for the detection of nuclear magnetic resonance. (a} Schematic arrangement; note that LZ is the radiofrequency coil, The 1/2 line ascertaing cancellation at the output of the signals from 6 and c. (b) A practical radiofrequency bridge circuit. For resonance conditions see Eqs. (7.44) of the text 278 43=6©7 0 Magnetic Resonance Experiments the 2/2 phase shift and ZL is the sample coil. The conditions for balanc Resistive balance: oC Cx {1 +C'/ Cy’) R’Rp = 1 Reactive balance: C4+C,4+ C2 (1 + Cy/Cy') = 1/Lw*, ( where Rp is the Ce resistance of the coil. The bridge is balanced cig i mode, whereas in Fig. 7.1 Ob it was balanced resistively. The sweep, iene ES from the 60-Hz line voliage, comespones to approximately 10-4 TAdivisiony 7 (a) ' (b) — Sweep=5 10-4 cm/sec = 1 gause/em {at the center} FIGURE 7.10 Results obtained from the nuclear magnetic resonance of protons using’ bridge circuit: (a) Dispersion curve and (b) absorption curve. The oscilloscope sweep was: as linear at 0.5 ms/cm, which corresponds to approximately 107+ Tfem at the center of: ahi SWEep. Sat at Pate tet tetany hey oh DUNT . “ae SS iat SAE eee ee ee Bee e ese erie Neanenaenctssarsnatatet IDR IRIES IOS aoe orcas 4° 4,467 BoP, 74 Experimental Observation of tha NMA of Protons 279 =~ Using a rotating coil flux-meter at the field position previously occupied = by the sample, the magnetic field at resonance is found to be Bo = 0.6642 + 0.0020 T, = and hence 2 y = 20 = (26.618 £0.08) x 10" rad/sT | (7.45) 0 . = in good agreement with the accepted value ¥ = 26.73 x 10’ rad/s-T. ~ Clearly, it is much easier to measure ratios of nuclear moments to high ® accuracy than to establish their absolute value to the same accuracy. To obtain the g factor of the proton—that is, the connection between ", Magnetic moment and the nuclear magneton—we recall that =glpn. i Thus from Eq. (7.4) yA y | =— = — — = 5.56+0.02, oun 2m unl h e where we used the derived value of y (Eq. (7.45)) and jzw/ hk from Eq, (7.2). =. We have measured the proton magnetic moment of the proton to an accuracy = of 0.4%. = 7.4.3. Measurement of T? = In this laboratory no pulsed NMR experiments were carried out. However, EASA = under certain conditions one can observe the free induction and its decay = with a CW apparatus. This happens if the field is swept rapidly enough => through the resonance, in which case wiggles such as those shown in Fig. = 7.11 appear. The interpretation follows the discussion of Section 7.3.4, Far from = resonance the field seen in the rotating frame is Bp, i.e., along the z axis. =. As resonance is approached the Bp field is canceled in the rotating frame >. and only Ay is present. This results in rotating the M vector into the x’/—y’ ~ plane. After the resonance is traversed the effect of Hy is again minimal, - but the magnetization remains in the x’—y’ plane, and it induces a signal at 260 7} Magnetic Resonance Experiments 45 2.25 sac 2xi079 i974 t t—-«— 0.21079 seciom {a) Linear sweep {b) FIGURE 7.11 Nuclear magnetic resonance signals of protons obtained with a 1 oscillator circuit. (a) The sample is water-saturated with LiF, (b} The sample is water-dopi with manganese nitrate. A linear sweep of the same speed is used in both cases. a frequency w(t) = y Bo(t), which differs from wp. The two frequencigg w(t} and wo beat against cach other, and this gives rise to the wiggles, Ont can clearly see that the frequency difference increases (the beating period shortens} as the field is further away from resonance. The effect that ig: relevant for our measurement is the exponential decay of the envelope:of the beat osciHations. : We still nst explain the wiggles that appear in Fig. 7.11a before the resonance 1s crossed. These are present because the spins have nat dephased ee by the time the sweep is restarted and continue to rotate in the x—y plang: ae Indeed they are absent from the trace of Fig. 7.116 where the water samplé ee was doped with manganese nitrate as compared to water-doped with LiF oe in the sample used for Fig. 7.1la. The shorter 72 in part (b) of the figuie ss leads to more rapid dephasing. oie If a linear sweep is assumed, the beat signal has the form ae 1 4H Je TE —— 1? 7.46): 2% ‘cos 3 ht | 746) Bs where ¢ = 0 when the resonance is traversed, Note also that the beat frequency increases with time since 2 pe _1l. da ‘Se Me ae ne From a measurement of the wiggle envelope, information about 7," can be. ee obtained, This is shown in Fig. 7.12 where the data are well fitted by amt: sdettotenandce se Reman SSeS TR 74 Experimental Observation of the NMR of Protons ~~ 28% aa Veet aia ww & 10° 3 Pr a ca ra} fan] in| =| x A 10' a) 0.4 a2 0.3 0.4 0.5 0.6 07 0.8 Time f¢ (ms) Be FIGURE 7.12 Sem#log piot of the aniplitude of the “wiggles” of the resonance signal = shown in Fig. 7.1la plotted against time. It yields an exponential decay of the amplitude oie with a time constant Ty = 2.4 x 107% s. ES exponential yielding TY =2.4x 10's, When we convert the measured value of T; into a magnetic field (see Eq. =:.(7.37)), we find that DORR RHR SAO nen estat Ce ca Le 2 5 2A Bg = Ty = 32x 10 T, “ ' =. namely, that an inhomogeneity of the magnetic field, over the size of the :. sample, of 0.32 G is sufficient to cause the wiggles observed in Fig. 7.11a. : We also conclude that 72 for this sample is longer than 2.4 x 107s. = 144, The Effect of T; = In Fig. 7.13 we show a very simple marginal oscillator circuit!” that is ade- quate for demonstrating NMR signals. The first transistor supplies constant 9p R. Singer and §. D. Johnson, Rev. Sci. Instrum., 30, 92 (1959). 262 7 Magnetic Resonance Experiments 5 pF mt r, (2N502) A P B(IN5@) 7, (2N247) 10 K roo FIGURE 7.13 A simple transistorized nuclear magnetic resonance circuit. the NMR signal increases with increasing RF power until the RF ample tude reaches approximately 0.5 V. Beyond this point the signal decreases because the sample is saturated. From a knowledge of the @ of the?=% coil one can convert the RF amplitude to the corresponding value of thé es rotating field H; and thus use the data to find the spin-lattice relaxation B time 7). Note also that once the sample is saturated there is sufficient magneti: x zation left in the x—y plane to begin showing a beat signal (wiggles) after Z passage through resonance (see Fig. 7.11b). For convenience the time scal@:22% on the oscilloscope trace in Fig. 7.14 was set to cover a full cycle of the ae 60-Hz sinusoidal sweep. 8 75 Elactron Spin Resonance 283 20 mV/cm (a) rf level 0.125 V 0.2 V/cm (b) rf 0.2 V 0.2 V/cm Netasy 0.2 V/cm (d) rf O.4V 0.2 V/cm (e) r§ 0.66 V Sample [s saturating secx 107+ FIGURE 7.14 Nuclear magnetic resonance signals from protons obtained with the cir- cuit shown in Fig. 7.13 as a function of the amplitude of the radiofrequency. Note that initially the output srgaal increases with increasing radiofrequency amplitude but at a Icvel of approxunately 0.5 V the sample is saturated and the signal begins to decrease, The signal of 0,5 V is shown in Fig. 7.! 1b. 7.5, ELECTRON SPIN RESONANCE 7.5.1. General Considerations So far we have discussed transitions between the energy levels of a proton or a nucleus in the presence of an external magnetic field. Transitions between the energy levels of a quasi-free electron in an external magnetic field can also be observed. We refer to this case as electron spin resonance (ESR) 244 =%7 Magnetic Resonance Experiments NMR. Namely for similar laboratory fields the resonance frequency i oe ee - the microwave region, ae serine find electronic states with J #0: this i is due to the fact that in the chon a binding of atoms into molecules, the valence electrons get paired off, so thakzies each atom appears to have a completely closed shell, For example, in NaGe? es . the sodium has a *S1/2 electron (n = 3, = 0) outside closed sheils, andzie4 the chlorine has a? P3/2 clectron hole (n = 3, 1 = 1) inside closed shell’2: Bes However, in the NaCl molecule, the sodium appears as a Na? ion,’ - hence presents a closed shell configuration, whereas the chlorine a paramagnetic. An sample is the compound copper sulfate (ColSOV), 1 — which the double valence results in a Cu2+ ion. For copper then = Ly 3 jon has?! = 2,5 = i, and, consequently, J #0, so that it does possess: é ] magnetic-dipole moment. In an external magnetic field, the ground. stats 2 ee Pee iat strong ‘and narrow resonance line, with a g factor very close to 2.00 (he fre 27 ee ae resonance is also observed in other materials where unpaired electrons may oo 7 eee 75 Electron Spin Resonance 285 2 NO: N NO, : a NO, Be FIGURE 7,15 Chemical structure of DPPH (diphenyl-picryl-hydrazil), (CgHs)2N- Be NCgH2-(NOp)s. Be. exist, such as crystals with lattice defects, in ferromagnetic materials, and ee. in metals and semiconductors, ge The much higher frequency of the ESR transitions is advantageous f=: because the energy absorbed from the microwave field for every transi- : fae tion is much higher than that in the NMR case, thus leading to a much E- jmproved signal-to-noise ratio. Furthermore the separation between the f- energy levels is much larger, so that they remain resolved despite their Ee large intrinsic width. ee The resonance condition is detected, as in the case of nuclear magnetic e-. resonance, by the absorption of energy, and for this reason solids and liquids ee: are much easier to study than gases with their very low densities. Much of ; fs our previous discussion on transition probabilities and relaxation mecha- “2: nisms is equally applicable to electron paramagnetic resonance. However, % the population difference between the energy levels (see Eq. (7.26)) is =: much larger because of their greater energy spacing. A difficulty with ESR a is that the width of the resonance line may be prohibitively large, since both =: the spin-lattice and spin-spin interactions are stronger than in the nuclear = magnetic resonance case. In order to reduce the line width, the sample may = be cooled to low temperatures (lengthens the spin-lattice relaxation time) =. and/or the paramagnetic fons are diluted in a diamagnetic salt (lengthens the = spin-spin interaction time by effectively increasing the distance between = the spins). ‘a When measuring electron pararmagnetic resonance lines in solids, 2 great = Variety of g factors are obtained. This is due to the differences in the “= coupling of the unpaired electron’s spin with the orbital angular momentum; =. the strength of this coupling depends very much on the position (in energy) %. of the adjacent levels of the ion as they are modified by the crystalline field. =. Purther, the electron paramagnetic resonance lines show hyperfine structure 286 7 Magnetic Resonance Experiments in a liquid solution. 7.5.2. The Electron Spin Resonance Spectrometer 10-GHz region (A; ~ 3 cm). Microwave components and “plumbing” ae co readily available. A schematic of the spectrometer is shown in Fig. 7.16and: at first appears quite claborate.?° However, the basic components showity eS iat & = 0.400 in. Bee The microwave source (block A) consists of a Varian X-13 klystron or be controlled by the KLSP modulator, and this feature is used io “Jock” the kKlystron frequency onto that of the external reference cavity shown i in: block C together with a phase shifter and tuner, There are also provisions e for measuring the wavelength. Detection i is accomplished i in block D, by o pare the sample signal with the reference frequency. Block E indicates: - the magnet power supply and a set of Helmholtz sweep coils, which are Q field, | 204 very simple ESR demonstration apparatus operating in the RF range, and thus at very weak field, is available from Klinger Educational Products. deg gan can REE ESR Ta eb ob ek NN Paar eo ed eae SSS as AMIN SETAE NCR RST ITHSPATEER ERLE RURAL HERRERA CRE SURES BRE RSE e wees, a SA ACCC NETS sh sphere heh eee eE nts SSN te a WS ANS ae Ne Sha : ae eC MCRAE! ras rer Sees Bets ace aS woke on Ne anne ees nt 4 . aa 2 hy a aie ome a ak aC Rr are eae aa Oe ee wey hgh My fas ate perarer thn: rs ltt ial tae eas ee ae eS s orate ta’, ont) i atte i os oe * Se REE 7 7: . ‘ " i. . ‘ * " ™ 1 om a . © . . ‘ ‘ Aa a F . 7 7 ioe 7 oa =a Fy F . . Cahbrated Pasig supply milenuaelar ee a a a nn rg 16 cb spiluier € FIGURE 7.16 Schematic of the X-band ESR spectrometer. 738 «=6p 7) «Magnetic Resonance Experiments We now elaborate on some of these components: without attermation and the wavelcngth A» in the guide js given by l 1 l 1 (m/a)* + (n/bY* — ; ~ 2 ae 2 4 of the guide; m and n are integers. Since a = 2.29 cm, & = 1.02 cm, and Ag ¥ 3.2cm we find that only the #2 = 1, m = 0 mode can propagate, and Ag = 45 cm. a In this mode, the electric field is completely transverse to the axis of th guide; this is called the TE19 mode. The field lines for the traveling Teg to the field strength. Top View Sids view Cross section al AB RHEE ~~ X Parspectrva y xe z FIGURE 7.17 Configuration of electric and magnetic field lines for a traveling wave in a: rectangular waveguide. A, is the wavelength in the guide. ee Rene nee ESS a —-—— Electric field --e—-— Magnetic field HSS NSE ice eee tea ist r we ee Sa 75 Electron Spin Resonance 289 ee f= (b) The Microwave Cavity and Sample. The cavity can be made from Bc: a part of the waveguide ending with a shorting stub, to set up a standing fe wave. The sarople is placed so as to be located in the middle of the magnet B= =:- polefaces, and then the (shorting) sliding stub is adjusted so that maximum ee B field exists al the sample. From the configuration of the standing wave Be pattem, maximum B field occurs at a distance x from the short, where De 1p . vm =(5 +3 J 4 ee ( 1) ; = with p an integer. Since the microwave field must be normal to By it is = preferable to place the guide in the magnet with its wide side parallel to = the polefaces. = (c) The Magic Tee. This is the heart of the bridge circuit, and is used to ; fe compare (interfere) microwave signals. It can be used in different configu- = pations but in the spectrometer used here it is set up as shown in Fig, 7.18. Let 2, be the reference field and £2 the signal field. The power at Dr = and Dy, are eee SAT " A — ,' INN ee Pa = |E\ + Ea)? =|E\|" + |Ba|? + 2Re (£) ES) a PL = [Ey — Ea|* =|, |? + |Ea|? — 2Re (Ey, E3) . Reference ‘ we ageeere RTS eae —- FIGURE 7,18 The magic lec used in the ESR spectrometer. A reference field and signal field are mixed within the tee to provide a sum or difference in the odtput arms of the tee. a py SASS ANS SOO TEE LIL LAW Pike PO EEO ENNELY WEEE APN OLE : Oh CRNA Saha a ee aaeeerstattatstotatat stat itetantar eter ete Fe ee i AU TE el ee ee le eo INS Nt aL St ones rote 200 7 Magnetic Hesonance Experiments S = 4Re (£1 £3). E) = Epe?®, Ep is real. Ey = Eg(l+ x). Fo is real. ; 3 Then recalling from Section 7.3.3 that x(w) = x’(@w) —ix”(@) we find: e that as = (4EpEo)(cos8 + x’ cos + x” sin 9]. (7.48 By selecting the phase of the reference signal, we can select the desired S curve. For @ — 0 we obtain 2 = (AER Ep)[1 + x'()]. Since only x’(@) is modulation dependent the signal follows the dispersive: curve (Eq. (7.35) or Fig. 7.5a). For @ = 1/2 we obtain 5 = 4(Ep Eg) x" (w), namely the absorptive part (Eq. (7.36) or Fig. 7.5b). If the reference phase: is not set properly the signal is a mixture of the two curves as given by: Eq. (7.48). " (d) Detection. One can use bolometcrs in the two arms of the magic tee.: These are devices where the resistance changes as a function of incident: power and are quite sensitive. It is, however, simpler to use in the magic lee microwave diodes similar to those used elsewhere in the spectrometer for diagnostic purposcs. ra (e) Lock-In Detection. If more sensitivily is required one modulates the Bo field and sends the difference of the signals from the two anms of the 2224 magic tee to the lock-in detector. When the modulation width is much,’ “228 less than the line width the detected signal represents the derivative of | the absorption (or dispersion) curve. This can be seen frorn the sketch of. oe 7.5.3. Experimental Results Results obtained by students are shown below. The magnetic field was ae modulated at 1 kHz and the lock-in detector was used. The modulation “22 a 7.5 Elactron Spin Resonance 231 SA yh.t%P, el eeltiets IN Detector current c >; AS} 5! a 3 v 5 o #5: a= ~o -= ge. FIGURE 7.19 Effect of small-amplimude field modulation. The output is proportional to "the derivative of the absorption curve and is maximum at the points of inflection, tei ae 5 a Sa ee > BP a en Cc ae Ao ty w ~{ -3 -5 0 20 40 60 80 100 FIGURE 7.20 Resonance signal for DPPH as a function of the magnetic field, A small modulation was applicd to allow lock-in detection, and therefore the signal gives the derivative of the absorption curve. amplitude was kept low so that the derivative of the absorbtion line was * observed. The field was swept through the resonance by slowly ramping _ the magnet current. The frequency was measured by using the wavemeter, and the magnetic field by using a Hall probe. Figure 7.20 shows the results for DPPH. The field measured at the two ends of the sweep*! was B(O0) = 0.3370 T and B(100) = 0.3480 T, 2'The number in parentheses refers to the markings on the x axis of the computer plot. 292 $%7 Magnetic Resonance Experiments The field on resonance is Bo = 0.3402 + 0.005 T, ‘ where the error arises from the error in the Hall probe calibration. The frequency was found to be vo = 9.578 + 0.010 GHz, the error reflecting an estimate of the accuracy of the wavemeter calibration, Thus hvo 1 9.978 GHZ no 4.0.03 (7.49) Z im good agreement with the accepted value 8DPPH = 2.0036. The width of the line is fairly narrow, of order AB = 8 x 10-4 T at full eee width, “ae Figure 7.21 shows data for a CuSO, sample under the same conditions, “238 The frequency is the same as before but the sweep of the fieldis much wider. = It extends from B(O) = 0.2690 T to B(100) = 0.3750 T. The central field is found to be | Note the large width of the linc. Bo = 0.3146 + 0.005 T, r EPR Se & i a “B 0 20 40 60 a0 100 FIGURE 7.21 As described in legend to Fig. 7,20 but for a Cu{SO,q)-7H20O sample. 4 7.6 References 293 h j 578 GH 8cuSO, = ids 9.978 GH? — 2.17+0.05, where the increased error is from locating the center of the line. This result lies between the known values of the two g factors of the Cu?* ion.?? What is strikingly different from the DPPH sample is the width of the line, which is A Bo = 290 x 10-4 T. This is a clear indication of the effects of the erystalline fields in broadening the energy levels of the Cu** ion. 7.6. REPERENCES A, Abragam, The Principles of Nuclear Magnetism, Oxford Univ, Press, Oxford, 1961. An outstanding work on nuclear magnetic resonance, where the treatment is theoretical and advanced, but very complete and clear. E.R. Andrew, Nuclear Magnetic Resonance, Cambridge Oniv. Press, Cambridge, UK, 1956. A sharter text conLaining experimental details as well; it is very useful to students in this course. C. H. ‘Townes and A. L. Shawlow, Microwave Spectroscopy, McGraw-Hill, New York, 1955. An extensive and comprehensive work on the subject, mainly treating che molecular specera obtained in gases. GE. Pake, Paramagneric Resonance, Benjamin, Elmsford, NY, 1962. D, J. E. Ingram, Spectroscopy at Radio and Microwave Frequency, Butterworth, Stoneham, MA, 1955. Very helpful for the study of paramagnetic resonance in solids and crystalline materials. E. Fukushima and 5. 8. W. Rodder, Expertmenral Pulse NMR, Addison-Wesley, Reading, MA, 1983. *2For a crystal the ¢ Factor depends on the onentation of the crystal axis with respect to the magnetic field. The sample used here was crystalline (powder), and therefore one cannot observe the two ¢ factors, gy and 2) . PRUNE ROR RNS che MH SR Sc Te CSUR Nh ss baht nies aS 1h ty SuRRA NER IR eS CHAPTER 8 Particle Detectors and Radioactive Decay 8.1. GENERAL CONSIDERATIONS The terms radiation and particle used in this chapter require clanfication, The term radiation here designates electromagnetic energy propagating in space (crossing a given area in unit time), but specifically of a frequency higher chan that of the visual spectrum, namely, X-rays and gamma rays. Visible, infrared, microwaves, and radiofrequency waves are not included. Because of the quantumm-mechanical aspects of the electromagnetic field, such radiation can be described by a flux of (neutral) quanta, the photons, with an energy E = Av and a momentum p = hv/c, where v is the frequency of the radiation. These quanta iteract with electric charges, and ". the probability for such interactions is of the same order as that for the interaction of two charges, The term particle here encompasses all entities of matter (energy) to which can be assigned discrete classical and quantum-mechanical properties, such as rest mass, spin, charge, lifetime, and so on. The use of 295 255 8 Particle Detectors and Radioactive Decay. the term “particle” is not always clear: for example, we speak of a hydropag= 245 molecule, whereas we refer to the nucleus of the hydrogen atom, the pro: E ton, as a particle. SinuJarly, the electron, the neutron, the (almost) massless: neutrino, the x meson, etc., are referred to as particles; the same termi frequently used for a fission fragment, a helium nucleus, or a heavy ion The visualization of a particle is that of a massive point describing a certai trajectory under the influence of external forces and initial conditions; thi provides a useful model for many calculations. Since particles have dimensions on the order of fermis ( 10-8 cm), the’ trajectory of a charged particle can become visible and be permanently ee z ee recorded. Thus, a particle detector, or radiation detector, is a device that: produces a signal (intelligible to the experimenter) when a particle or phos 222% ton arrives; if the device reveals to the expertmenter the whole trajectory: aes of the particle, it is called an imagze-forming detector. All detectors are based on the electromagnetic interaction of the charge Of the incoming particle with the atoms or molecules of the detector. The dif-: 48 ferent types of interaction (ionization is the most common) and the different: principles of amplification of this interaction distinguish the different types: of detector. Neutrons, however, are detected through the interaction of the” charged particles of the detector to which they transfer energy. This occurs either through elastic collisions of the neutrons with protons (hydrogenous’ materials}, or through neutron capture in certain nuclei, or through the production of fission by the neutron: for example, n+!°B—>?Li+a. In the following discussion we will be concerned with signal-producing devices, which we classify as follows: (a} Gaseous ionization instruments, encompassing the ionization cham- ber, the proportional counter, and the Geiger counter, (b) Scintillation counters, (c) Solid-state detectors, and (d) Other detectors. Such detectors can be designed so as to respond to the passage or arrival of a single particle or quantum. They can also be used as integrating devices 1High-energy electron-scattering experiments (which serve as a sort of microscope} 2: have, however, revealed muchabout the electromagnetic structure of the proton and neutron. oe ‘ 2 : ne a ' a ; : oe af “a re a Pr ; : : ; Be am i - a a fe ee a Ped . 7 . 1 Fated ve ae . - . "a — ee +e “a 1 ae 5 ore 9.1 General Considerations 297 = (as is frequently done with ionization chambers), giving a signal propor- = tional to NE, where N is the total number of particles crossing the instni- = ment per unit time and E the average energy deposited by each particle. = Jn evaluating a detector, the following properties are taken into ". eonsideration: = (a) Sensitivity, which defines the minimum energy that must be depo- :. sited in the detectar so as to produce a signal; related to it is the.signal-to- noise ratio at the system’s output. (b) Energy resolution, in certain detectors, which are large enough to stop the particle; the signa] may be proportional to the tnitral energy of the particle. In other cases the velocity of the traversing particle can be mea- sured, as in Cherenkov counters, or in dE/dx (ionization per unit length) detectors. :. (c) Time resolution, which charactenzes the time lag and time jitter ' from the arrival of the particle until the appearance of the signal, and the distribution in time (duration) of the output pulse; related to it is the dead time of the device, that 1s, the period during which no (correct) signal will be generated for the arrival of a second particle. (d) Efficiency, which specifies the fraction of the flux incident on the counter that is detected. It usually is fairly high for charged particles, but can be as low us a few percent for neutral particles and for photons. Particle detectors play a most important role in nuclear physics, and in many of the expenments described in this text some type of particle detector is used. Just as the spectrograph was the paramount instrument of atomic physics, so the Geiger counter and, later, the Nai scintillation counter have been the paramount instruments of nuclear physics.” In the following séctions, we first present a brief discussion of the interaction of charged particles and of photons with matter. Then gaseous ionization instruments are described with specific emphasis on the Geiger counter. This is followed by a descnption of the scintillation counter and the measurement of nuclear gamma-ray spectra. The following section deals with solid-state detectors and the measurement of the specific ionization of polonium alpha rays in air. Other detectors are mentioned, and some specific experiments using these detectors are described. “It is interesting that the first particle detector ever to be used (by Rutherford in his alpha-particle scattering experiments in 1910) was a scintillating screen, a technique that came again into prominence after 40 years. 298 «= 8s «Particla Detectors and Radioactive Decay Finaily, note that precautions should be taken when handling radioactive sources, We recommend that the reader review the material on radiation safety in Appendix D before undertaking the measurements destribed-j in this chapter. 2 $8.2. INFERACTIONS OF CHARGED PARTICLES AND PHOTONS WITH MATTER 3.2.1. General Remarks As already mentioned the interaction of charged particles and photons with matter is electromagnetic and results either in a gradual reduction of energy of the incoming particle (with a change of its direction) or in the absorption of the photon. Particles such as nuclei, protons, neutrons, and 7t-mesons, are subject to a nuclear interaction as well, which is, however; of much shorter range than the electromagnetic one. The nuclear interac- a tion may become predominant only when the particles have enough energy to overcome Coulomb-bamer effects. A nuclear mean free path, which is approximately 60 gicm*, is the distance over which the probability Jor a a nuclear interaction is of order unity. Heavy charged particles lose energy through collisions with the atomic electrons of the matcrial, while electrons lose cnergy both through col- lisions with atomic electrons and through radiation when their trajectory is altered by the field of a nucleus (bremsstrahlung—see Section 8.2.6): Photons lose energy through collisions with the atomic electrons of the-:: material, either through the photoelectric or the Compton effect; at higher’ : energies photons interact by creating electron—positron pairs in the field of a nucleus. & A brief review of definitions wil! be helpful. (a) Cross Section. We define the cross section, o, for scattering froma “24 single target particle as oes: _ scattered flux (3 1) : Q incident flux per unit area’ Oe Thus o has dimensions of area (usually cm?) and can be thought of as the Be area of the scaticring center projected on the plane normal to the incoming u beam. If the density of scatterers is n (particles/cm?), there will ben dx 22 eal ahr hehe oe Sree eh a Sattar tn el let SRN SRT . 7 = at . * 8.2 interactions with Matter 299 (a) (b) — we ie —f— — : Flux fis Flux 5 a — __ | | Ci et Ie ox | |. a FIGURE 8.) Scattering of an incoming fiux of particles by a target: (a) Area covered by flux is larger than the target area and (b) area coveted by flux is smaller than the target area. scatterers per unit area in a thickness dx of material, and the probability dP = 1,/1Io of an interaction in the thickness dx is _ a(Io/S) = where 5 is the area covered by the scattering material and /p is the total flux incident on the target; thus Jo/S is the flux per unit area as shown" in Fig. 8.la. The result of Eq. (8.2) is not surprising since dP must be proportional to n and dx: aP (Si dx) =ondx, (8.2) adP xndx, o is then the factor that transforms this proportionality into an equality. Nuclear cross sections are on the order of 107*4 cm? (one barn), as expected given the geometrical size (cross section) of the nucleus Ogeom = HR? = 3.14 x 10776 A2Bem?. (b) Differential Cross Section. For a single scatterer we define* da(6,$) _ flux scattered into clement d82 at angles 6, d dQ incident flux per unit area , It follows that 20 t do d ——— §] a = ' | 70 sin@édé@ =a 3 Occasionally confusion arises because the area of the incoming beam may be smaller than the area presented by the target as shown in Fig. 8.1b. The definition of Eq. (8.1) is valid in either case and always leads back to Eq. (8.2). 4See the discussion on “solid angle” in Section 9,1. 300 «88 Particle Datectors and Radioactive Decay particle emerges with variable energy, then do(@,@, E) dQdk __ flux with energy E, within dZ, scattered into dQ at angles 9, 6 incident flux per unit area It follows that ia d*a(8,¢, E) ap _, 4a@, @) 0 dQdE dQ ° where the integration is over all possible energies of the scattered flux. Eq (8.2) —dI(x) = f(xodP = I(x)on dx; thus a = = —ondx, I(x) =Ipe77™*. If we designate by P(x) the probability for scattering in a length x, ve have P(x) = 1 — (probability for survival in a length x} @7onx —KEX = |-— =l-e*"’, where « = on is the absorption coefficient. Similarly A = l/on, whiel has dimensions of length, is called the absorption length, or mean free pall The density of scattering centers n is given by n= pNo/A if we consider scattering by nuclei pee ne = pNoZ/A if we consider scattering by electrons (8.4)25 nn = pNo if we consider scattering °Y nucleons, a number, respectively. AANA noe a ete tate Fra Sa a Ya Ya 0S ODDS Bd OE ‘ OR ¥, 7% OA A v9'9,9 + ,9,7 A 2 ea ee ele es SHRINE NN nV ENEMIES CUN TINNY UNH MOSER ANIM SRN NN * ety 1 LALA . ei en eee neonatal gel he o c at J ‘, ef? * at ee “s A ee ed et 8.2 Interactions with Matter 301 Often we wish to express the absorption in terms of the equivalent matter traversed, namely, £ = g/cm?. Then the thickness of the material can be expressed by d&, where dé = pdx. The mass absorption coefficient is defined by K =, ‘ (8.5) p so that the fraction of a beam not absorbed is I —=¢ (8.6) Ip Similarly, if the region of interaction ts very thin, the scattered flux ts given directly by N I, = Ipon dx, for example, for nuclei. i, = tp sade, _ 8.2.2. Energy Loss of a Charged Particle - When acharged particle collides with atomic electrons, as we have already - seen in the Frank—Hertz experiment (Section 1,3), it can transfer energy to them only in discrete amounts. It can either excite an electron to a higher atomic quantum state or impart to the electron enough energy so that it will leave the atom; the latter process is the ionization of the atom, Since in our present considerations the incoming particles have considerable energy, the process of ionization is by far the prevailing one, and we will use this term in the discussion. Let vs consider then an atomic electron at a distance b from the path of a heavy charged particle, of charge ze, mass M, and velocity v, as shown in Fig. 8.2a. If we assume that the electron does not move appreciably during the passage of the heavy particle, we can easily obtain the impulse transferred to it due to the electric field, E, of the passing heavy particle: “+00 +o0 6 = Funds =e f E (1) at =e | E(t) —dx = = [ E\ (x) dx. ax v —oo —o ee, ee 302 8 Particle Detectors and Radioactive Decay (a) EN oh ss mens b, eat a a em aS ee a be FIGURE 8.2 (a) Aparticle of charge ze, mass Af, and velocity v passes by an electron with; an impact parameter b. (6) The differential number of electrons with an impact parameteg 25 b in the interval 2b is given by the volume of the cylindrical shell 2rbdb dx. We use only the component of the electric field normal to the particle’ a trajectory since the longitudinal component averages to 0 when integrated: from —oo to +00. However, from Gauss’s law, integrating over a cylindex4 of radius b, coaxial with the trajectory (see Fig, 8.2a) we have | rs too Qe 47 ze = ge - dS = | E |, 2nbdx anid | BE, dx = — hs hence 2ze* 4 — Ub a ‘Since the electron was originally at rest, its momentum after the collisiét p = i,, and the energy transferred is Se ‘ in inet 2 24 2m mp“ b+ cs Thus & is a function of the impact parameter &. To obtain the total energy lost by the heavy particle per unit path length, we must count how mani electrons it encounters and average over the impact parameters. : From Fig. 8.2b we see that in a cylindrical ring of radius b, width di and unit height dx, there are contained? n,2mbdb dx electrons; hence “: Annedxze* db teeta tat thas SS “ 4 a a dE (hb) = *) my? b and Se dE 4nz2e* [bh es —-—— = wae She in| | ; (8.8 ie ax nu min ies Sn, is the electron density as also given by Eq. (8.4). SSL ETE RAN anh ahh SS RHA AH AN MNS SRN Mi SBA MEER oD Pane en $.2 Interactions with Matter 3 where because of the logarithm we had to use finite limits on & rather than 0 and oo. The finite limits are mposed by physical considerations: for bax we consider the distance where the time of passage of the heavy particle’s field becomes of the same order as the period of rotation of the atomic electron in its orbit. Thus t=-=- or Bmax = — (8.9) ™ For the minimum value® we equate b to the DeBroglie wavelength of the electron 3 bain = - =. (8.10) p We then obtain (8.11) The frequencies of the atomic electrons v are, however, different for = each orbit, so that a suitable average must be taken; we thus replace (fv) : with an average ionization potential /. Finally, inclusion of relativistic : effects and a precise calculation give df dar z*e* Im? dE _ 4nz*e .| mu -5°| (8.12) = —— nr, | h —=————_ ax mu* Idi — B*) for the energy loss of heavy particles due to ionization. In Eq. (8.12), 8 = u/c, and we see that the energy loss is only a function of the velocity, v, of the charge ze of the incoming particle, and of the = electron density, 7., of the scattering material. Note that im Eq. (8.12), m is : the mass of the electron while the mass of the incoming particle does not = appear at all. Before further investigating Eq. (8.12), we should note the following effects: (a) Equation (8.12} was derived on the assumption that the incoming =<: particle is not deflected, and thus tl is valid only for heavy particles; for =: electrons the term in the parentheses must be slightly modified. An alternate approach is to set 6,3, sach that maximum energy is transferred to the =: electron. Because of momentum conservation we have pinay = 2mnv leading to bai, = gad 2 a Ze" fm”. 34 = = 8s Particle Detectors and Radioactive Dacay mn Electrons also lose energy throu gh their i interaction wi ith the mt that accelerated charges radiate. This Tadiation, called remssuahing? is discussed in Scction $.2.6. ie (c) For extremely relativistic particles, v © c, 8 = 1, Eq. (8.12) prediegs a continuous mse in d E /dx Proportional to In ) where y = E/ mer é i/d- This is due to volarization of the medium: the electrons that are being: y into motion by the field of the incoming particle move so as to reduce, ue where For silver bromide 7’ * 48 eV. {d) For low-energy particles we obtain from Eq. (8.12) where Mf is the mass of the incoming particle and £ its kinetic endings © The above expression (when applicab!e} is useful since a measurements ee dE/dx and of & identifies the incoming particle E E & } a eM. ax fe) In image-forming devices and particularly in nuclear emulsions ns the density of developed silver bromide grains can be used as a meses sure of the particle’s velocity because of the dependence of Eg. (8.12): an Eq. (8.13) on 8. However, the density of the track depends only on en nety ner Bs See J. D. Jackson, Classical Electrodynamics, 3rd ed., Section 13.1, Wiley, New Yo 1999. s SO ohn ia 8.2 Interactions with Mattar 305 : transfers <5 keV, since when an atomic electron acquires more energy, its own track becomes visible and separated from the primary particle’s track; :. such electrons are called knock-ons or delta rays. The energy-loss expres- gion for energy transfers <5 keV does not exhibit at all the relativistic rise =. of Eq. (8.13), but for high values of +, stabilizes at a plateau 1.2 times the = minimum value. The energy loss of a heavy particle in a typical absorber, such as nuclear : emulsion, as a function of the logarithm of its anetic energy (in units of : yest energy) is given in Fig. 8.3. Strictly speaking, this curve holds only : for a given absorber and all singly charged particles, since we know from =. Eqs. (8.12) and (8.13) that dE/dx is a function only of the velocity of the =: incoming particle and its charge. (Note that K.E./mc* = y — 1, which has : 2 a one-to-one comespondenice to §.) However, the general behavior of this Ee curve holds for all absorbers. We do recognize four regions of interest: (a) near the stopping point fe where a Bragg curve is applicable (see Fig. 8,32); (b) the low-energy region = where the 1/v? dependence of Eq. (8.12) dominates, and tends asymptoti- Be: cally toward the value 1 /c?; (c) the relativistic region, where because of the =~ rise of the logarithmic term, a minimum appears approximately at y = 1; = and {d) the screened region in which Eq. (8.12) becomes applicable. Had e:- polarization effects not been included, the rise of the dE /dx curve in this &: last region would be steeper than indicated in Fig. 8.3. The lower curve s:- in Fig. 8.3 (energy transfers <5 keV) is applicable to the grain density in =. nuclear emulsions. Ee: liv? a Stopping Dependence ARelativistic Scraaning Bee region fg region pecs: (Bragg = E Bec CUITVE) i, . = Energy transters <5 keV a r 1 A I =o at i 10 10° 19° agra” y—t = FIGURE 8.3 The universal energy-loss curve for a singly charged particle plotted in ee = MeVi(g—cm~*) against y — [. Nate the upper curve for the total energy loss and the lower ae Pay a we . ae . peer ss a pie +e ee curve for energy loss invalving only energy transfers smaller than 5 keV. 206 §& Particle Detactors and Radioactive Decay horizontally by? m/my, in such fashion thatd E /dx}m, = dE ‘/dx ls wh y —I: 3 es o i z 4 “ate This is shown in Fig. 8.4, which gives the absolute value of energy tae a -dFE {as cin MeViig on) in air for protons (curve 1} and 7 -MesOns ctor? oe “ae Further, if we consider particles of different z, the energy loss will ditter oe vy the ratio (2 /22)°. In this fashion v we obtaiu curve 3 in Fig. 8.4, the Hee eo Z = pNo— PINOT and ad& = pdx Thus dk 9 ——— = Nna— I dE 0 42 FB ) a ey oe ae eee ee | charged particle in any materials is 2 MeV/(g-cm~*). =e ee i | ore) ae cee ae aed pein ae ot pe eds Ry ae sti ha uate ty ys hasty " Soret oO et ae eae SI IE I ia aaa a ar aaa ane ao se " iJ oe 200 Sor Ios 600 Set | LLDS. 00 LIN Cases T_T a TiN | 1 ‘. aan) | NH shit by (2 2, U4 meni =e RON ae - - Me ° FCI SS SSR Riil - S MeViig—cm-2) S 2 NL NSS S 7 Hi Hoes SEB SS i n SER: ee Penal Shitt by 2% 22 9.460 HE 4 i" L SSCS ne eS ee iia im — SS Kinetic energy, Mov 6.1 0.2 0.3 PIGURE 8.4 Energy-loss curves for different charged particics in air and in lead. Note how all the curves are related to each other. | ‘ 8 308 4868s «OParticle Detectors and Radioactive Decay a hee 8.2.3. Range of a Charged Particle Since the exact expression for the energy loss ofa charged particle is known, itis possible by integration to find what total length of material an incoming: particle of given energy will traverse before coming to rest; this is called its range R, and we can set © sts Hi he a Sak a al is a ret - the mass of the incoming Oe 1 f° dE ’o f{B) 1, __ _M R= dx = -—— ——_— => F f ar? Mn Fi(B)— 2?ne =f AB” Pn, a of electrons of aluminum (in aicm?) R —0.543E — 0.160, E> 0.8 MeV, where £ is the initial kinetic energy of the electron {in MeV). As suppested above, it is highly preferable to express the range in granis::* per squared centimeter, because then the dependence on the absorber mate: rial is slow (since n,/p == NoZ/A), resulting in a larger range (in g/cm’) in heavy elemenis. ie Figure 8.5 gives the range (in g/om* ) of protons, -mesons, and pa 2 fs Nouselativistically we have the simple relation dE = Mc* df. " *See, for example, the compilation by the Particle Data Group (2000; see Section 8. 2 7 i nd J ane |e ‘aval es SSeS ot te q =a Range g/em? : cat ee ——T a Te E AT as a cee See ria ee op sons | > Kobe | 0.003, Zar ee i Lt | 8 _——— wot res ee le 10 109 1000 Kinetic energy, MeV FIGURE @.5 Range curves for different particles in air and in lead. Note how the different curves are related to each other. ve a 310 4 «=6©8 «Particle Detectors and Radioactive Decay multiplying the ordinate (first) by mg/m, = 4 (due to the different mas and then by (zp /Zy)? = 1/4, hence leaving it unshifted. Finally, the range of protons in lead is also given. The concept of range loses its meaning, however, when the amount of material that the particle must traverse before coming to rest is on the order of a nuclear mean free path as explained in the introduction to this section.!9 8.2.4. Multiple Scattering In discussing the passage of a charged particle through matter, we have 24 neglected up to now its interaction with the electric field of the nucleus, because indeed the energy transfer to the nucleus is minimal. However, 222 when a particle of charge ze, mass m, and velocity v passes by the vicinity -°4 of anucleus of charge Ze, it will be scattered (Fig. §.6) with the Rutherford: cross section 2 al , do 1 feZe\? 1 a <= (2) —__. (8.16) 222 a2 4\mu? ) sin? 0/2 : ns Pe tel atta ta Pret he eT ee ee ty showing that the probability for small-angle scattering is predominant.-For’ 32 such small angles we approximate the angle of deflection by EE a OEE a AP EE (8.17) 22 p pub oe where p is the momentum of the particle and bis the impact parameter, = 2 During its traversal of the material, the incoming particle suffers many “=: small-angle scatterings. It can be shown (hat the resultant scattering angle 9 , = after traversal of a finite thickness of material D, has a Gaussian" distri- 2 bution about the mean @ = 0; the probability for a scattering through an 2 angle @ within the interval d@ is 5a j 1 /@\* The standard deviation iso = v 6? (the root mean square scattering angle). | 3 ‘For heavy ious, energy loss due to collisions with the nuctct must also be considered. See Chapter 10. : sy Ah ark ere Lere arr The aeagarananseaniaanaannaannnae ram nnanansnoary apa ern amannnninanpinan sini e k ms Pochette he ee ar te ee 8,2 Interactions with Matter 311 FIGURE 8.6 Deflection of a charged particle when passing in the vicinity of a nucleus. Note the scattering angle @, For the mean square scattering angle we have — 8nz*Z*e4 agup where ao is the Bohr radius. We further simplify Eq. (8-18) in order to exhibit the dependence of 62 on the incoming particle’s charge z, velocity B, and momentum p, as well as on the material’s thickness D, the density of nuclei nm = pNo/ A, and the atomic number Z: we obtain Ve = 3 apy D2 ni, where F is a slowly varying function of the parameters of the incoming particle and the scattering matenal (it contains the logarithmic term and constants). Furthermore 1/(Z*n) is proportional to the “radiation length” Lead Of the material (defined in Section &.2.6 below), which is frequently tabulated, so that we finally wnite — 21.2(MeV 62 = |Olms = et €), (8.19) *. Where |@|mms is in radians, and p must be expressed in MeV/c; € is a small = correction’* depending both on the scattering material and on 8/z of the = incoming particle. When we are interested in the rms projected angie, the = numerical factor in Eq. (8.19) must be replaced by 15 (MeV/c). \2Calculated from Moli¢re theory; see U.C.R-L. Report 8030 by W. Barkas and A. H. ; Rosenfeld for tables of e. 312. «8 ~Particle Datectors and Radioactive Decay s a aerate SS see 8.2.5. Passage of Electromagnetic Radiation (Photons) . through Maiter seat As mentioned in the introduction to this section photons lose energy or ate: absorbed in matter by one of the following three mechanisms: (a) Photoelectric effect, which predominates at low energies, (b) Compton effect, which predominates at medinm energies (below a a few MeV), and (c) Pair production of electrons and positrons, which is dominant in the high-energy region. The relative importance of these processes and the energies at which they set in are best seen in Fig. 8.7, which gives the cross section for the interaction of a photon as a function of its energy (in units of the electron’s rest mass). We wil] now briefly consider each process separately. (a) Photoelectric Effect. We speak of the photoelectric effect when: 224 the photon is completely absorbed and all its energy is transferred to an 2222 atomic electron. Consequently the photon must have enough energy to. excite the bound electron from its quantum state to a higher state or into = the continuum; the latter process (ionization of the atom) is much more : probable. Since the binding energy of the inner electrons in atoms is on * the order of kiloelectronvolts, as the frequency of the photon is increased - and it reaches the value of the binding energy of a particular shell, - a new “channel” opens, and we expect a sudden rise in the absorption ~ cross section. Apart from the onset of new channels, the overall variation of the photoelectric effect is a rapid decrease as the third power of the pho- — ton frequency (as v—‘/*), thus resulting in the curve shown on the left in Fig. 8.7. The cross section for the photoelectric effect is derived in Heitler (1954),!4 from which we give the nonrelativistic value for the ejection of one electron from the K shell, when the photon energy is not too close to (Note that 2 = 1 clectrons are said to be in the K shell, m = 2 in the L shell, n = 3 in es: : the M shell, etc. : l4w. Heitler, The Quantum Theory of Radiation, 31d ed., pp. 207 and 208, Oxford Univ. Press, Oxford, 1984. “ = 7 ‘= Es * i _- “a a . a a a oe ee ia . - a wa my ere ee a eee eee nae a Par vel a = Sy = is we ~ ee Pe eee er eee a ail oa Pa a rere ee a - ieee et ra aie “i . a Ser Ps arate ee pate at gee ae ae . ata ‘une fe at watt nth ae ore vr is pete ee ie 5, 8.2 Interactions with Matter 313 a | Pair production Photoelectric (v4) SThomean | 0.04 0.7 1 10 +100 y=huime? FIGURE 8.7 The cross section for the interaction of photons with matter as a function of their energy (expressed in units of the electran’s rest mass), the absorption edge, 2 hv |? op = OT aay? | (cm’). (8.20) Note the dependence on the Z of the nucleus, indicating that £ shell and higher-shell ejection is less probable because of the screening of the nuclear charge. Here of is the classical Thomson cross section, which is derived from the simplified assumption of a plane polanzed elecromagnetic wave scattering from a free electron (it is assumed that the displacement of the electron ts much smaller than the wavelength); we obtain Sr [ez ]° 8x 5 i | =" (8.21) where rp = e”/ mc? is the classical radius of the electron = 2.8 x 1073 em. Note that the Thomson cross section is independent of the frequency of the incoming photon, (b) Compton Effect. In the Compton effect, the photon scatters off an atomic electron and loses only part of its energy. This phenomenon, which is one of the most stinking quantum effects, is described in detail in Section 9.2; the cross section for Compton scattering ts given by the Klein— Nishina (K—N) formula, shown in an expanded scale in Fig. 8.8. The energy of the photon is given on the abscissa in units of the electron rest mass!> y = hv/mc*, and the ordinate gives the ratio of the Compton cross section oc to the classical Thomson cross section or. \SNot to be confused with the usual definition of y for a charged particle y = E f/m c2, introduced in Eq. (8.13). 314 «$8 Particle Detectors and Radioactive Decay 0.01 0.1 1 10 #00 1000 y=hyime? FIGURE 8.8 The ratio of the Compton scattering cross section, ac, to the constant Thomson cross section, op, as a fonction of photon energy expressed in units of the electron’s rest 11ass. | We give below the asymptotic approximations to the (K~N) Comptom Scatiering cross section: we For low energies: 26 oc = or (1-2y + S74] y =hv/me* «1 For high energies: ac = 3 opt (1 2Y + 5) y =hy/mc* » 1, (8.22) 8 Y 2 : (c) Pair Production. In pair production a photon of sufficiently high energy is converted into an electron—positron pair. For a free photon con: servation of energy and momentum would not be possible in this process, sO pair production must take place in the field of a nucleus (or of another electron), which wil] take up the balance of momentum. Clearly the thresh- old for this process is 2rc* (where m is the mass of the electron), hencé’ 1022 keV. The cross section for pair production rises rapidly beyond the24 See threshold, and reaches a limiting value for kv/mc? ~ 1000 given by!® 2g Zz ,[28. 183 2 ; ee a — jn — - . B29 )ene Opair 137 "0 E In 21/3 a (ero) gy l6See Heisler (1984), p. 260. BZ ee uv on Senco SSeS Re ee a ir te a ttl eo Da Bl a = ht % *. Ppa ar a hel ese Po enki DD De DD Per ae hr Dt CRO RA SOR oS OL eS LE Lh aC aC Sa aE CeO i a ee ee i a ere ee red ear rai he he aa wT AE, 7 ss wo, na raters . ar bk Beek ed esc tcitiiths Sarees SRI gers 8.2 Interactions with Matter 315 Siace both the photoelectric and Compton effect cross sections decrease as the photon energy mses, pair production is the predominant interaction mechanism for very high-energy photons. It is advantageous to introduce the mean free path (L pair) for pair production; when a photon traverses a material with density of nuclei n, ee 1 AOpis — (28/9)(Z2n/ 1372 In(183/Z!/3)’ where we have dropped the small term 2/27. Thus, the attenuation of a beam of 2g photons will proceed as L pair (8.24) T(x) = [pe /# pair, (8.25) In conclusion, Fig. 8.9 gives the total absorption coefficient for a photon traversing lead as a function of its energy (im units of the electron rest mass). Note that kp = op2n because there are 2 K -shell electrons per nucleus Kc = OCNe electron density Kpair = Opairl density of nuclei. The dashed curves in Fig. 8.9 indicate the relative contributions of each of the three interaction mechanisms. Absorption coafiicient (cm *) O.f 1 10 160 1000 FIGURE 8.9 The relative contmbution of the three effects responsible for the interaction - of photons with matter. The absorption coefficient in lead is plotted against the logarithm of photon energy (in units of the electron’s rest mass). 3150S 8s Particle Qeteactors and Radioactive Decay $.2.6. Interaction of Electrons with Matter (Bremsstrahlung) Since electrons carry charge, their interaction with matter must follow: along the lines given in Section 8.2. Because of their small mass, however, their interaction with the nucleus results in significant energy loss by radi- ation; this process, called “bremsstrahlung,” becomes the dominant mode, of energy loss for high-energy electrons. We can obtain an estimate of the cross section for “bremsstrahlung” fron a classical nonrelativistic model. Consider an electron (charge e, mass m7, and velocity v) passing by the vicinity of a nucleus of charge Ze, and let vis assume that in the collision process the nucleus does not move (Fig. 8. 6). The scattering angle of the electron is given by Eq. (8.17), and the change in the velocity vector of the electron is | Aus ce (8.26) 8 mub pe) The radiation formula for an accelerated charge!’ dE 2¢ — Pith= = oO | (6)° — (8 xB)’ |. (827) 7 So for our case, since B is normal to B, Qe. Se dE(t) = <— [pl at. (8.28) 22 3c mat By a general theorem of Fourier analysis, if 2) 2 p+oo E= =< | \A(t)2 dt, 36 Jago then also wee face? de, 3 ¢ Jose ee ee i re wate Se en, eR ee he ee ie are a ye Ld bd ete be a] eT oh Coa Sot as whe ek a7 F etnta ty! ST ta ae ag ee at al 1 wabytatat ae a a ee a ON tn atta) Lr rene 4 i" t aaa oe Seer 17See Jackson (1999), p - 666. In Eq. (8.27) y was set equal to 1; similarly Bq. (8.28) & shanld include a term (1 — 6°) = 1/y”, which was also set equal to 1. se y De eh Aan ste Ved bee ee a ee bk ee . en . B.2 Interactions with Mattar 317 where A(@) = — {~ Atte’ di (8.29) Jon F060 ) is the Fourier transform amplitude of A(t). Using then Eq. (8.29), we obtain in analogy with Eq. (8.28) the frequency spectrum of the radiation!® > 22 4 e? 4 dE(w) = — | Aw@yP + |A(—o) | dw = = |A(w)2 dw. (8.30) To evaluate dE (w) we must perform the integral indicated in Eq. (8.29) with A(t} = |B|. We assume that the acceleration AB occurs in a very brief interval of time, on the order of t = a/u, where a is the charactemstic distance over which the force is appreciable!?; then AB wt < 1 +00 i A(w) = [pla dt = sn" (8.31) ] /2m J—co wt > 1. Ifwt > 1, there will be several oscillations of the exponential term over the region where |p| is different from zero, and the integral will average to ZeI0. The integral results in a rectangular spectrum for the emitted radiation, as shown in Fig. 8.10, with 2e7 Az 2e4 dk _ eS Fy =) Bee chm 2b? ot< | (8.32) w = wt> il. Next we integrate over all impact parameter b to obtain the total radiated energy at frequency w when the electron passes by a nucleus bmax x(o) = | SO on db, s2) min = where we can set byay = @ = Tv and in view of wr ~ 1 we also let bmax ~~ v/s; from classical constderations (see footnote 6) Ze? Dmin = —>- mov 18Recause A(t) is real, A(w) = A*(—w). I9See, for example, W. K. H. Panofsky and M. Phillips, Classica! Electricity and oo Magnetism, p. 304, Addison-Wesley, Reading, MA, 1955, 318 = 8COPaarticle Detectors and Radioactive Decay f b- ae ae =A da) FIGURE 8.10 Idealized bremsstrahlung spectrum resulting from the sudden scsi z ]Af| of a charged particle. ee cc sate , The cross section Opens, giving the probability of emission of a photon ra) energy Aw in the interval d(ha), is related to x (w) through oS (hw) Oprems(w)d (fiw) = x (w) dw, peeetie Md resulting in the classical nonrelativistic bremsstrahlung cross section - 16 Ze? ( e® \* ey? 1 mu- Me — 3 moe —} ~—In{ >3— }- 8. a} eee The average energy loss per path length, —d#E/dx, is obtained by inte: ee grating over all photon energies (the square pulse) and multiplying by th density of nuclei: : dE “Ty = fr (A) Opremsd (hw) = (fiw) (Remax) Obrems- the a 4 HAE ty SESS SEARS Substituting 1/137 = e7/fe, rq = e*/mce?, and (Kamax) = Eo, the ener of the electron, we obtain dE\ 1622 fey? (mv : — TE in | —-—— }. 8.34 (fF jie 3137-7 (5 3) " (z Faw) ee : Equation (8.33) is a fait approximation; the correct quanturn- -mechanical #2 result, including the screening of the nucleus by the atomic electrons, ; is: ee given by” Se maaan dE z2 183 2 ee _ ( ) la a (410 75 5) ; : ye | > dx Z1/3 + 9 se A Sto 20 See Heiter (1984), p. 253. uy ast cinerea . toate a, Sete tte ath RC CEE &.2 Interactions with Matter 319 ” The mean free path for bremsstrahlung by an electron, called the “radiation =. Jength,” is defined as 1 1 = sSsSFs (8,36) NOprems 4(Z2n/ 137) rf In(183/Z4/9) La = which is obtained from Eq. (8.35) by setting Lyaq = dx, when —dE/ Eg = 1; the term F (small as compared to the In) was dropped. = * To show at what electron energies bremsstrahlung becomes important, we note that (dE/dx)rma ZE(MeV) (2E/ax }ioniz 800 , This is shown in Table 8.!, where we give for some common absorbers, Lead, as Well as the electron energy at which bremsstrahlung loss becomes =., equal to ionization loss. Equation (8.36) is amazingly similar to Eq. (8.24), by which we defined L pair - We have 7 brad = 9» pairs a indicating that in matter the mean free path of a high-energy electron is of the same order as the mean free path of a high-cnergy gamma ray; this is the e reason for the phenomenon of the electromagnetic cascade, first observed in COSMIC rays. Ifa very high-energy electron is incident in the atmosphere, it will soon (after approximately 330 m) emit one or more high-energy gamma rays. f These gamma rays will soon again (after approximately 330 m) produce = electron—positron pairs. Each of the secondary electrons and positrons will! again radiate, and so on, until most of the energy of the primary electron TABLE 8.1 Radiation Length of Electrons in Different Materials Electron energy for Material brad (dE /dx)yaa = (2 E/dx)igniz Aur 330 om 120 MeV Alaminum 9.7 cm 52 MeV Lead 0.52 cm 7 MeV a oe) Cr ee] 320 «68s Particle Detectors and Radioactive Decay FIGURE 8.11 Formation of an electromagnetic cascade. Note that high-energy electrons’ = (positrons) radiate gamma mys and the gamma rays later convert into electron—positror: pairs and so forth. (or gamma ray} has been transfcrred to many less energetic electrons: 2 (Fig. 8.14). Te In another connection we have already used L yg in Eq. (8.19) for mul- : tiple scattering; from Table 8.1 we sec that in heavy materials scattering. 34 will be much more pronounced. Note that multiple scattering is the same: for particles of the samc momentum. Thus, at low energies a light particle 22: mn scatter much more than a heavier particle of the same kinetic energy 2: = /2Tm). This is clearly seen when observing the tracks of low~ chery protons and electrons in an image-forming device; the former ones. z are, in general, straight, whereas the latter ones suffer multiple scattenng S through Jarge angles. a Para) saat va 8.3, GASEOUS IONIZATION DETECTORS; = THE GEIGER COUNTER iS 8.3.1. General As mentioned earlier, most particle detectors are based in one form or’: another on the energy lost by the charged particle due to ionization of the x medium it traverses. In a large class of instruments the detecting material : is a gas; the ionization potentials are on the order of 10 eV, but on the 22 average, for example in air, the charged particle loses 30 to 35 eV for each .:; electron-ion pair formed.”! By collecting the free charges that were thus Ss 2\ This is due to additional interactions such as excitation and elastic scatlering. oy *—a gia * * “. walls act as the negative electrode, and positive voltage is applied to the -. central electrode, Under the influence of the electric field, the electrons are :. collected at the center while the positive ions move toward the walls. It is ee m raat 8.3 Gaseous lonization Detectors; the Geiger Counter 321 insulat nsulator = CG R = FIGURE8.12 Diagrammatic arrangement of a cylindrical Geiger counter; the central wire is charged to B* through Rc while the cylindrical envelope is held at ground. The output - sjgnal appears across Ry. ae created, it is possible to obtain an electrical pulse, signaling the passage of ge: . the charged particle. The simplest type of gaseous detector consists of a cylindrical chamber with a wire siretched along its center, as shown in Fig. 8.12. The chamber desirable to collect the free charges before they recombine in the pas; this .. is mainly a function of the pressure of the gas and of the applied voltage,” If, however, the voltage is sufficiently raised, the electrons gain enough -, energy to ionize through collision further atoms of the gas, so that there is a ==. significant muldplication of the free charges originally created by the pas- g:. sage of the particle. In Fig. 8.13, Curve | gives the number of electron-ion = pairs collected as a function of applied voltage when an electron (mini- mum 10012ing) traverses the counter; Curve 2 gives the same data, but for = amuch more heavily ionizing particle. Thus the ordinate is proportional to the pulse height of the signal that will appear after the coupling capacitor C (in Fig, 8.12). Referring to Fig. 8.13, we see the following regions of operation of L a a gaseous counter: in region H the voltage is large enough to collect all . the electron—ion pairs, yet not so large as to produce any multiplication. A detector operated in this region is called an ionization chamber. As the : - voltage 3 is further raised, region Ii is reached, where multtplication of the <:° Original free charges takes place through the interaction of the electrons as : aa they move through the gas toward the collecting electrode. However, aver 221 is also, of course, a function of the specific gas or mixture of gases used, 32208 «Particle Detectors and Radiaqactive Decay BES saa oa ‘ io" Gaiger-Miller f counter | es Recombination - Region of before collection linnitedt 1"° praportonaity lonization Preportional chamber counter 408 Number of fons collected 3, 10? 0 250 560 750 +000 Voltage, volts FIGURE 8.13 The number of electron-ion pairs collected when a charged particle tra- verses a gaseous counter of average size plotted against the voltage applied between the electrodes, Curve | is for a minimum ionizing particle, whereas curve 2 refers to a heavily ionizing particle. Note the three possible regions of operation as (a) an ionization counter, (b) a proportional counter, and {c) a Geiger counter. a considerable range of voltage, the total number of collected electron-ion pairs 65 ramp pryenal.to the original ionization caused by the traversal of the charged particle. A detector operated in ims régmir outlecLa are portional counter, ithas an advantage over the ionization counter in that t signals are much stronger, achievable gains being on the order of 10* to L Finally, further increase of the high voltage leads to region IV, where ve large multiplications are observed, and where the number of collec’ electron-ion pairs 3s independent of the original ionization. This is region of the Geiger—Miiller counter, which has the great advantage very large output pulse, so that its operation is simple and reliable. Inde *3The proportionality does not have to be a linear function of the applied voltage. rstniihnininsnsiyyminanny sinnninnhsngenngnnmnaten a ey teeta SAMARAS Tanna ee ee ee eee ea * ' Pat he tat ae Sea ai ea ae ee oe ee eee en en en ne eee EEE ea betaty tate ne erate geet at abate ty Hat gtylyty fala lefa talk el elale aa Par ar ai i Sa aC BE SE LC a Oa A OR Ce kia i a Aa 8.3 Gaseous lanization Detectors; the Geiger Counter 323 at such high voltages, once a few electron-ion pairs are formed the elec- trons produce more ionization at such a rapid rate that regenerative action sets in, the whole gas becomes ionized, and a discharge takes place. At that point, the resistance between the central electrode and the chamber wall becomes negligible, and the counter acts as a switch that has been closed between the high-voltage source and ground; this discharges capacitor C through resistor Ry (Fig. 8.12). Since C was charged at Bt (on the order of 1000 V), very large ontput signals may be obtained. For example, if the number of electron-ion pairs collected is 10!° (as given by Fig. 8.13) and C = 0.001 pF, we obtain 1.6 x 107! x 10/8 ya 2 ON aly, (8.37) By scaling this result according to the graphs in the figure, it is easy to appreciate the difficulties involved in the amplification of proportional- counter and iomzation-counter signals. The disadvantages of the Geiger counter are the loss of all information on the ionizing power of the charged particle that traversed the counter, and the long timc necessary for restoring the gas to its neutral state after a discharge has taken place. However, the simplicity and good efficiency of the device for single-particle detection have made it a very common nuclear radiation detector. 8.3.2. The lonization Chamber The main difficulty with ionization counters is theix very low signal output. If they are used, however, in an intense flux of radiation as an integrating device, high signal levels can be reached; in that case the output signal corresponds to the total number of electron-ion pairs formed (per unit time) by the radiation. In this fashion ionization chambers are frequently used for monitoring X-ray radiation or high levels of radioactivity; in such applications they are far superior to Geiger counters, since the rates are so high that a Geiger would be completely jammed. When an absolute measurement of the created free charges is made, as with an electrometer, ionization chambers may also serve as standards of ionizing radiation. Most coromercial instruments, however, amplify the output pulse and are directly calibrated in roentgens (or fractions of roentgens) per hour. For use in the laboratory an ionization counter 394. SB SCOéParticle Detectors and Radioactive Decay Model 2526 (“cutie-pie”) manufactured by the Nuclear-Chicago Company: : is suggested for radioactivity surveys and as an X-ray monitor in the range: ‘ of 0-2500 mB/h. Below we describe a very rudimentary “student-type” ionization cham=: ber that was used in this laboratory for measuring the range of alpha’: particles emitted by 7!°Po. Figure 8.14 is a sketch of the apparatus; it: consists of a flask with a 5-in, outer diameter, its inside wall having beer: coated with a conducting material (such as aqua-dag or silver). A rubber: stopper inserted at the mouth of the flask acts as a support, electrical insu-": lator, and vacuum lock. Through the stopper is fastened a brass rod at the: tip of which has been attached a 20-~Ci 7!9°Pa source,*4* which is thus: located at the center of the flask. A 180-V battery is connected between the : flask walls and the rod supporting the source, and the ionization current is: measured with a Keithley electrometer. The energy of the 20D alpha rays is 5.25 MeV, and their range in: air at stp is 3.93 cm; hence the alphas stop before reaching the walls’: of the flask and deposit all their energy in the gas. By using the number of: approximately 30 eV per electron-ion pair, mentioned at the beginning ot Section 8.3.1, we would expect per alpha particle a total of 5,25 x 10°/30 = 170,000 electron-ion pairs (the true number in this case being closer to 110,000), Since 20 pCi = 20 x 107° x 3.7 x 10! = 7.4 x 10° alpha particles/s if all the electrons were collected, the ionization current should be P= 1.6x 107% x74 x10 x 11x 1 =13x 108A, (8.38) | which is readily measurable. If now the flask is slowly evacuated, the alpha particles will traverse a : longer path before stopping; however, as long as the alphas stop in the gas, 2" the same number of electron—ion pairs is formed and the ionization current should remain flat and independent of pressure. When the density of the 2% air in the flask becomes so low that the alphas reach the wall before losing =: all their energy in the pas, fewer electron—ion pairs are formed and the 2:4 ionization current will drop monotonically with decreasing pressure. 24See Appendix D. part SUS TMM SSSD wet eee Pare et ed Peay a at a ee a ar 1 oe ae oe wan rT 1 reba a! Pera Tteedetete stetene uti SCUMIESIATAT ad egagenec abet ot SS re oe oe ee a ae a a at MBB OE TN DBT ee ER A A he ee To silver coating Manometer To pump | FIGURE 8.14 Asimple arrangement for the determination of the range of alpha particles in air by measuring the ionization current as a function of chamber pressure, 326 #8 Particle Detectors and Radioactive Decay 0.9 P,=51.5+0.2 cm a & T=25C Atm pressure= 76.3 cm Haxtorly) 0.7 6.6 20 25 30 35 49 45 50 55 60 65 P (cm Hg} FIGURE 8.15 The results of the measurement referred to in Fig, 8.14. The ionization ee current is plotted against residual air pressure and a decrease in current begins at P32 51.5 cm Hg. This corresponds to a range of 4.02 cm in air at stp, | Data obtained in this fashion by a student are shown in Fig. 8.15. Indeed. the expected qualitative behavior of the ionization current is observed; froxn B 5 the breaking point we conclude that at a pressure of 51.5 + 1 cm Hg, the range of 2!°Po alpha rays in air is R = 6.14 cm. Hence at stp (760 mm Hg 15°C) 51.5 ,. 288 P Tstp Bs = —- * — = 6.14~x — = 4,02 a 8 R sto RX Pup r 760 0 * 302 + 0.1 cm S in good agreement with the accepted value of Rap = 3,93 cm, example of a low-efficiency integrating ionization chamber. 25Some loss is also due to self-absorption in the source, and the geometrical solid ang 3 is only 27. 5 Paes eda” = bl atea at a” 9 iat =a . i mead a" = tae a" = rane . PM = ia! ate en le bee: section for 5-keV quanta is on the order of 6 x 10-* em?, so that if the =: counter represents approximately 50 mg/cm? of material, the efficiency for Ren: pamma-ray detection might be as high as 8.3 Gaseous lonization Detectors: the Geiger Counter 327 83, 3. The Proportional Counter : We will not describe in detail the proportional counter?® but only give =:. the results obtained by a student using such a detector in connection with es the experiment on the Méssbauer effect (see Chapter 9), The advantage = of proportional counters lies in the detection of very low-energy X-rays ee or gamma rays, which can hardly penetrate a scintillation crystal, and = when in addition good energy resolution is required. This is the case in => the Mdssbauer experiment from °’Fe, where it is necessary to identify a 14.4-keV pamma ray in a strong background of 123-keV gamma rays and a 5-keV X-rays. The proportional counter used was*’ Amperex type 300-PC. It was filled : with a xenon methane mixture at a pressure of 38 cm Hg. The equip- = gent used for amplification and pulse-height measurement”® is shown in =: Fig, 8.16, and the counter was operated at 2100 V. Figure 8.17 gives the me results obtained, where the number of pulses is plotted against the discrim- jnator channel. The large peak at Channel 12 is the 5-keV X-ray; the small :. peak at Channel 26 represents the sought-after 14.4-keV gamma ray. As we know from Section 8.2.5 (Fig. 8.7) the predominant interaction =: of low-energy gamma rays in the gas is the photoelectric effect. The cross 6x 107 x 50 x 1073 x 2x 6x10 x a 10%. = Using the data from the 5-keV peak, we obtain for the resolution of this ee: proportional counter, AE/E = 1.7/12 = 14%, =. where for AE we chose the half-width of the peak at half-maximum (after =» background subtraction). 26For an extensive discussion of proportional and ionization counters, see the Encyclope- a dia of Physics, Vol. 45, Nuclear Instrumentation fT, Springer-Verlag, Berlin, 1958, articles Eby H. W. Fulbright, pp. 1-50, and by S. C. Curran, pp. 174-221. 27 Manufactured by the Amperex Corporation and obtainable from Scientific Sales, Inc., Be J Long Island, N.Y. 28For a more detailed discussion of pulse-height spectra see Section 8.4. 328 & Particle Detectors and Radioactive Decay \ | Proportional _| Cathode 4_ j counter 4 | follower 4 es a Se Single channel analyzer aa cl ee a ee 1 1 | IDiscriminator!- ss Amplitier Mg I | ape il RCL Mod. 20506 0 5 10 15 20 25 30 35 Channel FIGURE 8.17 Pulse-height spectrum of the low-energy gamma radiation from 57 Re ag , obtained with a commercial proportional counter. The pronounced peak at channel 12 is: in the Méssbauer effect. 8.3 Gaseous lonization Detectors; the Geiger Counter 329 8.3.4. The Geiger Counter; Plateau and Dead Time It has been pointed out in Section &,3.1 that a gaseous counter operates in the Geiger region when ihe voltage between electrodes is sufficiently large; that is, the traversal of a charged particle initiates a discharge in the gas, and as a result a pulse appears at the output that is independent of the original ionization. If the voltage is further increased, spontaneous discharges occur, making the device useless as a particle detector. Because the principle of operation is simple, Geiger counters are sumply constructed, the geometry of Fig. 8.12 being typical. For certain applica- tions, the thickness of the walls is an important consideration, and Geiger counters may be built with special thin windows (usually mica of few mg/cm”). Glass envelopes for Geiger counters are fairly common, and various pressures as well as mixtures of gases are used, Another important consideration for Geiger counters is the “quenching” of the discharge imtttated by the traversal of a charged particle. Until the gas is returned to ifs neutral state, the passage of a charged particle will not produce an output pulse; this is the period of time durmg which the counter is “dead.” The quenching of the discharge can be achieved through the external circuit (for example, in Fig, 8.12 the charging resistor Rc will introduce such a voltage drop that the discharge will extinguish itself), through the addition of special impunties {such as alcohol) to the gas of the counter, or by both methods used together. The circuitry necessary for the operation of a Geiger counter 1s also extremely simple. A single stage of amplification and pulse shaping is usually sufficient to drive any scaler. In order to operate a Geiger counter properly, the high-voltage source must be set in the “plateau” region (Fig. 8.13, region [V), where a similar output is consistently obtained for all charged parucles traversing the counter, We may then define the efficiency of the detector as the ratio of the number of output pulses over the total flux traversing the counter; since the pulse heights are all equal in the plateau region, we do expect _ the efficiency to remain constant in that same region. Clearly any parti- cle detector should be operated in a region where the efficiency is “flat” = witb respect to variation of operating parameters. The efficiency of Geiger . counters is 90% or higher for charged particles, but for photons it is much » lower, being only on the order of 1-2%. It is difficult to make absolute efficiency measurements for Geiger coun- = ters, A “standard” calibrated source of radioactive material may be used, * and the output count compared with the expected flux from a knowledge eh a ie) 3H 4 86B 60Particle Detectors and Radioactive Decay the square root of the number of counts, and thus the measurement Soul be interpreted as 1000 + 31 = 1000 x (1 + 0.03) counts or in common parlance, 1000 counts give 3% statistics. The high ge should he well stabilized, usually to a few parts in one thousand. Figure 8. 18 gives the plateau found vy a student for the RCL*® ty a ia} a ae ee discharge region begins at 1400 V. te The slope of the plateau, from Fig. 8.18, is ne ee) wee vie ae 150/3200 = = 5% per 100 V. ae Pred mentioned. Indeed, once a discharge has been initiated, the counter i not register another pulse unless the discharge bas extinguished itself, and: until, in addition, thc counter has “recovered”—that is, retummed to a neutral state. During the rccovery period, the counter will generate an output pulsé;= but of a smaller-than-norma! amplitude depending on the stage of recovery: = 291f this measurement is repeated many times, in 68% of the cases we will obtain N — o oe N > N +o, where WV is the average of ail measurements. See Chapter 10 for the definition, ES of c. roe ee Radiation Counter Laboratories, Inc,, 512 West Grove Street, Skokie, Ill. 8.3 Gaseous Ionization Detectors; the Geiger Counter 331 5500 Counts/min 500 | y) 1000 1100 1200 1300 1400 Volts FIGURE 8.18 Plateau curve of a Geiger counter. Note thal the plateau region extends for 250 V and has a slope of the order of 5% per 100 V. Horizontal scale 100 psec/cm Vertical scale 5 V/cm FIGURE 8.19 Multiple-exposure photograph of oscilloscope traces obtained from a Geiger counter exposed to a high flux of radiation, Note the effect of the “dead time” of the counter and the gradual buildup (recovery) of the output pulses. ee a ad ' Ar 3320-8: «~Particle Detectors and Radioactive Decay This phenomenon of recovery can be clearly seen in Fig. 8.19, obtained. by a student. The Geiger counter was exposed to a high flux of radiation: the trace of an oscilloscope is tiggered when the output pulse appears: The horizontal scale is 100 ys/cm so that the shape of the output pulse and its exponentially decaying tail can be seen in detail. If now a second particle arrives within 1 ms of the previous one, it will appear on the same oscilloscope trace since the scope will not trigger again until the sweep ig completed (the screen is \Q cm wide). The picture shown in Fig. 8.19 was obtained by making a multiple exposure of such traces. The correlation of. pulse height against delay in arrival time and the exponential dependence: of the recovery are clearly noticeable. If we consider that the counter is inoperative until the output is restored to 63% of its original value (1 —1/ e); the data of Fig. 8.19 give a value for the dead time t on the order of = 400 ys. (8.39) Pulses, however, seem to appear after an interval t = 300 ps. (8.40) The dead time of a counter may also be obtained by an “operational” technique, such as by measuring the counting loss when the detector is subjected to high flux. If the dead time is 7 (s), and the counting rate FR’ (counts/s), the detector is inoperative for a fraction Rt of a second; the true counting efficiency is then 1 — Rr. Consider two sources 5S, and $2, which when placed at distances from the counter 0, and Dy» give a true rale (counts/s) R,, 2. The counter, however, registers rates R; < Rj, Ry < Ro due to dead-time losses, and: when both sources are simultancously present, it registers Ri. < Rj + Rj due to the additional loss accompanying the higher flux. Now, Ri = Ri — 17) , ppt R, = Rr(i — Rt) Be 12 = (Ry + Re) — Ript). Se ee) ee We solve by writing "ae “ee Rio R Ré ae _ 1 2 meee 4 = 4 ee {| — Riot 1-—Ryt \—-Ryt we ae . urecay tthe eta eeeccnent Sasa MtaSCahtabehe® ASA 8.4 The Scintillation Gountar 333 which reduces to a quadratic equation in t with the solution Pa /1 — Rig (Ry + RA — Ryy)/ RRS 12 This can be expanded in the small quantity (R) + R, — R},) to give the approximate expression T= oa (RARE Rip) ! (8.41) 2R; Ri We now apply Eq. (8.41) to data obtained by students with the same counter used for Fig. 8.19. In practice, source §; is first brought to the vicinity of the counter and R{ is obtained, next 52 is also brought in the area and Ri, is obtained, and finally S; is removed and R, is measured: thus no uncertainties due to source position can arise. They obtain R, = 395 + 3 counts/s Ri) = 655 ~ 3 counts/s R, = 334 + 3 counts/s, yielding t = 282 + 20 ws, in better agreement with Eq. (8.40) than with Eq. (8.39). The rather long dead time of the Geiger counter is a serious lim- itation restricting its use when high counting rates are involved; the ionization counter and proportional counter have dead times several orders of magnitude shorter. 8.4. THE SCINTILLATION COUNTER 8.4.1. General As we saw, in gaseous-ionization instruments, the electron—ion pairs were directly collected; in the scintillation counter the ionization produced by the passage of a charged particle is detected by the emission of weak scin- tillations as the excited molecules of the detector remrn to the ground state. The fact that certain materials emit scintillations when traversed or struck : by charged particles has been known for a long time, Ratherford being the : first to use a ZnS screen in his alpha particle scattering experiments. 334 «= 8s «Particle Detectors and Radioactive Decay The scintillation counters used currently were developed | in n the 19505 3 multiplier that responds to the light pulses. Anthracene or stilbene crystals ee make excellent scintillators, but organic compounds embedded in trang parent plastic, such as polystyrene, are now widely used because of ease i282 handling and machining and availability in large sizes. Such materials arg commercially available*! under the general description of “plastic scinti#s2= lators.” The active materials are compounds, such as “PPO,” 2-5-dipheny[=2% oxazole, or diphenylstilbene, or others, and are also available in liquid form: Organic scintillators have an extremely fast response, on the order of/3%4 10—? s, which can be matched by good photomultipliers. On the other hand 2 because of the low ents and low , their efficiency for gamma-ray ee i | or ree) se iil in 105). Inorganic crystals have an excellent efficiency for gamma-ray come version, due to their high Z; trom Eq. (8.20) we recall that the photoelectric 4% effect is proportional to Z° and from Eq. (8.23) pair production is propor: 44 tional to Z*. However, the light output from ; inorganic crystals is spread: 24 over a much longer time interval, on the order of 10~*s, Such inorganic 4 . wnt crystals arc also available commercially,>* appropriately encased since they. are damaged by humidity; they come in sizcs up ta several cubic inches. ° The light output of scintillators is proportional {as a matter of fact,: linear) to the energy lost by the particle that traverses the detector; thus, by. pulse-height analyzing the electrical output of the photomultiplier, the scin-* tillation counter may be used as a spectrometer. This procedure is discussed : in detail in the following section, where it is seen that energy resolution on. “4 the order of 10% or better is achicvable. . The mechanism of emission of the photons in the scintillator material is rather involved. Table 8.2 gives a chart of the processes involved in the 4 emission of light in organic and inorganic crystals. In inorganic materials “2 it is the migration of the electrons through the lattice (until they excite an Se impurity center) that is responsible for the long duration of the light pulse. “32 Even though the efficiency for transferring the energy lost by ionization “ to the photons in the visible region is on the average low, € * 1.5%, a scintillator still provides ample light output. Consider the casc of a “: plastic scintillator 1-cm thick, traversed by a minimum-icnizing particle: - dE/dx =2x 10° eV per g/cm’; if we take the average photon energy % ASNtnbeeeos eiabatahia 31 Bor example, from Pilot Chemicals Inc., 36 Pleasant St, Watertown, MA. 32 For example, from Harshaw Chemical Corp., Cleveland, OF. eter ng We AS SEERA URS AC LS SUNS TABLE 8.2 The Series of Processes Leading to the Emission of Light When a Charged Particle Traverses a Scintillator Material* Inorganic scintillator (Impurity activated) Holes Electrons Drift to impurity center and ionize it. Emission of thermal radiation Yonized impurity center» Capture with emission of thermal radiation Excited impurity center Electron drops into metastable state of impurity center ' Thermal energy Radiationless tran sition raises electron to excited state Organic scintillator Electronic structure of molecule is excited Loss of energy to Dissociation yibrational states Emission of light quantum \ Energy may be transferred to other molecules Radiationless transition Emission of light * After J, Sharpe, Nuclear Radiation Detectors, Methuen, London, 1965 (Courtesy of the Publishers). 336 «= 8s Particle Detactors and Radioactive Decay as 3 eV, we obtain 104 photons. The efficiency of a photomultiplier cat ode for converting photons into electrons is on the order of 0.1, and the: geometric efficiency for collecting the photons onto the photocathode'j usually high, so that on the order of 1000 electrons are released. Witl modem techniques, however, it is possible to detect the release of a © photoelectrons, 0 or even of a single one. Bs that traps and guides the light due to total internal reflection at its suk a faces, At the surfaces where the lightpipe is joined to the scintillator or #6: the photomultiplier, optical contact is achieved by the use of either vig ae cous fluids or special glues. 33 Obviously the whole assembly must be light: a ie sen egtae the scintillator, hghtpipe, and phototube. i Because of its great stability and ease of operation, as well as because of its time and energy resolution, the scintillation counter has become thit#22% most frequently used detector in nuclear physics, especially for hi gh-energy 2 particles. rs 8.4.2. Experiment on the Determination of the Energy of Gamma Rays with a ScintiJlation Counter If atoms are quantum-mechanica] systems and a typical manifestation of: this fact is the emission of spectral lincs of light, it should be expected that: nuclei, when excited, would emit similar line spectra. Since the nuclear radius is three to five orders of magnitude smaller than, that of atoms, the forces that bind the nucleus (against the repulsion of the: positive charges confined in its volume) must be correspondingly stronger: than the forces that bind the atomic electrons to the nucleus. As a conse: quence, the energy levels and the quanta of energy emitted in a nuclear tran-: sition are also orders of magnitudes larger than those of atomic transitions Indeed, the quanta of electromagnetic radiation emitted in a nuclear transi tion fall in the gamma-ray region, and new techniques are needed for their: detection and for the measurement of their (wavelength) energy. O25 SO Se aN oH Sint mh ae — mete, ta ‘at wana vate 33m the first category, Coming 200,000 centipoise fluid or clear vacuum grease; in the 2 latter, R 363, PS 28 acrylic glue, ctc. i ce nate chek rath genase 8.4 The Scintillation Counter 337 Further, because of the larger spacing between energy leveis, it is not easy to excite anucleus from its ground state by the simple means of electric discharges or arc sources such as are used for atoms; instead, beams of neutrons or high-energy garmma rays, or high-energy charged particles, are required, However, in distinction to atomic transitions where the de- excitation probability is on the order of 108/s, some nuclear transitions have a very small “decay” probability, as small as 10~"/s, corresponding to a lifetime of 100 days. Thus, it is possible to excite a sample of nuclei inside a nuclear reactor, or by subjecting them to cyclotron bombardment, or by other means, and subsequently bring them to the laboratory for measuring their spectrum or for other uses. Indeed, some of the nuclei that have very long lifetimes can be found in nature in their excited state; these are the naturally radioactive elements. We now know that the appropriate detector for measurements of the energy of gamma rays is an inorganic crystal. When a gamma ray of energy <1 MeV enters the detector, it will interact either by the photoelectric effect or the Compton effect. In the former case it is fair to assume that the ejected photoelectron will deposit all its energy in the scintillator; in the Compton effect, however, the scattered photon may or may not convert in the scintillator (depending on the size and geometry of the detector). The pulse-height spectrum for gamma rays of a given energy will con- sist of a peak at an energy corresponding to that of the gamma ray and a continuum below the peak, corresponding to Compton-scattered gamma rays that escaped from the crystal before tatally converting. This can be seen in Fig. 8.20 and those that follow. Clearly the larger the size of the 2 crystal, the larger the percentage of the output counts that will lie in the = photopeak; thus, the gamma-ray line will become more pronounced. Most of the data reported here were obtained with a Nal—Tl activated 2 crystal,** 2 in. in diameter and 2 in. wide, coupled directly to a photo- =~ multiplier tube.7> (Photomultiplier bes and high-voltage bias schemes =: are discussed in Appendix E.2.) The output pulse is fed to an Ortec*® Sane I ENE ES SRSA Model 570 amplifier, and tts output is fed to a Canberra multiport mul- ee tichannel analyzer (MCA). The MCA is controlled and read out through a 4B icron Corporation, http://www.bicran.com/. SThe crystal and phovoruultiplier tube assembly is a commercial package from Canberra =, Industries, httpyAvww.canberra.com/, Model 802-3. The photomultiplier tube “base” was * constructed from a commerctal socket and simple components. 3¢hetp://www.artec-online.com/, 338 «=©6©8s«~Particle Detectors and Radioactive Decay 200 150 Counts per minute over 32 bins 0 id 0 1000 §=6©2000 3000 4000 5000 6000 7000 98000 Pulse height (Channels) Counts per minute over 32 bins 0 0 1000 «=862000 3000 4000 5000 6000 7000 8000 Pulse height (Channels) FIGURE 8.20 Pulse-height spectrum of ®°Co gamma rays obtained with a Nal crystal; along with the decay scheme of ©°Co. The upper spectrum was taken with a 2-in.-diameter crystal detector, while the bottom was taken with a 3-in. crystal. The Co source was relatively weak (less than 1 Ci when these data were taken) and the source-to-detector. distance was 10 cm. The decay scheme is also shown. 8.4 The Scintillation Counter 339 5+ (5.26 yr) Gg >go% 4+ 2.506 Mey (Z=27) 2.82 MeV 2+ 1.393 MeV a O+ SOK; (7=28) FIGURE 8.20 (Continued) GPIB interface, in this case using a laptop computer. Spectra acquired in this way are histograms with 8192 = 2)3 bins. (Adjacent bins were added together to reduce the statistical fluctuations from bin to bin. This is easy to do with the reshape command in MATLAB.) The conversion of bin number to photon energy depends on the combined gain of the photomultiplier and the amplifier, and must be calibrated with sources of known photon energy. The following figures give the results obtained by a student. Figure 8.20 gives the spectrum of Co and shows two distinct peaks, which we attribute to gamma rays emitted in the de-excitation of °Ni from its 2.505-MeV level to the 1.333-MeY¥ level, and from that leve] to the ground state according to the decay scheme also shown in the figure. For comparison, we also show a spectrum taken with a 3-in.-diameter and 3-in_-wide crystal. As a measure of the energy resolution, we may consider the full-width of the peak at half-maximum, which is on the order of 480 channels, hence a resolution of 480/6000 ~ 8%. We also notice a significant background for pulse heights lower than that of the peaks, which is due to Compton-scattered gamma rays that subsequently escaped from the crystal. This background is much less severe for the larger crystal. Figure 8.21 gives similar data for a sample of !37Cs; here the 0.662- MeV gamma ray represents the de-excitation of !*’Ba. Again we notice some Compton background and an energy resolution on the order of 10%. Figures 8.22 and 8.23 give the pulse-height spectra from **Na and '33Ba, respectively. For the 7*Na, the peak at 1.277 MeV arises from the de-excitation of **Ne; the larger peak at 0.511 MeV arises from annihila- tion radiation. Indeed, from the level diagram of **Na decay, we notice that a n _o) = Oo r 712+ (30 yr) 04 250 Cs 94.5% ‘ (Z=55) @ 200 11/2— 0.6616 MeV g (2.65m) E 450 1.18 MeV o 5.5% o 1/2+ 0.281 MeV 2 100 D Oo 3/2+ 50 197Ba (7=56} 1?) 0 1000 2000 3000 4000 5000 Pulse height (Channels) FIGURE 8.21 Pulse-height spectrum of !3’Cs gamma rays obtained with a Nal crystal, and the associated decay scheme. a ee PARP Ot Ne OC at CORE ARE CRT Ne ae a Coe nat Ca era he Tar eer S Gee oe Pk OL HO OL be Pa i ot ie beh BO at OC OE UN POON CL oe RC oe eh SS CEES Ss Mie A, ay i Lat ha hl aC fat eb lhe ha a eh aE a ee fd er a ee EE Ed EE EE Ede Ee EEE Ee RE RP eS arr irs arr ire rarer eres ae ea 3” MRT SCA EERE NRTA CLE NEM WEE SCE Eaaeeeaaat fet be p 6 ee eS 6 TTC ES SOS. EO cen aes ens nae RARE CCCI OOOO ODOOOOOOOL eee Ren Ee PSST SSS SSESEM SCS SCIENCE MEE! SEM WE SEREMC WC ESRC RE DERE EERE RE NT Ere eae ee eee ee HEP e este eb 6 ats eb ae SSO OIO COOOL Sea OT SST UE TOT SPOTS UU ar Oar Tao OE RETEST MEU UL SPIE EPIE USE OL OESE ULOE SESE ACME SEAS IE SAE SESE AE SIE AE AEE SESE EEN EC NECA Se CIC OCT SE SESE SESE SEC IL AC SCSI ICSE SESE SCR COERL LAER SCROLL IC rie Bo tate a ela ata gee ee eee TT eee et vies aes 3+ (2.60 yr} 22Nia {Z=11) 1.2768 MeV 2+ 2.84 MeV Counts per minute over 32 bing D 2 O+ t eisieeniehiemes seem @ ah] o - .. Nea (Z=10) 0 1000 «aoe i (CO COOO aD Pulse neight (Channels) FIGURE 8.22 Pulse-height spectrum of **Na gamma rays obtained with a Nal crystal, and the associated decay scheme. Note that the 51 1-KeV line is due to positron annihilation. ¥ Let 342 #868 Particle Detectors and Radioactive Decay Counts per minute over 32 bins it) S00 1000 1500 2000 2500 3000 3500 Pulse height (Channels) and 302 keV. positrons are cmitted; the positrons are usually stopped in the walls of th source container, or in the crystal face, and as they come close enough:t an electron they annihilate into two gamma rays, each pamma ray sharing: the energy of the electron—positron pair?’ iti 18 one” of these gamma a (MATLAB provides a useful utility command, ginput, for inecactive : idcnitying the peak position in spectrum plots using the cursor on vos ses continuous instead of being a sharp line as is the case with gamma-ray a spectra; in addition, electrons may lose variable amounts of energy before: ae reaching the scintillation crystal, so that unless special precautions are: ee taken, the energy resolution is usually poor. 3? See also the detailed discussion in Chapter 9. 38Note that they are emitted with a relative angle of 180°. SOAS a nT hh Sia iniainh Sanh hs SSNS Bearer enh nner ceene ane ee eens enmmitneematnet 8.4 The Scintillation Counter 43 Peak channel ib) 500 1000 #500 y-ray energy (keV) =" FIGURE 8.24 Plot of gamma-ray energy against the central channel of the photopeaks =" appearing in the spectra of Figs. 8.21 through 8.23. The detector response is obviously quite =| finear over this range. Note also that for a zero photon energy, there is a “pedestal” of a = few hundred channels. This ensures that none of the spectrum is last below the range of the = multichannel analyzer. In interpreting gamma-ray spectra some care must be taken since spu- = cious peaks due to instrumental effects or physical effects da appear. :. First, there can be peaks arising from the emission of X-rays, following :, photoejection of K-shell electrons either in the source or in the shielding. = Also, a peak may appear due to photons that backscatter (by 180°) in the =. photomultiplier window or elsewhere; then the Compton-scattered elec- = tron escapes, but the scattered photon becomes converted in the crystal. | For '37Cs with its 0.662-MeV gamma ray, the backscattering peak appears = at 0.185 MeV and can be identified in a carefully measured spectrum. Another spurious effect occurs when an incoming photon of energy £ ejects a K-shell electron from the iodine of the crystal, but the emitted X-ray eo @8CAPES without converting in the detector. The ejected photoelectron has = an energy E— Ex, Z where Ey is the energy of the K shell of iodine, namely, 29 keV, and will give rise to a peak not coinciding with the true photopeak. This so-called 344 #88 Particle Detectors and Radioactive Decay calculated.?9 8.5. SOLID-STATE DETECTORS 8.5.1. General ey ond ee We have seen how the gaseous ionization counters and the scintilla ey tion counters are widely used for the detection of radiation and charged : particles. It is also possible to use semiconductor materials for the detect: oe tion of charged particles, especially those of low energy; such detectors aré:=2 appropriately referred to as “solid-state counters.” ” : In a general sense, we can think of this type of detector as a solid. Be State ionization chamber, having two basic advantages over a gas-filled ae ionization chamber: eos Be i ob (as compared to approximately 30 eV in a gas) so that stronger signals st better statistics can be achieved. wee possible to stop, in the detector, particles with energies typical of nuclear a: interactions. Consequently a very large number of electron-ion pairs are | formed, leading to very good energy resolution. A 1-MeV proton stopping: in a solid-state detector will create 300,000 electron-ion pairs, while thé:# Same proton traversing a proportional counter of 2-cm thickness would, only release approximately 30 pairs. ae Sites: 9 See the Encyclopedia of Physics, Vol. 45, Nuclear Instrumentation H, p, 110. 40The scintillation counter is also a detector in the solid state! 68.5 Solid-State Detectors 345 In practice, however, it must be possible to collect the free charges (those =: created by the passage of the charged particle) before they recombine; this :: qpight be done, for example, by the application of an electric field in the e detector material. This requirement ts very difficult to meet with any of the = ordinary crystals. Clearly, the material must have a high resistivity, since = otherwise current will flow under the influence of the field, masking the = effect of the pulse produced by the passage of the particle; on the other = hand, in high-resistivity materials, the mobility of the free carriers is very =; low and the recombination probability high. Even though some results have been obtained by using diamond as : a detector, semiconductor materials come much closer to fulfilling the = sequirements mentioned above. Very pure material (an intrinsic semicon- :* ductor) is used to achieve the necessary high resistivity, on the order of = 10’ Q-cm, and the detector is operated at low temperatures. Such devices sc are called “bulk semiconductor detectors.” SEES STREETER REN ERS RENEE Wee uueatnnohnninn nine ui aman emnnieemenmmannutnn meme eee enantio rue imme ena EEE RTE A great improvement occurs when a semiconductor junction*! is used as : the deiector volume; a device of this kind is called a barrier-layer detector. The junction is made by either of the following methods: (a) Diffusing a high concentration of donor impurities on a p-type : material, usually silicon, thus creating an n—p junction. (b) Utilizing a thin p-type surface formed by oxidizing n-type silicon = or germanium when it is exposed to air. This surface is so thin that it is : usually coated with gold to provide a good electrical contact; thus we have = a p—n junction. In either case the operation is similar, but the junction is always reverse biased. Below we will briefly discuss the diffused junction (n—p) type of detec- - tor; Fig. 8.25a is a reproduction of Fig. 2.20, and gives the configuration of the energy bands at an n—p junction, electrons being the majority carriers in the left, or 2, region, and holes the majority carriers in the right, or p, region. : Electrons may oot move to the right, since the conduction band is ata higher : (negative) potential, and holes may not move to the left, since the valence - band is now at a higher (positive) potential; as a consequence there is some repulsion of majority carriers from the junction; Fig. 8.25b shows their :- density distribution. We note a “depletion zone” in the region marked §—T. 41 Semiconductor junctions were discussed in 2.4.2, and the reader may find it useful to : review that matenal. 3446060 8 sOParticle Detectors and Radioactive Dacay distribution of impurity cenicrs on the two sides of the junction. (d) Distribution of space Be charge on the two sides of the junction. Z the junction; that is, these centers which may be expected to be ionized bY the passage of a charged particle. To the left the donors have given electrons: to the conduction band and are left positive; to the nght the acceptors have = a a ee nC ee ae BIBIBT ELAR ETREAT RLS ET B.5 Solid-State Detectors M? “ acquired electrons from the valence band and are left negative. However, = these impurity centers are neutralized by the majority carriers, so that the free (space) charge distribution is the sum of Figs. 8.256 and 8.25c, as = shown in Fig, 8.25d. Thus we see that space charge exists in the region of the junction, and as a consequence an electric field (the so-called barmer) exists as well, and extends over the depletion zone. If an electron-ion pair is created in the : depletion zone, the electric field is such as to accelerate the negative charge ’ toward the n region, where it will have high mobility (being a majority carrier); similarly, the hole will be accelerated toward the p region. Thus a good collection efficiency is achieved. Figure 8.26 shows the same junction under reverse bias, 8.26a being :. the same as Fig. 2.21. Figure 8.26b gives, as before, the density dis- tribution of majority carriers, which are now further removed from the = junction, and Fig. 8.26c is exactly the same as 8,25c, giving the density of = jmpurity centers. Figure 8.26d, however, which gives the space-charge dis- = oibution, shows that the ionized impurity centers have reached saturation and extend beyond the junction. Thus, most of the applied bias voltage appears across the depletion zone, which now is much more extended; the limit to this increase in sensitive detector depth is set by the breakdown . yoltage of the semiconductor material itself, In a diffused junction, such as used for a detector, the concentration of donors in the n-type material is much larger (about 10°) than the : concentration of acceptors in the p-type material. Since the total free charge must be the same on both sides of the junction, the space-charge distribu- - tion is asymmetric, as shown in Fig. 8.27b. Figure 8.274 gives some of the physical dimensions in a realistic diffused junction; we note that most of = the “sensitive volume” is in the p-type material. “. §.5.2. Practical Considerations in Solid-State Detectors CSR SKS Stat cn SSR ETS REO eR SoS CT - From the previous discussion we have seen how a semiconductor junc- > tion may provide the appropriate electric field within a solid so as to = collect electron-hole pairs produced by the passage of a charged particle. - Multiplication such as occurs in the proportional or Geiger counter never - takes place in a solid, except under special conditions (“avalanche detec- . tors”). To achieve good resolution in a solid-state detector one must always = collect all the electron-hole pairs produced. Thus the sensitive volume of 348 & Particle Detectors and Radiaactive Decay ly = ef one Bias (b) (¢) {d) FIGURE 8.26 The n-—p semiconductor junction under reverse bias, The plots in (a), (bys: 2 (c}, and (d) pertain to the same distributions as described in the legend to Fig. 8.25 but under reverse bias. Note the increase of the “depletion zone,” 5’T’, = SS ahs 8.5 Solid-State Detectors 349 107*cm neat aa ~_ 0.2 om (Typical) a = — Particles incident on Sensitive volume 4 this surface 1077 to 107*' em AYALA SAAD ADA A p (a) ' FIGURE 8.27 Arrangement of an n—p semiconductor junction for use in a solid-state -. detector. (a) Actua] dimensions. (b} Distribuuon of the space charge. ' the detector must be longer than the range ofthe particle detected; it is alsa desirable that the dead layer at the entrance side be as thin as possible. Since detectors with sensitive yolumes*? of a length of 3 mm have been : achieved, the use of solid-state detectors has been extended to particles of * energies as high as 30 MeV. The resolution in energy is usually extremely : good—that is, on the order of 0.25% for alpha particles (see also Fig. 8.31). : The overall size of the detector is restricted to a few cubic centimeters, due :, to the available semiconductor crystals; on the other hand, small size and the absence of need for a photomultiplier are a great advantage. It is also possible to use solid-state detectors, not as total absorption : ‘counters, but as dE/dx devices, in which case the p region is also made :thin and no electrodes are placed in the path of the particle. Such detectors ~. have been made to respond to high-energy (minimum ionizing) particles 42The sensitive volume or barrier depth can bé obtained from a nomograph, as given by : gd. L. Blankenship, “Proceedings of the Seventh Scintillation Counter Symposium, institute 2 of Radio Engmeets, N'Y," Nuel. Sci. '7, 190 (1960). MATA 350 & Particle Detectors and Radioactive Decay Discriminator and scaler 5 pF Preamplifier Amplifier FIGURE 8.28 Typical setup for use with a solid-state detector including a feedback: preamplifier. : as well, Semiconductor devices are also very useful for the detection of: gamma rays. In general due to their smalj size, the ratio of counts in the?) photopeak as compared to background counts is smaller than that for a.2225 scintillation crystal; however, the resolution is excellent, reaching one part? in a thousand,*? oS : In practice, the construction of a solid-state detector is an art, and the 22: attachment of electrodes to ensure good ohmic contacts may be quite? 2222 difficult. When germanium is used, cooling to liquid nitrogen temper-. atures may be required, while silicon gives good resolution at ambient temperature. The output signals are small, the voltage being determined 2222 by the capacities of the junction and of the amplifier input; the former 2 depends on the length of the depletion zone and the area of the detector. “25 If we assume a typical capacity of 200 «KF, then for 1-MeY energy loss, =: the signal voltage is BE ya 216% 107'? x (10°/3) ~€ 200« 10-1 , It is necessary to use a charge-sensitive preamplifier because the capacity 22 C depends on the applied bias; thus if voltage is directly measured, severe 2" variations in gain occur when the bias is changed. Leakage current in the 22% 2.5x 1074 ¥. (8.42) 3 : crystal and amplifier noise set the limits of the smallest detectable signals. 2 Most of the hardware for solid-state detectors as well as the detectors 22 themselves are now commercially available“*; Fig. 8.28 shows a typical =: a i eo Be ah setup with afeedback preamplifier. A surface-barner silicon detectorisused =: 43.G. T. Bwan and A. J. Tavendale, Can. J. Phys. 42, 2286 (1964). “4For example, from Oak Ridge Technical Enterprises, Oak Ridge, TN. wytctlelededadeletet ity hii leleretetetate ts SE RS SSSR asa ns SS B.5 Solid-State Oetectors 351 and operated at room temperature. Figure 8.31 gives the response obtained from polonium alpha particles of different energies (after attenuation in air). Another type of solid-state detector, called p—i—n (positive-intrinsic- negative material), consists of a layer of intrinsic crystal placed betwcen p- and n-type material. It has the advantage of a much longer sensitive volume. 8.5.3. Range and Energy Lass of 7Po Alpha Particles in Air In Section 8.3.2 a description of the method of obtaining an estimate of the range (and hence energy) of *!"Po alpha particles in air, by means of a crude ionization chamber, has been given. With solid-state detectors, it is possible to improve on these measurements, as well as to study the rate of energy loss of the alpba particles as a function of their energy. A collimated *!°Po source and the detector are both placed in an evacu- ated vessel at a fixed distance of 15 cm, as shown in Fig. 8.29. Then air is allowed into the vessel, and as a function of the pressure we measure: (a) The number of particles counted in the detector, and (6b) The pulse-height distributton of the output signals, namely, the energy of the alpha particles when they reach the detector. In measurement of type (a), the same number of alpha particles should be reaching the detector until the pressure 1s raised to the point where the amount of material (g/cm* of air) between source and detector is equal Slits Pot 16 Detecior SOUS Signal out Nae To pump and gauge FIGURE 8.29 Arrangement for the measurement of the cange in air of *!9Po alpha particles. Note mounting of Lhe solid-state detector and source inside an evacuated chamber. 352 8B Particle Detectors and Radioactive Decay Counts/s 0 0.5 1 1.6 2 25 3 3.5 4 Effective distance (cm) FIGURE 8.30 Data on the number of counts from a *!°Po alpha source reaching ti solid state detector as a function of pressure in the experimental chamber. Note that the: corresponding effective distance in centimeters of air at stp is indicated, The dashed curve is: the derivative of the solid line; it indicates the “straggling” in the range of the alpha patie to the range of the alpha particles; beyond that pressure the counting rate, should abruptly fall to 0. Note that since the relative position of source and: detector is not altered, the solid angle AQ does not change, and the only: variation arises from the increased multiple scattering; this, in turn, may: result in some loss of particles from the beam. “S These considerations are indeed borne out by the results obtained by ai student and shown in Fig. 8.30. Here the ordinate to the left gives the counts: per second while the abscissa gives the pressure of air in centimeters of mercury, or, equivalently, the effective distance of air at stp. The dashed: curve to the right ts the derivative with respect to distance of the counting: curve and gives the range (and so-called range straggling) of 7!9°Po alpha, particles. We obtain a mean range of R = 3,72 +£0.06 cm and an extrapolated range R = 3.82 + 0.06 cm, which might indicate some systematic discrepancy from the accepted value: for the extrapolated range of 3.93 cm. is 8.5 Solid-State Detectors 359 2000 a re Vacuum 71800 + 32.8 6m Hg - Counts/min Discriminator channel FIGURE 8.31 Distribution of output pulse height of the solid-state detector for five different pressures. Note the gradual decrease of the energy of the alpha particle. Turning now to the measurements of type (b), Fig. 8.31 shows the dis- tribution of the detector pulse beights as obtained with the single-channel ~ discriminator (described in connection with the scintillation counter). Each peak corresponds to a different pressure, and we thus uote that the alpha - particles reach the detector with progressively less energy when they have lraversed more grams per squared centimeter of air. We set the pulse height ~ abtained in vacuum equal to the full energy of the 2!°Po alpha particle, namely, 5.25 MeV, and use the linear characteristic of the solid-state detec- ' far to obtain the energy of the alphas as a function of material traversed. The results obtained by a student are given in Fig. 8.32 (solid curve). If the derivative of the energy curve is taken with respect to distance, we obtain the energy-loss curve, dE/dx, as a function of distance, as shown * by the dashed curve in Fig. 8.32. Such a curve is called a Bragg curve, _ and shows a 1/E dependence*® as predicted by Eq. (8.12); for the parti- ~ cles KE = 5 Mv* and the influence of the logarithmic term of Eq. (8.12) is - Minimal. As the particle reaches the end of its range the energy loss d E /dx “ drops rapidly to 0. . 43We might plot the dE/dx curve against energy by making use of the data of the energy : clrve to express the distance from the stopping ppint in energy units. 354 8 Particle Detectors and Radioactive Decay es “1 ) = ee ee 4.5 . al in ro np tA dE/dx MeV/em {air at $tp) Energy of o at detactor (MeV) in ca —_ o tn 4) 0.5 4 1.5 2 25 3 3.5 4 Effective distance of air stp (cm) FIGURE 8.32 Plot of the residual energy of a polonium alpha particle when it reaches ine oe detector as & function of air pressure (plotted, however, in terms of the equivalent amouny se fat roe = Fig. 8.31. The dashed curve represents the derivative of the solid (energy) curve; ts, gives the energy loss per unit length. {tis called the “Bragg curve.” an average loss of 36 e¥ for the production of one electron-ion pair in air = 4°For historical reasons, the standard unit for decay rate is the Curie = 3.7 x 10/9 ~~ per second, This is the number of decays per sccond in one gram of radium. The modern (SH: ee unit is the Bequerel, defined as one disintegtation per second, so 1 Bq = 1/(3.7 x 10!) hg For more details, see Appendix D. anaes 8.6 Nuclear Half-Lifa Measurements 355 = proportional to the number N of nuclei in the sample at any particular :” gime. That is, R=—=-)N. | dt = The proportionality constant is called —A, the minus sign reflecting the =: fact that the decay causes the number of nuclei to decrease with time. This = differential equation has a simple solution, namely ° N(t) = Noe", =! where No is the number of nuclei present at f = 0, Obviously, 4 charac- =" terizes the lifetime. The larger A is, the faster the sample decays, and the = shorter the lifetime is. There are two definitions we use for the lifetime. =: One is the mean life: t= r = The other is more practically minded, and measures the time it takes for <° the sample to decay to 1/2 its original number. This is called the half-life, : and itis determined by solving N(t) = No/2 fort. In 2 ij2= oa = 0,693 7. : References usually quote the half-life, but not always. Be sure when you = look up a lifetime, that you are getting the half-life or mean life. A good source of information on nuclear decay half-lives is the National Nuclear =. Data Center at Brookhaven National Laboratory and available at the Web =. site http://www.nnde.bni.gov/nnde/nudat/radform.html. Obviously, we must resort to some sort of trick to obtain a sample nuclei : with short-lived states that can be measured. One trick we will use is the : chemical separation of barium from cesium. However, we will also create oh new isotopes using a type of nuclear reaction called neutron activiation. : In neutron activation, reactions with neutrons are used to create radioactive :. isotopes from stable nuclei. Neutrons are produced using a plutonium- “ beryllium (PuBe) source, which is safely packaged away so you cannot = get near it, and allows the neutrons to irradiate samples inserted into the container. Plutonium decays by a-emission, that is, 239py 5 251 +a, 356 «=©=s 8-—s Particle Detectors and Radioactive Decay and the & particles react with the berylhum a+ Be > ?C+n2, releasing neutrons. These neutrons are slowed down by collisions with protons {in all the paraffin, ahydrocaron surrounding the source), peace discussed in Section 3.9. The data presented here were taken with = different setups, including a multichannel scaler plug-in board for a desktoj running LogeerPro. All one really needs is an interface and software . ze counung pulses from the Geiger counter (or other detector and clectromict counting again for the same fixed period of time, and so on, A graphical 7 display of the data as it comes in is very useful, and generally part of amy commercial package. ee In what follows, we discuss the analysis of three radioactive isotopes with varying half-lives. A key point is the presence of some sort 0 “background” signal, in addition to the primary radioactive decay. (Sucl backgrounds are always present, at least at some level.) In the first example (116Ty decay), the half-life is rather long, and a method for estimating the background level “by hand” and for incorporating its effect into the sys~ tematic error is outlined. In the case of }°""Ba decay, a fitting technique that allows one to determine the background precisely and find the half life with its corresponding random uncertainty is discussed. Finally, we discuss radioactive silver isotopes, which present a combined signal from two radioactive isotopes, cach with relatively short half-lives. 8.6.1. Production and Decay of {!In ‘a te ot ‘ao errr a You can produce !!°In using neutron capture on a piece of indium. Indium is a very common metal used for soldering compounds, and all of natural: indium is the isotope '!7In. The decay scheme for '!In to ''®Sn is shown in Fig. 8.33. Note that the ground state has a very short half-life, only 14 5+ 0.06 MeV (64 m) 1+ 11% 116, (145s) (2=49) 40% A+ 2.80 MeV 49% —— 2.52 MeV q 4+ 2.38 Mev 2+ 2,22 MeV 2+ 1.29 Mav ae 8Sn (2=50) FIGURE 8.33 Decay scheme for !'®Jp, 3.3 MeV You will be detecting 8~ decay of the excited state, 60 ke V above the ground state, The decays proceed mainly to a couple of states at around 2,3 MeV, and the available energy is 3.3 MeV, so the §~ typically have energies up to a megaelectronvolt or so. These are easy to detect in a Geiger counter. Irradiate the piece of indium for an hour or so. Remove it and place it on the Geiger counter platform, close to the counter window. Take data for an hour or so, setting the multchannel scaling program to count for intervals of something like a minute. It is probably a good idea to make a semilog plot of the data, and estimate the half-life by hand, just to make sure the result looks about right. To do a better job, you can easily fit the data to a decaying exponential. Just use the MATLAB function polyfit to fit the logarithm of the number of counts versus channel to a straight line. In fact, this is a case where you can accurately write the random errors of the points, since they are governed by a Poisson distribution. That is, if there are N counts in any one channel, then the random uncertainty in V is SN = 4/N, and the random uncertainty in the logarithm of N is 5log N = 1//N. A sample of data on indium decay is shown in Fig. 8.34. Each channel represents 30 s. The simple fit described above is shown by the dashed line. Note that the fit is not really very good. You can see that more clearly if you plot the difference between the fitted function and the data points. In fact, this is nol too surprising since you expect some background radi- ation from other radioactive isotopes in the piece of irradiated solder. You can try subtracting a constant value (representing the background counts) 358 «868 «~Particle Detectors and Radioactive Decay eee Number of counts per channel * td ” sy * Stas ar 0 50 100 150 200 20 300 350 Time (Channels) a FIGURE 8.34 Data and fits for the decay of !!®In, The dashed line is fitted to a decay- ~:2: ing exponential, while the solid line includes a constant background of 17 counts. The. a muitichannel scaler recorded duta every 30 s; that is, each channel represents 30 5, S from the data before you fit it, and see whether it looks better. By cal- cs culating the x* function, you can even optimize the background term by = minimizing x°. Z The MATLAB program shown in Fig. 8.35 was used to do exactly this. = After reading in the values of channel and counts, the user is asked for a number of background counts. Then this value is subtracted from the data, and care is taken to make sure the value is not less than 1. (Remember, you are going to take a logarithm.} Two fits are done, one that is unweighted (using polyfit} and one that is weighted according to the Poisson uncer- tainty in the points (using linreg). The results, including the x*, are printed and plotted. By trying various backgrounds, you find that the lowest x (i.e., the “best fit”) is found for 17 background counts. You can even esti- mate your systematic uncertainty by looking at how much the lifetime varies as you move around in x* near the minimum. This can be large if the minimum in x” is shallow. For this particular data set, we find that t = 160.7 + 2.0+ 10 channels, ee ee Ca Sessa iat . 7 LF, Cues) Ceca a rae_a a hae A % LOAD AND EXTRACT DATA POINTS load indium. dat chaneindium(: ,1); data=indium(: ,2); ra % PREPARE DATA FOR FITTING LINE TO LOGARITHM bkgdeinpot (*Background counts *); dnet=max (data~bkgd,1) ; ndof=length (data)-2; edata=sqrt (data); ldata=log(dnet) ; eldata=edata./dnet; Y i, UNWEIGHTED FIT coefa=polyfit(chan,]data, 1); fita=exp(polyval (coefa,chan)); chisqa™sum{ ((dnet-fita} ./edata) .*2); fprintf (’Unweighted fit:\a’); fprintf(’ tau™%é6.3e\n’ ,-1.0/coefa{i)); fprintf<’ chisquare/dof=“6.3f\n’ ,chisqa/ndof); vA 4, WEIGHTED FIT (coefb, scoefb,1fitb] =linreg (chan, ldata,eldata) ; fitb=exp(1fitb); chiagb=sum({(dnet-fitb) ./adata) .*2); fprintf(’Weighted fit:\n’); fprintf(’ taumZ6.3e’ ,-1.0/coefb{2}): fprintf(’? unceart%46.3e\n’ ,scoefb(2) /coaefb(2)°2); fprintf(’? chiequare/dof=%6.3f\n’ , chiaqh/ndof) ; FIGURE 8.35 A MATLAB program (1.c., m-file) used to fit indiam data. The program asks the user for a number of background counts, then carries out the fit, and reports the results, including the x~. Although the background level can be fitted automatically using nonlinear fitting techniques, this program gives onc a feeling for the sensitiviry of the x? to the background level. where the first uncertainty is random and the second is systematic. Since éach channel is 30 s, we determine that I t12 = log2 x t x min/channel = 55.7 + 0.7 + 3.5 min, which agrees well with the accepted value of 54 min. In fact, it seems we may have overestimated the systematic uncertainty. 4.6 Nuclear Hali-Life Measurements 359° 30 06 «Particle Datectors and Radioactive Decay Actually, this business of adjusting the background term to minimize x? : can be done automatically in MATLAB, That brings us into the world of: nonlinear fitting, and we will do that next. : 8.6.2. The Half-Life of °’" Ba Now we will measure the half-life of another short-lived isotope, !°7"' Ba.’ The background is very clear in this case, and we will use that to go a step‘: further in our data analysis techniques. This isotope does not need to be produced in the neutron oven. ea Recall the decay scheme of '5’Cs in Fig, 8.21, The daughter nucleus, 2 137 Ba, is produced in its ground state only 5.4% of the time. The rest of the: time it is made in the excited state, called '?’"Ba for “metastable,” which: decays by y-ray emission, but with a relatively large half-life (for y decay):: Be of around 2.5 min, Of course, '°7""Ba is produced all the time, as the very), 2% long-lived '3/Cs decays, so you cannot isolate the 137" Ba decay without: somehow separating it from the !7’Cs. SS You can make this separation because chemically, cesium is very diffe~ 33% rent from barium. By passing a weak acid solution through a !37Cs source, 232 barium is captured and comes out in solution. Some cesium comes through 22:2 as well, but most of the radioactivity of the solution is from !°7"' Ba. Simple :: kits are available*’ for carrying out this chemical separation. It is best if: you squeeze the drops through slowly, enough to fill the smali metal holder : in about 30 s. Then place the holder in the Geiger counter tray, and start: the dala acquisition program. Realize that you are working with radioactivity and hydrochloric acid. :: Do not be careless. None of this is concentrated enough to be particu-:” larly dangerous, but you should take some simple precautions. Disposable: gloves are located near the setup. It is also a good idea to wash your hands: : soon after you are finished, : You should choose a dwell time that allows you to get a relatively large. 25 munber of points in each channel, but many channels over the expected: 2224 decay timc of a few minutes. You should he able to get several hundred": counts per bin in the first bin or two, and a background of less than 20 counts 22% per bin. (The background level will be clear after counting for a balf-hour. ee You might need a few tries to get all of this where you want it. ee: _—____ ee ‘7 For example, from ‘TEL-Atomic, Inc., http://www-telatomic.com/. ne cnet ee ee ae Se ae ore a eee ee ere Ae De ee ed Re ee TRSRERSTES ESS BETS ECSU ATESUS MSHS AES ES WUT Ta Dan Da Daan aaa Dae Da ae Dae alee a ce 8.6 Nuclear Half-Life Measurements 361 You can use the program in Fig. 8.35 to fit the data and adjust the back- ground counts, but that is tedious. In this case, since the background will be very clear, you can determine it precisely by averaging over the last many channels, and subtract that number from the data before fitting. However, MATLAB gives you the ability to fit things all at once. What you need to do is minimize the x? function numerically, and MATLAB gives you a numerical minimization function called fminsearch that can do this. You need to minimize y? as a function of thrée variables, two for the exponential fit and one for the background value. First, write a simple m-file called expcon.m, which calculates the function you are going to fit to the data: function y=expcon(x,N0, tau, bkgd) y=N0*exp (-x/tau) tbkgd; and then write another called fitexpcon.m, which calculates x7: function chisgr=fitexpcon (pars, xdata,ydata, edata) chisqr=sum({( (ydata-expcon(xdata,pars(1),pars(2), pars(3)))}./edata) .*2); Do not forget that for these data, the array of uncertainties edata 1s just the square root of the counts, ie., edata=sqrt(ydata). (If any of the channels has zero counts, then set edata equal to unity.) Play around with some values of pars(1,2,3) so that you have a good starting point. (Just plot the data points, and then overplot the function éxpcon until it looks kind of close.) Then type the command fminsearch(@fitexpcon, pars,0,[],xdata, ydata, edata) and you will get the best-fit values returned. (Check the help documentation for details of the arguments for fminsearch.) Exactly this procedure was followed to ht the data shown in Fig. 8.36. The fit achieves a minimum y? fora lifetime t = 3.80 min, corresponding to a half-life t}2 = 2.63 min. The random uncertainty is determined, as shown on the right in Fig. 8.36, from the values of t that increase the minimum x7 by one unit. These x7 “data” are fitted to a parabola, and we determine the uncertainty in t to be 40.10 min. Consequently, we 362 & Particle Detectors and Radioactive Decay 300 250 Fit ta the form N,exp{~t/r)+B Minimum x*=103 Number of data points=100 2 200 Cc 43) Mes oO 2 150 2 aed Fs) (5 100 50 Time (Minutes) 3.65 3.7 3.75 3.8 3.85 3.9 3.95 Lifetime rt (Minutes) FIGURE 8.36 An example of a nonlinear fit, The data are from the decay of 137mpRa EE ; including some constant background. The MATLAB function fminsearch was used to “34 make the fit. The plot on the top shows the best-fit curve, while the lower graph shows the 24 x* minima found by fixing the decay lifetime to various values. The random error in the © lifetime is determined from the values that increase x? by one unit. 8.6 Nuctear Half-Life Messurements 363 find that fyj2(19" Ba) = 2.63 + 0.07 min. This is in good agreement with the accepted value of 2.55 min. Note that the radioactivity you detect from /°’"'Ba decay is y radiation, which is not detected very efficiently by a Geiger counter. You might try using a Nal(TI) detector instead, keying in on the particular y-ray in ques- tion. This should greatly increase your counting statistics, as well as reduce the background. 8.6.3. Radioactive Silver Isotopes Natural silver is pretty much evenly divided between two isotopes, 9’ Ag and ' A», Neutron activation captures a neutron equally well on these two isotopes, producing the two radioactive isotopes '8Ag and '!"Ag, Both of these decay with a relatively high momentum A that is easy to detect, but one isotope has a half-life of 24.4 s and the other of 2.42 min. You might want to look up the decays to get more details. Take a piece of pure silver foil and cook it in the neutron oven for at least 10 min. Quickly take it out, put it in the Geiger counter, and start the program. Do not forget that the lifetime of the shorter-lived isotope is only half a minute. It should be clear frorn the raw data that there are two lifetime components from the decay. Representative data taken by students is shown in Fig. 8.37. The dwell time was set to 2.5 s, but in order to get better statistics in each channel, the MATLAB function reshape was used to add every four channels together. Error bars are added to the data points using the errorbar function. The points are fitted to a double exponential decay, completely analogous to the way we fitted a constant plus an exponential to the /°?"'Ba data. The only difference is that the m-files for the fit function and for the y* are changed slightly. The best fit yields half-lives of 26.9 s and 3.53 min. The shorter half-life is in good agreement with the accepted value. The longer agrees much less well, but this 1s not surprising, No background term was included in the fit Jeading to an overestimate of the half-life), and the statistical accuracy of the longer decay is clearly marginal. The ambitious student can explore these points using the techniques discussed for !!“In and !37"Ba decay in the previous sections. 30440 8s«éParticle Detectors and Radioactive Decay Counts per channel 6 8 8 & & ho o o 100 200 300) 400 600 Time (Seconds) FIGURE 8.37 The decay of meutron-activated natural silver, fitted to the sum of two: decaying exponential functions. The plot was made using the MATLAB function arrorbar:: In addition to the best-fit curve, we show the two individual exponentials separately, 8.7. REFERENCES By necessity the discussion presented in this chapter is not complete;: Below is a selective list of references (including those already mentioned: in the footmotes to the chapter) that the reader may consult for additional information. : On interaction of radiation and particles with matter: E. Fenni, Nuclear Pirysics, Univ. of Chicago Press, Chicago, |950. J, D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, New York, 19672, W. Heitler, The Quantum Theery of Radiarion, 3rd cd., Oxford Univ. Press, London, 1954, On gaseous and scintillation detectors; ncutron detectors: W. J. Price, Nuclear Radiation Detectors, McGrmw-Hill, New York, 1958. B. Rossi and H. Staub, Jontzation Chambers and Counters, McGraw-Hill, New York, 1949. J. Sharpe, Nuclear Radiation Detectars, Methuen, London, 1955. Encyclopedia of Physics, Vol. 45, Nuclear Instramentation I, Spriuger-Vertag, Berlin, 1958. J. B. Birks, The Theory and Practice of Scintitiation Counting. Pergamon, New York, 1964. On solid state detectors: 1. M. Taylor, Semiconductor Particle Detectors, Butterworth, London, 1963, Paha ha aah Lead ba Lt ek ba Ce ee ey TSTENTN NNN MRNA AN HN eRe eR Ss a Ly oe eee eee a ey a areata heh rie heat be REAR Oe Be ee ee a ee ee ee a ae COGS mye ONS Sates 8.7 References 365 There are a number of good introductory textbooks on nuclear and particle physics, Some examples are: K. 8. Krane, /nsroductory Nuclear Physics, Wiley, New York, 1988. This is a good basic book with some discusston of experiments and experimenta] methods. S. 5. M. Wong, Introductory Nuclear Physics, 2nd ed., Wiley, New York, 1998. A bit higher level Wen Krane, bul a thorough survey of the undedying physics of nuciei. D. Griffiths, {ntroduction te Elementary Particles, Wiey, New York, 1987, An excellent undergraduale level discussion of particle physics. s D, H. Perkins, [Introduction to High Energy Physics, 4th ed., Cambridge Univ. Press, Carobridge, UK, 7000. A modern, up-to-date version of a classic book. Many of the details of detectors, materials, and the statistics of nuclear process, as well as an excellent summary of particle physics, can be found in: Particle Data Group, Review of particle properties, Eus. Phys. J. C18, 1-878 (2000). ee i ae ae rs a a ar a rT a aN a ater ea Sad a a ar a Nar Ma ar a aa a a a ta a ar a eh boa. CHAPTER 9 Scattering and Coincidence Experiments 9.1, INTRODUCTION Ever since Rutherford performed his original experiments on the scattering of energetic alpha particles from atomic nuclei, scattermg has become increasingly more powerful as a tool for investigating the forces between elementary particles. By now it is familiar to the reader that an electron, under the influence of the attractive electromagnetic force of the nucleus, may be found in a bound state. The classical analogue of this situation is the motion of the plancts around the sun under the influence of the gravitational force; they describe elliptical orbits. In general, a scattering experiment probes a system by sending a pro- jectile “into” it, and then studying what “comes out” of it. Surnilariy, correlation or “coincidence” experiments can probe a system by looking at what comes out simultaneously in two or more directions. In this chapter, we will study some types of each of these measurements. 367 368) 02«=6 9) Scattering and Coincidence Experimants The experiments in this chapter make use of radioactive sources. We rec- S ommend that the reader review the material on radiation safety in Appendix D before undertaking these measurements. é The concept of “solid angle” is important for understanding the formal- ism dealing with cross sections. The solid angle is a three-dimensional » generalization of the familiar planar angle A@, which is the length of a =: circular arc As divided by the radius r of the circle, ie., AQ = As/r.! Solid angle AG is the area AA of a piece of a spherical surface, divided by the square of the radius, 1e., AQ = AA} r*, Planar angles are mea- sured in radians and solid angles are measured in steradians. Just as a circle 2334 subtends a planar angle of 2x to any point included in the circle, a sphere?! subtends a solid angle of 42 to any included point. TSB Solid angle is a useful concept whenever we are dealing with some sort:: of detector intercepting radiation which spreads out in all directions from':::222 a source. Ionizing radiation and elementary particle detectors are just one 2:22 example, but you would encounter the same thing in fields like optics or 224 sonics. To be explicit, let dA be a vector whose magnitude is an area dA in3222 some plane, and whose direction is normal to that plane. Let n be a unit" oes vector pointing toward the source, which is a distance r away. Then ) dA aA da—- ant | 1). where dA. is just the perpendicular component of the area. A spherical: surface is most convenient since all surface elements are normal to the: direction to the ccnter. In spherical coordinates (7, @, p), where 0 < 6 < 9” is the polar angle and 0 < ¢ < 2m is the azimuthal angle, a differential::: element of the surface has area : dA = width x height = (r sin@ d@) x (r d@) = r* sin@ dé dd, so the infinitesimal solid angle is just | : dQ = sind do d¢. (0.2) You will encounter this equation many times in physics. : We can easily apply this lo the common case of a “detector face,” normak to the direction of the incident radiation, as shown in Fig. 9.1. Let the “detector face” be a circular area with radius R located a distance d from a 2 source. There is perfect azimuthal symmetry, so we immediately integrate 22% 9.2 Compton Scattering 369 FIGURE 9.1 Calculaung the solid angle of a circular face. over @ to pet aQ. = 27 sin é dé and integrate from @ = 0 to Onax = tan !(R/d) to get AQ 1 Fra —_= a sin 6 ad, An 2 a=0 where we have written the fraction of the total solid angle as A&2/4z7. This integral is done most easily by a change of variables to 2 = cosé@ with pz ranging from cos Oma, = d/Vd* + R* to 1. Since du = — sind dé, AQ I ] d — = du = —|1———-——- }. 93 Ar L 7 Al PaaS | 0%) For d = 0, AQ@/4nr = 1/2, that 1s, the surface covers one entire hemi- sphere. Ford —» oo, expand Eq. (9.3) to first order in R/d to find AQ/42 = R*/4d* or AQ = (2 R*)/d?, which is just what you expect from the basic definition of solid angie. 9.2. COMPTON SCATTERING 9.2.1. Frequency Shift and Cross Section This section deals with the scattering of electromagnetic radiation by free electrons. As mentioned in the introduction to this chapter, it is the scatter- ing of electromagnetic radiation from various objects that makes it possible for us to “see” them. However, as the frequency of the radiation is increased beyond the visible region, the light quanta have energies comparable to, or larger than, the binding énergy of the electrons in atoms, and the electrons can therefore be considered as free. 3700 9s Seattering and Coincidence Expariments hele direction implied by the scattering. If, on the other hand, we think of 2 incoming radiation as being represented by a beam of photons, we nese: 7 only consider the scattering of a quantum of energy & = hv froma nee electron: then, because of energy-momentum conservation, the with the experiments of Compton, & The frequency shift will depend on the angle of scattering and can bé: easily calculated from the kinematics. Consider an incoming photon-0f: energy E = hv and momentum Av/c (Fig. 9.2) scattering from an electroit: (at rest) of mass m; p is the momentum of the electron after scattering,: r and Av’ and Av’/e are the energy and momentum of the photon after the: scattering. The three vectors hv/c, hv'/c, p must lie on the same plans and energy conservation yields : hv + me* = hy’ +f p2c? + mct. From momentum conservation we obtain hv = hv'cos@ +cpcos@ 0=hv'sin@ —cpsing. ‘See, for example, J. D. Jackson, Classica! Electrodynamics, 3nd ed., p. 694, wie, New York, 1999. : ee oo - {ANNAN PER Stew ht SSE Me utets a ch tsdote ioe ante airtime ates Ty tet A, oy MA ay. Seba NN were TaTATaT Seer aa ~* ateat-= LEAS 9.2 Compton Scattering 371 ~ Here @ is the photon scattering angle, and @ the electron recoil angle, - To solve the above equations we transpose appropriately, square, and add = Eq. (9.5) and Eg. (9.6) to obtain h?y* —2h* vv’ cos +h? v* = cp. By squaring Ea. (9.4) we obtain h2y* 4+ hey? — 2h* vv’ + 2hme?(v -y)= c? pt. = and substraction of the two above expressions yields y— vp! / = 4 —cosé). (9,7) Vv me We can recast Eq. (9.7) into two more familiar forms: (a) to give the = shift in wavelength of the scattered X-ray beam A Ai =i’ —’ = —(1 —cos 8) (9.8) mc a or (b) to give the energy of the scattered photon E 1 + (E/mc?)(1 — cos 6) From Eq. (9.8) we see that the shift in wavelength, except for the angular dependence, is a constant, the Compton wavelength” h/mc = 2.42 x 107! cm = 0,0242 A. B= (9.9) For low-energy photons, with A >> 0.02 A, the Compton shift is very = small, whereas for high-energy photons with A < 0.02 A, the wave- length of the scattered radiation is always on the order of 0.02 A, the - Compton wavelength. These conclusions can equally well be obtained from Bq. (9.9), where the energy shift increases when E/mce* becomes large. For E/mc* >> 1, E’ is independent of £ and on the order of E’ © mc’. =. Hence A! = c/v’ = ¢/(E'/h) ~ c/(mc2/h) = h/me as stated before. Oh al be i) ee SOARED RDNA As an example, in this laboratory gamma rays from !37Cs are scattered from an aluminum target; since E = 0.662 MeV, we have E/mc* = 1.29, so that backscattered gamma rays (@ = 180°) will have E’ = £/3.6, 2The mass of the electron pte was used in evaluating A/ mec; by using the mass of the > pion, or another particle, we obiain the pion Compton wavelength, and sa forth. 32 26s Scattering and Coincidence Experiments which is less than 30% of their original energy. It thus becomes quité easy to observe the Compton energy shift as compared to X-ray scattering, where, if we assume } = 2 A, AA/A = AE/E =0.01. In the original experiments Compton and his collaborators observed (especially for high 4 materials) in addition to the frequency-shifted A-Tays, scattered radiation nor shifted in frequency. The unshifted X-rays, are due to scattering ftom electrons thatremained bound in the atom”: in this = process the recoiling system is the entire atom, and we replace in Eq. (9.8):) 22338 m by ma (where ma ~*~ 2000 x A x m,), resulting in an undetectable wavelength shift, AA’ = 1077 A. | Next we are interested in the differential cross section for the scatter- ing of the radiation from the electrons. Classically this is given by the Thorson cross section,* which can be easily derived: consider a plane wave propagating in the z direction with the E vector linearly polarized along the x direction. This is incident on an electron of mass m, as shown in Fig. 9.3. The electron will experience a force F = eF = eEqcosort, and its acceleration will he a eFo v = — coswt. m : According to Eq. (8.27), the power radiated by this accelerated electron will be (nonrelativistically, in SJ units) ° dP i é 32 dQ 4n Ane. : where @ is the angle between the direction of observation and the E vec-~. tor of the incoming wave. Using the expression for J, we can write for |: Eq. (9.10) averaged over onc cycle a dP\ 1f 2 \* 4, (aa)=3 (aan) €9 Ec sin @. Finally, from the definition of the cross section (see Section 8.2.1.a) we “222 have SER sin” ©, (9.10) . do — energy radiated /(unit time — unit salid angle) dQ — incident energy / (unit area — unit ume) 3 similar situation is discussed in the following section on the Mdssbauer effect, where : the nucleus remains bound in che lattice and the recoiling system is the entire crystal. 4See also Section 8.2.5. §.2 Compton Scattering 373 y FIGURE9.3 Classical picture of the scattering of electromagnetic radiation by an electron; this leads to the Thomson cross section. Here the denominator is clearly given by the Poynting vector fy = = J— Eg = ~eqc kg. (f) 2V wo 0 am SOe ‘i Thus we obtain 2 do e in? @ (9.11) — = | ———_ | sin‘ ©, dQ 4x éqme* where e- —___ = © An egmec? 9 has dimensions of length, and ts referred to ag the “classical electron radius” ro = 2.82 x 107% cm. Finally, we average aver all possible directions of polanzation of the incom- ing wave and use the angle @ measured from the direction of propagation of the incident wave to obtain da (= *) ae 5 (9.12) 374 «©6969 Scattering and Coincidence Experiments When integrated over all angles, Eq. (9.12) yields the Thomson cr08 section ; &r 4» : CT = 3 O° (9.1 (This result was given without proof in Eq. (8.21).)} Several objections can be raised to the simple cross section given by fa Eq. (9.12) or Eq. (9.13): (a) it does not depend on frequency, a fact nigg2222 supported by experiment; (b) the electron, even though free, is assumed not to recoil; (c) the treatment is nonrelativistic; and (d) quantum effects are not taken into account. Indeed, the correct quantum-mechanical calculatioti: for Compton scattering yields the so called Klein—Nishina formula” da 2 | +. cos? ¢ i] — = foe e-eeaeorrene Oe eee dQ °° 2 [l-ty (1 —cosé)}* 2 _ 6 . pot x }1+ (= cose)" _ (9. 14} (1 + cos?) (1+ y (1 — cos 6)] where rp and 6 were defined previously, and y = Av/mc*. The cross section has been averaged over incoming (and summed over outgoing polarizations, By integrating Eq. (9.14), the total cross section can be obtained. We will not give the complete result here, but the asymptotic expressions have already been presented in Eq. (8.22). : A comparison of the Thomson (Eq. (9.12}) and Klein—Nishina cross: sections, including the results obtained in this laboratory for y = 1.29, ig shown in Fig. 9.8. We remark that although the Thomson cross section is’ symmetric about 90°, the Klein—Nishina cross section is peaked forward: strongly as y increases. This is due to a great extent to kinematical factors. associated with the Lorentz transformation from the center of mass to the: laboratory; note that the center-of-mass velocity of the (indicent gamma ray + free electron) system is | v=cB=cy/ity}), where as before y = kv /mec’. ee The experimental data are in perfect agreement with the results of 2222 Eqs. (9.9) and (9.14), which are among the most impressive and convincing“: oe 5See for instance F, Gross, Relativistic Quantum Mechanics and Field Theory, Section: ::: ze 10.5, Wiley, New York, 1993, See! 9.2 Compton Scattering 374 successes of quantum theory. In the following two sections we will describe the experimental verification of these predictions. 9.2.2. The Compton Scattering Experiment As with any scattering experiment, the apparatus will consist of: (a} The beam of incident particles, in this case photons, (b) The target (containing the electrons from which the photons scatter), and (c) The detector of the scattered photons. The beam of photons is obtained by collimating the gamma radiation from a '3’Cs source. An intense source is required in order to get an appreciable counting tate for the scattered photons. As shown in Fig. 8.21 137s (!37Ba) emits a gamma ray of energy 0.662 MeV, and the detection techniques have been discussed in Chapter 8. Figure 8.21 also shows the pulse-height spectrum of the gamma radiation from '3’Cs, as obtained with standard equipment; the same detection equipment is used in this experi- ment with the only difference that heavy shielding ts needed to prevent the detector from seeing the intense 13’Cs source directly. A schematic of the apparatus is shown in Fig. 9.4. The lead pig A is fixed and holds the source, which can be introduced through the vertical hole (V), Another lead shield B contains the detector and can be rotated about the center, where the target is located. The lead assemblies are rather heavy (approximately 100 Ib) and some provisions must be taken for adequate mounting. For the source, a 7-mCi !37Cs sample was used, which was properly encapsulated before being shipped to the laboratory. It should always be transported in a Jead container, and when transferred into the lead pig A, it must be handled only by the attached string. The source holder (A) bas a collimator (/) drilled horizontally, subtending a solid angle on the order of 0.03 sr. Of interest to us will be the density of the photon beam at the target, and the expected value is 3.7 x 1029 x 0.007 1 ea [.3 x 10* photons/em?-s, Ayr r2 where we use a source-to-detector distance r = 40cm, for the data presen- ted here. 3% = 9s Scattering and Coincidence Experiments ye ee catlared photons the '37Cs source is not directly visible to the detector at forward angles. (c) Use of a. fines mn Been n target when measuring Compton scattering at large angles. By such placement the scattered 2575 photons do not have to traverse very Jarge amounts of the target material. ; 9.2 Compton Scattering 377 In contrast to the scattering of alpha particles, there is no need to enclose the beam and detector in vacuum or to use a very thin target. We know that gamma rays do not gradually lose energy when traversing matter as a charged particle does, but their interaction can be characterized by a mean free path. For the 137Cs parma ray we find that 4= 4.7 cmin Al, X = 0.92 cm in Pb; this corresponds to 104 cm of air, so that the interaction of the photon beam ia the air of the apparatus (approximately 100 cm) is indeed negligible. Also, the target thickness can safely be a fraction of a mean free path before the probability for multiple interactions becomes considerable. Aluminum targets - in. thick are quite adequate for this experiment. Some special mention must be made of the geometrical shape of the target. We may use a flat target (such as an aluminum plate), in which event the cross section is obtained by considering the interaction of the =: total beam with the number of electrons per square centimeter of the target”; alternatively, we may use a target of circular cross section (such as a rod), in which event the cross section is obtained by considering the interaction of the beam density (photons per square centimeter) with the total number of electrons in the target.’ When using a plate, it is advisable to rotate : it so that it always bisects the angle between beam and detector, since otherwise the scattered photons may have to traverse a very large amount of material before leaving the target (see Pig. 9.4c). Ip that case, however, the amount of scattering maternal in the beam path varies as | / cos(@/2), = and this correction must be applied to the yield of scattered particles. These e effects are obviously eliminated when a target of circular cross section is :* wsed. In addition, the scattering point is better defined even if the beam is : only poorly collimated. On the other hand, accurate cvaluation of the flux =. density at the target is difficult. The results presented here were obtained = by using a 3 in.-diameter aluminum rod as the target. An interesting refinement of the technique is made by observing the recoil electrous in time coincidence with the scattered photon. However, Se ek RRS Rn SRM ~ the kinetic energy of the recoil electron is y(1 — cos@’) 7. = — — * es E—Ek 1+ y(1 —cosé@) ®See Fig. 8.1. See Fig. 8.1. 3780s 9s Scattering and Coincidence Experiments which at its maximum value (0 = 180°) ts T (electron) = 0.662 x (2.58/3.58) = 475 keV. The range of such an electron in aluminum is only 150 mg/cm? (see, for example, Feather’s rule, Chapter 8, Eq. (8.15)), which corresponds to: approximately 0.06 cm. Thus, the recoil electrons will, in almost all cases; ae stop in the target. On the other hand, if a plastic scintillator is used as thé: 252 target, and is viewed with a photomultiplier, the recoil electrons do produce a signal that can be easily detected. As mentioned before, the detection system consisted of a commer: cial Nal detector. The dimensions of the crystal were 3 in. diameter < 31 in, thick. Data was acquired with a multichannel analyzer, with a GPIB. gs | ee a ey difference between the target i in and out spectra 1s also plotted. os a By measuring the pulse-height distribution at various angles, we obtain ue the energy of the scattered photons as it is given by the position of the pho _ ee 8 of energy for several different Nal crystal dimensions. 9.2.3. Results and Discussion from the online library at http://www.bicron.cam. §.2 Compton Scattering 379 (a) w a zz c 8 0 1000 2000 3000 4000 5000 Channel} (b) 1100 1000 : + 900 . = *=Target in 800 +S oe ®=Target out j=. 4=100° “ D oO = 3 c = f=) Oo re) 1000 2000 3000 4000 5000 Channel : FIGURE 9.5 Pulse-height spectrum gamma rays in the Compton scatleting apparatus. : The plots (a), (b) show data acquired for 120 s both with the target rod in (solid points) and out (open circles) of the beam. At @ = 30°, the detector intercepts some fraction of - the primary beam, and the rate is considerably larger than at @ = 100°. In addition, there =: are large signals due to K-shell X-rays and Compton backscattering in the lead shiclding = at both scattering angles. However, in each case, these background signals subtract cleanly = away, leaving a pure Compton scattering signal from the aluminum target The subtracted plots are shown in (c) and (d). SPADA ES Sok Oe Re Sb a AN oe wen a viptetetetete'y se! et Pf AOE .. ‘ it) = 3300 9s Scattering and Coincidence Experiments (c} goo g=30° subtracted 700 : G00 ' > I z = | @ 400 | i |] = I Z at iP ft © 300 | Mai 200 | | | ae | : 0 0 1000 2000 3000 4000 5008 Channet {dj | 700 i @=100° subtracted ae xe +f _ ee: . + ee: 2 pan 5 ee 5 } He uy ee = y res 3 Bd | ee t ge t ae ‘, $45 6 atgets ¢. 2000 3000 4000 SOX) Channel FIGURE 9.3 Continued ares See _, oe eee Before beginning measurements of Compton scattering, itis worth whilecZz to measure the beam profile of the '3’Cs source. This is best done by ot SN are negligible if the total rate is less than several kilohertz.) Then by moving 22 9.2 Compton Scattering 381 Figure 3. Source distance 60 cm Peak total-ratio o of 10 1456 20 25 30 Energy (MeV) SHS EARS TRS Percent absorption ot CONN PL TTT AAC INED | Energy (keV) = FIGURE 9.6 Detection efficiency plots for Nal crystals of various dimensions, from = httpi/Mwww_bicron.cam, Shown are the peak-to-total ratio and the intrinsic absorption = efficiency, all as a function of energy for various crysta) dimensions. be het be he hh eB a har hae SERS ?: photon beam. For our measurements here, however, we will simply assume : the calculated beam flux for a measurement of the differential cross section. Compton scattering data are taken by accumulating pulse-height spectra : at various angles, both with the target in and out, for fixed periods of times. » In order to minimize the effects of gain drifts, and other changes over » longer times, it ts best to take the “in” and “out” spectra tmmediately 382 § Scattering and Coincidence Experiments TABLE 9.1 Summary of Compton Scattering Data Angle Peak Counts Counts | -Peaktoml = == da/d@ (°) channel = Gn) (out) 5’ (MeV) ratio = Efficiency (1077? omer 20 «4300 «« $28,161 508,714 0614 0.47 0.865 55.2 30 3732 97,663 B1,2L 6.564 0.50 0.890 42.9 40 3384 «=. 20,856 «14,566 0.508 0.53 0.930 35.8 60 2810 16,382 8062 0.402 0.57 0.960 23.9 80 2258 16268 6251 0.320 0.65 0.990 18.0 100 1922 «17,482 7632 = 0.263 0.72 0.999 15.8 Note. Each spectrum was acquired for 120 s, one after the other. (For example, see Fig. 9.5.) Data taken by students arez:z2 summarized in Table 9.1. In this table, £’ is the photon energy as calculated: Bs from Eq. (9.9), and is used to look up the peak-io-total ratio and the intrinsic’: efficiency from Fig. 9.6. Radioactive sources are used to calibrate the analyzer channel in terms z of photon energy. (See Fig. 8.24 and the associated text. It is advisable ton222 carry out a calibration both before and after taking Compton scattering data, “24 in order to check for gain shifts.) In this experiment, it was determined that;: Energy = 0.1527 x Channel ~ 34.96. Then, using the photopeak values summarized in Table 9.1, we determine: the scattered photon energy E’. In Fig. 9.7, we plot the inverse of the: a measured photon energy, 1/&’, against (1 —cos 6). According to Eq. (9 Dye: cae a straight line should be obtained, since : 1 — — i = (1 — cos ?). This is indeed the result, and the slope of the line gives 1/mc* with an Ye ; intercept at 1/#. From a least-squares fit we obtain : ? — §05 ++ 12 keV in very good agreement with the known value of the electron mass. We: thus conclude that Eq. (9.9) is very well venfied and that our explanation: : of the Compton frequency shift is firmly supported by these data, & a ca 5 F 3 , o a) o 5 ° RH =a ob ie o = me R. G fe: Ga A € 5 r o 5 a rr | a reer wee rH ae ae ae ae wae . rene Sees ae vat ae aa of ae ee explained before, we integrate the counts under the photopeak. The results: 9.2 Compton Scattering 383 Slope=1.98 MeV! O 0.2 0.4 0.6 0.8 1 1.2 1—coasé FIGURES.7 The results obtained forthe energy (frequency shift) of the Compton scattered gamma tays. Note that 1/2 is plotted against (] — cos @), leading to a linear dependence. The slope of the line gives the mass of the electron. are also summarized in Table 9.1. To obtain the cross section we note that da __iyield dQ (dQ)Nig The detector solid angle is given by tal aQ = _— — 6.4% 1077 sr, Fr where r 1s the distance from the target to the detector. For the total number of electrons in the target, we have d\? ON N=z| = hp—Z, 2 A where? ad = diameter of target = = in, = 1,91 cm h = height of target = 4 cm re. "The height of the target is obtained by estimating the length of target intercepted by =; the beam. 364 48690: Scattering and Coincidence Experiments p = density of aluminum = 2.7 gm/em? No == Avogadro’s number = 6 x 108 A = atomic weight of aluminum = 27 4 = atomic number of aluminum = 13, thus N = 8,9 x 10” electrons. Q For Jo, the flux density at the target, we use the previously obtained value Q Ig = 1.3 x 104 photons/cm?-s, : 2 and the data acquisition time for each spectrum is 120s, so that finally da corrected yield dQ (6.4.x 10-2) x (8.9 x 10%) x (1.3 x 104) x (120) corrected yield ~ 8.89 x 1029 : The values of the differential cross section obtained in this fashion are: Z given in Table 9.1, and are also plotted in Fig. 9.8. The solid line in Fig. 9:83 Renae #0 dofda (1072? emessr) fy fi. on m go Le | oS ao ho a) 0 20 40 64 6 0 120 140 160 18 Scattering angle @ FIGURE %.8 The results obtained for the scattering cross section of !37Cs gamma tay as a function of angle. The solid line is the prediction of the Klein—-Nishina formula for that particular energy; the dotted line is the Thomson cross section. a atl or 9.3 Mossbauer Effect 385 Be gives the theoretical values for do /dQ derived from the Klein—Nishina * formula (Eq. (9.!14)) for y = 1.29, while the dashed curve represents the Thomson cross section. The agreement of the angular dependence of the experimental points with the theoretical curve is indeed quite good and clearly indicates the inadequacy of the Thomson cross section for the description of the scatter- : ing of high-energy photons, while confirming the Klein—Nishina formula. On the other hand the absolute value of the experimental cross section is = subject to some uncertainty due to the way tn which the flux density Jp and ee PAPRIRTIA ees nt in :. total number of electrons NV were estimated. Nevertheless, the agreement = is good. * 9,3. MOSSBAUER EFFECT - 9.3.1. General Considerations =: In the Compton scattering experiment, Wwe could visualize the scattering eee “. process as if it were a collision of two billiard balis in which the incom- = ing photon maintained its identity but suffered a change in momentum i and energy. The phenomenon of scattering can, however, also be visu- alized us the absorption by the target of quanta of the incoming beam, = with the subsequent re-emission of these quanta; this was the model a we used in the derivation of the Thomson scattering cross section in & Section 9.2. Since we know that emission of quanta of energy A(vg — vy) in the visible spectrum is due to transigions of atoms from a state of 8B — a we : musi also expect that when quanta of this energy A(vg — va) are incident = on an atomic system in state w, they may be strongly absorbed, with the z consequent raising of the atom from state a to state 8. Evidence for such : strong absorption is obtained by detecting radiation of frequency (vg — ve) :, emitted from the absorber in all directions; it is due to the atoms that, : having absorbed a quantum from the beam, were raised to state 8 and then : underwent a spontaneous transition back to state a, emitting the quantum * h(vg — vq), but with equal probability into all directions. Such radiation 4§ called “resonance radiation” and was first observed by R. W. Wood in “ godium vapor in 1904. A schematic of the apparatus is shown in Fig. 9.9. : An absorption cell was illuminated by sodium light, and at right angles to : the incident beam the sodium D lines were observed. 386 4 «=6©9) «Scattering and Coincidence Experiments Resonance radiation To spe ph or Cofimator detactor ar fb | seo A | g eis ——_s WA ae Le or) ne Fitters Primary beam ae Z Absorption Na lamp cell eB (Na vapor} ae nn FIGURE 9.9 The arrangement of an optical {atomic} resonance radiation experiment oe Here the sodium D lincs afe incident on a cell containing sodium vapor; it is then possible: ted Ro ae SS to observe, at right angles to the incident beam, the appearance of the D Lines. Let us note two facts: (1) Since the alom must be in state @ when the radi: : ation is incident, a is usually the ground state of the atom.!9 (2) The incident 2 __ hne.'To understand this, consider a system R originally atrest; R undergoes: a transition from 8 — a, where the energy difference between siates es and £ is : energy Av, and momentum /2v_/c; ve is to be determined. From Fig. 9.104: we see that to conserve momentum, the emitting systern K must recoil with: momentum A#v,./c; therefore it will have energy (nonrelativistically) =< (Ave)? 2c? p= absorb (and re-emit) radiation in order to yield observable resuits. In very special ae “ metastable state, to which a large fraction of the atoms can be transferred (by some oth: si Means), can serve as State @.. o 9.3 Méssbauer Effect 287 _ _ tric K=O hfe os =<) Ae C} + =A - nufe it om A yo {a} (b) FIGURE S$.10 The effect of momentum conservation (recon effects) in the emission and absorption of nuclear garmma rays. (a) A system # orginally at rest emits 2 gamma ray Av; it must recoil with a velocity ug = (Au/e)/mpR. (b) A system RX moving originally with a velocity vj = (Av/c}/m, absorbs a4 gamma ray Av; alter the absorption the system will be at rest. (c) Derivation of the first-order Doppler shift for an observer moving with velocity v. To balance energy, we must have Er thy, = hv, leading to hve = hv(l —x + 2x7 +--+), (9.17) where x = hv/2me?* will generally be small. Similarly for a system R’ originally at rest in order to be raised from level a > 8, where Eg — Ey = hv, it must absorb a quantum of energy hv, = hv +x — 2x? +.+-), (9.18) If the emitted quanta were strictly monochromatic, then it is clearly not possible for a free system R to absorb a quantum Ay, emitted by a similar free system R’, since hv, # Ave (Fig. 9.1 1a). We know, however, that spectral lines have a certain width! Av; in Fig. 9.116 the emission and absorption lines are shown appropriately cen- tered about Av, and #v,, but with a width Av. If then the two line shapes overlap, it is possible to have resonant absorption. lithe minimum or “natural width” of a line is determined from the lifetime t of the transition 6 — a; from the uncertainty principle AF Ar % A, and thus Av * 1/7. Other contributions include the “Doppler broadening” due to the thermal motion of the atom or nucleus, collisions, external perturbations or imperfections in a crystal latrice. 388 02= 4 «Scattering and Coincidence Experiments lat 2g om Overlap (a) (b) FIGURE 9,11 Ladication of the energy shift of an emitted or absorbed gamma ray due bo 32-2 the recoil of the nucleus, (a) The sitvation when the line width is very narrow in comparison 2:42 to the recoil energy: no resonant absorption can then take place under normal conditions, 22 (b} The situation when the Hine width is on the same order as the recoil encrgy; note that 2222 resonance absorption can now lake place and it will be proportional to the convolution of oes the two line shapes. ees oh ie This is true for alomic systems: here hv ~~ 2 eV, and for tytn mce* *% 10° eV; thus x ~ 10—%. The width of atomic spectra lines, however, "2 is on the order of Av/v * 107°. Thus ae Av Ay —— wz 1976 es 107? |. ( v }> (55 ) For nuclear gamma rays, Av * 104~10® eV; also, in general, nuclear 2: lifetumes are longer than those ior atomic systems, so that & Av oy ww 10719 — 19-8, Thus we see, 1 contrast to the situation for atomic systems, !? that Av —10 Av ~ 7 (S* ~10 «(s+ ). making resonance radiation impossible. Fs In the preceding discussion we assumed the that the emitting and absorb- ing nuclei were at rest. We could, on the other hand, think of imparting tothe : absorbing nucleus (by some means) enough velocity in a direction opposite . : to that of the quantum (Fig. 9.10b) so as to satisfy Eqs. (9.17) and (9.18). ° 12For example if r = 1079 s, then AE © 6 x 107? eV. Further, nuclear gamma rays are subject to broadening influences much less than atomic Lines. 9.3 Maisshauer Effect 389 For example, if hv = 10* eV, and the nucleus has A 100, and we wish that h — =mpy hve=(me2)u (9.19) we find for the velocity 3x 1019 » 104 , — ] . 100 x 103 3x 0 cm/s Such velocities can be obtained in the laboratory by placing the samples on the rim of a centrifuge and orienting the incoming beam toward one of the tangcnis. It then becomes possible to observe nuclear resonant absorption. Nuclear resonant absorption would also occur if both the emitter and absorber were so massive that momentum could be balanced with negligible enerpy being given to the recoiling system, that is, if the denominator 77 in Eq. (9.16) became infinite. Indeed, R. M6ssbauer showed in 1958 that for atoms bound in a crystal lattice, a nucleus does not recoil individually!? but the momentum of the nuclear gamma ray is shared hy the entire crystal. This can be understood if we consider that the binding energies of the atoms in a lattice site are on the order of 10 eV, whereas the recoil energies, given by Eq. (9.16), are always less than 1 eV. Since, however, the nucleus is now part of a larger quantum-mechanical system, there exists the possibility that the energy available from the de- excitation of the nucleus 8 — a@ might not all be given to the gamma ray, but might be shared between the pamma ray and the lattice, in the form of vibrational energy, Lattice vibrations—the so-called emission of phonons—ate a quantized process, and the lowest energy phonon that a single nucleus can emut has E=kT, where T = ©p is a characteristic temperature for the crystal, the Debye femperature. Thus, if the recoil energy of the free nucleus, as given by Eq. (9.16), 1s Ex < k@p, it 1s not possible for the lattice to become excited into 4 vibrational mode, and the total energy of the transition is taken by [it is customary to say that “the nucleus does not always recoil individually,” in order to account for the instances where the nucleus transfers energy to the latlice as explained in the following paragraph. 390 «9 Scattering and Coincidence Experiments the gamma say. The probability of recoilless emission of the gamma ray is then given by 2kOp Equation (9.20) holds at absolute zero, and for finite temperatures we - may use 2 ae f =exp (-5) , (9.21) : fo Here 1/A* = (2xv/c)* is the square of the wave number of the entit- ted gamma ray and {x*) is the mean square deviation of the atoms from *: their equilibrium position and is proportional to T. As an example, for the ~: 14.4-keV line of >’Fe, Er = 0.002 cV and Ap = 490 K; hence f =o 8% = 92%, We therefore see that in certain materials (°7Fe being the most suitable) the Mossbauer conditions are met; recoilless emission and absorption can take place, and consequently nuclear resonance radiation can be observed, It has been explained earlier (Eg. (9.19)} that we could cumpensate for the recoil of the nucleus by moving the absorber in a direction opposite to the incoming gamma ray (so as to make the total momentum of the nucleus-plus-gamima-ray system zero). It follows then that if the absorption is recoilless, such motion of the absorber would destroy the resonance condition. In recoilless emission (absorption) the gamma ray has energy Ey, = hvo in the system, which is at rest with respect to the nucleus; if the nucleus is moving in the laboratory with a velocity v in the direction of the gamma ray, the laboratory encrgy of the gamma ray E,, is given by a Lorentz transformation 1 Ey = (Ey + upy) = Ey ==, where B = v/c. For 6 < | we obtain to first order AE Uv AE = E, - y = BEy or “ETB Hs f=exp (-376) ——(9.20)' ae ar ia ea ae . ree. 1 . 1 at an a ee | aa : et ee LF . “t ™ a ard faye . Sota ara ae ef Ct he he re hat het belle = oy RHC eC ie er TG HOSE Ws AL ere ee Le oe ar hr AOR STOEL eens X oo 9.3 Muissbauer Effect 2391 Thin absorber = Source & 100 Detector at cc o Counter a sO = 1 £ i c | Sy a —-7-2012 > Velocity (mms) (a) {b) Overlap region § fF reg O96 2S =n OH Ss? Emitted jine 25 depends on ag ' source velocity = o j ov Absorber ine (a) FIGURE 9,12 The Mossbauer resonant absorption experiment (a) Diagrammatic view of the equipment. (b} The probability for transmission of a gamma ray as a function of the source (or absorber) velocity when oo hyperfine structure is present. (c) The width of the iTansmission curve is a combination of the shape of both the source and absorber lines. which, written as Av/v = v/c, is the first-order Doppler shift of a wave emitted (absorbed) by a moving observer (Fig. 9.10c). To obtain a quan- titative estimate we consider again the 14.4-keV line of >’Fe, which has a lifetime t ~ 107? s and hence Av/p = 45 x 10—}3, Thus, velocities on the order of v = c(Av/v) ~ 1.5 x 107? cm/s will be sufficient to destroy the resonant absorption. Such velocities are easy to achieve and control in the laboratory. We therefore measure the transmission of the 14.4-keV gamma ray through an °’Fe absorber as a function of its velocity. Altematively we can leave the absorber stationary and move the source. A possible experimental arrangement, indicated in Fig. 9.12a, consists of an °’Fe source, an °’Fe absorber that can be moved at a constant veloc- ity,!* and a detector for the 14.4-keV gamma rays; we measure the rate of transmitted gamma rays. At zero velocity the transmission is low because of 14The velocity, however, is varied in the course of the experiment. 392 9 Scattering and Coincidence Experiments resonant absorption; as the velocity of the absorber 1s increased, however: the resonance is destroyed and the transmission increases, leading to a typi- cal curve as shown in Fig, 9.12b. We may think of the incoming gamma ray as scanning over the absorption line as a function of the velocily, and there-: fore the observed absorption is a measure of the convolution of the two lines. as shown in Fig. 9.12c. In this way we “trace out” the natural line width for this nuclear gamma ray, and measure energy deviations of one part in 1613 (v = 0.06 mm/s}. This represents a highly precise measurement and this is why the Mésshauer effect is an important tool in many physics applications: 9.3.2. The Apparatus and Some Experimental Considerations In this laboratory the Mossbauer effect was observed using the 14.4-keV. garmma ray of ?Fe, which follows the decay, by electron capture, of °’Co (see Fig. 9.13). Basically the apparatus required for the experiment consists of (Fig. 9.12) (1) the source (with or without appropriate collimation), (2) ©: the absorber and a mechanism for moving the absorber or the source at : constant speed, and (3) the detector for the 14.4-keV gamma ray. Prom 22 Fig. 9.13 we note that the 14.4-keV line of interest will be accompanied *’ by a 122-keV gamma ray as well as by a weaker 136-keV line. There is : also a strong background present from the 6.5-keV X-ray of ?’Co, which : follows the electron capture from the K shell. The source used was 1 mCi © of °’Co plated and annealed onto an ordinary iron backing.'° The detector is chosen so as to provide pood efficiency and discrimina- - tion for the 14.4-keV gamma ray, A xenon-methane proportional counter, _ followed by a single-channel discriminator, was used. In Fig. 9.14, curve (a) : pives the pulse-height spectrum of the gamma rays emitted by the source, — while curve (b) gives the same spectrum after the gamma rays have tra- versed a 0.001 in. absorber. The shaded area represents the “window” selected on the discriminator, so that only gamma rays within these energy hmiuts were recorded by the scaler. The absorber in this case is usually a thin steel foil, but it should not exceed 0.001 in., since nonresonant scattering increases so much as to smear out the 14.4-keV line. Further, natural iron contains only 2.17% of °’Fe, so that poor signal-to-noise ratios result. It is possible, however, !5 Purchased from Nuclear Science and Engineering Co., P.O. Box 1091, Pittsburgh, PA. 9.3 Missbauer Effect 393 Co” 12 270-day Electron caplure 5/2- 0.136 MeV 0.122 MeV 91% - . we 9.01437 MeV r=1.4e10"’s — Noabsorber -~— With absorber ee Transmission (Counts/s-channel) ' ‘ ‘ ‘ ‘ ‘ Channel number FIGURE 9,14 Pulse-height spectrum of the low-energy gamma rays of >’ Fe as obtained with a proportional counter. The solid curve has been taken without the absorber in place, whereas the dashed one has been taken with the absorber in place. The shaded region indicates the discriminator window used for observing the Méssbaver effect. to obtain absorber samples enriched in *’Fe, and in the present experiment, such a foil (of 1 cm? area) was used; the *’Fe concentration was 91.2% and the thickness 1.9 mg/cm? (approximately 0.0001 in.). The motion of the absorber can be achieved either by purely mechanical arrangements, or by a transducer of some type. Examples in the former 394 89 Scattering and Coincidence Experiments FIGURE 9.15 An amplifier circuit capable of driving a speaker coil for use in the 2424 Missbauer experiment. Beason category are a plunger driven by an appropriately shaped cam (logarithmic: spiral ry = &@) or the rim of a wheel rotating about an axis that is not normal to the surface of the wheel. In ali cases of mechanical motion, special attention must be paid to decoupling the vibrations of the driving motor from the absorber. For the present experiment, a device of the latter category was chosen, namely, a loudspeaker driven by a sawtooth current (see Fig. 9.15}. The source was mounted on the core of the speaker and the absorber was kept stationary. The driving waveform was obtained from the horizontal sweep of an oscilloscope after amplification. | To calibrate the speaker, a micrometer screw was mounted in a special :: manner above the speaker. By listening, the experimenter could discern :223374 when the screw touched the speaker, giving results to within +0.003 om) 22 out of a maximum travel of 0.2 cm. Assuming that the speaker is linear with current, the calibration shown in Fig. 9.16 was obtained. The small: variation in solid angle with the change of source—detector distance does: : not affect the results obtained. It is also advisable to gate the scalers so *: as to count only during the linear part of the motion (and in the desired 222: direction). 9.3 Madssbauer Effect 395 Speaker calibration 0.274 em/i0d mA Micrometer reading (cm) Q 100 200 300 400 500 Current to speaker (mA) FIGURE 9.16 Velocity calibration of the speaker used to provide the motion of the source in the Missbauer expenment. 9,3.3. Results and Discussion In Fig. 9.17 the results obtained by a student are given; the abscissa gives the velocity of the source in millimeters per second, and the ordinate, the counting rate at the detector. It is clear that maximum absorption occurs at zero velocity, in accordance with the hypothesis of recoilless emission (and absorption) of the gamma ray and the conclusions reached in the previous sections. The full-width at half-maximum for the zero-velocity peak as obtained from Fig. 9.17 is Papp = 0.70 min/s. If the two curves shown in Fig. 9.12c are assumed to have a Lorentzian shape, then the apparent width [app can be reiated to the true line width [° through Tapp/ T° = 2.00 + term correcting for absorber thickness. Thus we find that [ (14.4 keV) ~ 0.30 mm/s ‘and 3% 9 Scattering and Coincidence Experiments 220 200 180 z 140 Transmission (counts/s) 120 & 0 2 4 6 8 10 Velocity (mins) FIGURE 9.17 Results obtained for the Méssbauer effect of >’ Fe using a>’Co source on . ordinary iron backing, and an enriched 5'Re absorber. which is in fair agreement'® with the accepted value of Av/v = 3x 107}3, . It is clear that in Fig. 9.17, apart from the zero-velocity peak, there also’ appear subsidiary peaks at v = 2.5, 5.5, and possibly also 7.5 mm/s. What, is the origin of these peaks, so reminiscent of the hyperfine structure of: atomic spectral lines? 2 Indeed this structure of the Mossbauer line is greatly dependent on the:: type of host material in which the absorber (or source) nuclei are embedded." In natural iron, there exist strong magnetic fields at the site of the nuclei; ag’: a result, the nuclear energy levels are split, giving rise to a “Zceman effect”: for the nucleus,!” Figure 9.18b shows the splitting of the ; excited state: lost of the discrepancy can be traced to the considerable thickness of the absorber: satis The probability for interaction is ziven by “ P =oofa(No/An, where ft = absorber thickness * 2 x 10-4 p/em?, No/A =6~x 104 FS? %& 1074/¢, on = the Méssbauer absorption cross section = 1.5 x 10—!® cm*, f = probability for recoilles§4 absorption, approximately 1, and a = concentration of the resonantly absorbing nuclei i322 the sample, approximately 1, Hence, for the present case, P = 30! ae 17 See Section 6.2 for a detailed discussion of the Zeeman effect. 9.3 Mossbauer Effect 397 Excited 3° stata (a) FIGURE 9.18 Hyperfine strucuire splitting of the nuclear energy levels of 57 Fe. (a) When stainless steel is used, the levels are not split. (b) In ordinary iron, however, both levels are split, giving rise to a hyperfine structure with six components. and the 5 ground state of °’Fe, and consequently the 14.4-ke¥ line has six hyperfine structure components. Figure 9.18a shows the same levels for stainless steel, where no splitting occurs. If both the source and absorber are not split, then clearly only a single peak will be observed, as in Fig. 9.12b. If the source is not split, but the ahsorber is, then as a function of velocity we will “scan” with the single line over the hyperfine structure pattern of the absorber. In this case there is no absorption at zero velocity (see Fig. 9.19a). Finally, 1f both the source and absorber are split, a complicated pattern emerges, depending on the degree of overlap of the individual components as the two hyperfine structure patterns are shifted one over the other; however, maximum absorption occurs at zera velocity (see Fig. 9.19b). In the experiment that yielded the data of Fig. 9.17, both the source and the absorber were split, so that a pattern of the type shown in Fig. 9.19b was obtained. Table 9.2 gives the relative intensities and known positions of the peaks as well as the positions obtainable from the results of Fig. 9.17. The apparent discrepancies in the known and observed positions are due in part to a small velocity calibration error. Materials like stainless steel, potassium ferrocyanide, sources made by diffusing 7’Co into chromium - metal, do notexhibit structure in the 14.4-keV line and give simple patterns. " Jn Table 9.3 we summarize some of the numerical values pertinent to the :* Méssbauer effect in >” Fe. 398 480969 Scattering and Coincidence Experiments Change in transmission, percent -§6 ~-4 =-2 0 2 4 6 0 2 4 6 8 10 Velocity (mm/sec) Velocity (mm/sec) (a) (b) FIGURE 9.19 The expected pattern of the Méssbauer line when splitting of the levels lakes, place. (a} Hither the source or absorber is split; note that the Méssbauer line is split into six components and no absorption takes place for zero velocity. (>) When both source anid absorber are split a complicated pattern results with maximum absorption at zero velocity: a TABLE 9.2 Position and Amplitude of Miéssbauer Peaks in >’ Fe, Including the Experimental Results Position Observed position Peak Amplimde (munis) (mms) 0 7 0 0 1 4 2.2 2.75 2-3 LS 43 5.5 4 2.5 6 7.6 (?) 5 3 8 — 6 2 10 — TABLE 9.3 Some Numerical Values Pertinent to the >’ Fe Missbauer Line Transition energy By = 14.4 x 10° eV Internal conversion coefficient azefy= 15 Lifetime t=14x 1077s : Relative width Avjv=3x loo m Recoil energy of free nucleus ER =0.19 x 10-2 eV Debye temperature (Massbauer) @p = 490 K & Probability for recoilless transition ¥ at room temperature f =0.80 Cross section for resonant absorption gg = 15 x 107!9 em? Natural abundance of >’ Fe 2.17% 9.4 Detection of Cosmic Rays 399 A very complete description of the Missbauer effect, including reprints of the most important papers, will be found in H. Frauenfelder’s The Mossbauer Effect (W. A. Benjamin, New York, 1962); this reference should be fully adequate until the student finds it necessary to consult the current literature. 9.4. DETECTION OF COSMIC RAYS ’ 9.4.1. Flux, Composition, and Detection of Cosmic Rays The earth is continuously bombarded by a flux of high-energy particles that originate outside of the solar system. These are mainly protons but the primary cosmic ray flux also contains a fraction of light nuclei. When these particles reach the earth’s atrnosphere they cause nuclear interactions so that at sea level we observe only the final products of the nuclear cascade. !8 The interaction of the primary protons with the oxygen and nitrogen nuclei of the atmosphere results in the production of secondaries including unstable particles such as #~ mesons, K*+ mesons, and others. These in turn decay by the weak interaction into lighter particles, including muons, electrons, and neutrinos. Electrons and high-energy y-rays also interact rapidly, giving rise to electromagnetic showers as discussed in Section 8.2.6. Since the earth’s atmosphere 1s equivalent to ten nuclear interaction lengths, all strongly interacting particles are absorbed before reaching sea level.'? What is observed (at sea level) is a “hard component” consisting of z* (muons) and a “soft component” consisting of et and low-energy y-rays. The total flux per unit solid angle around the vertical, crossing unit horizontal] area ts I.i x 107/ m?-sr-s, (9.22a) :. where 75% of the flux is in the hard component The angular distribution =. is approximately cos” @ (with 6 = 0 at the zenith). It is also useful to know a 184 goad reference on cosmic rays in general can be found online from the Particle Data = Group at http://pdg_lb!.gov in the “Reviews” section under “Astrophysics and Cosmology.” 19Some of these unstable particles were first observed and studied at high mountain = altitudes or by baloon-bome detectors. Today such sobnuclear particles are produced pro- => fusely by particle accelerators, but cosmic rays are still used for the study of the very highest my EIKEPBIes. 400 9 Scattering and Coincidence Experiments Pulse from PMT Inverter nm (must be cantained with the gata) LG 105/A — Discrim- Coinci- Discrim- inator dence inator Set to qanerata ~50 nsec gate pulse Height of the Cosmic stretched pulses ray Proportional to area of scintillator pulse FIGURE 9.20 Typical layout of a cosmic ray telescope and electronics. Prov measuring the pulse height in one of the counters (not discussed in the text shown, that the total flux crossing unit horizontal area is 2.4 X 107 /m?-s, The mean energy of the muons is 2 GeV and falls off on the h as E~, Cosmic ray muons can be easily detected by measuring the coir rate between two scintillation counters placed vertically one al other as shown in Fig. 9.20. By increasing the distance between ters one can restrict the solid angle acceptance and also study the distribution of the flux. Plastic scintillation counters have the a of large area so that the cownting rate can be several per second. counter is placed in coincidence with the two-counter telescop located physically in a different location (as in Fig. 9.21} one sul coincidences.”? These occur because several cosmic rays arrive at time over the area covered by the telescope and the third “roving” 20These are true coincidences after any accidental effects (Section 9.5.1) subtracted. 9.4 Detection of Cosmic Rays 481 Roving counter FIGURE 9.2] Arrangement of counters for measuring cosmic ray air showers (top view). * Namely a “shower” of cosmic rays occurred. One finds that the rate for such showers is 1/300 of the telescope rate, given a typical counter area of 0.2 m? and a displacement of 3 m. We will describe an experiment in which cosmic ray muons are also detected by simply using a 5-gal tank of liquid scintillator, viewed by a 2-in. photomultiplier tube. Muons traversing the tank give a large signal so that it is possible to use the singles rate, without the need to form coincidences. However, the PMT high voltage and the discriminator must be set carefully. The dimensions of the tank are ¢@ = 28 cm diameter and h = 35 cm height from which we can estimate an effective horizontal area of 2 x [x (d/2)*] ~ 0.12 m*. The singles rate 1s of order 25/s, in reasonable agreement with Eq. (9.22b). 9.4.2. Time of Arrival of Random Events The arrival of cosmic rays is a random process,”! so we expect it to follow the distributions discussed in Chapter 10, In particular when the expected number of events in a given time interval is small, the observed number should obey the Poisson distribution. Let r be the average event rate, namely the average number of events per unit time. Then the probability of observing n events in the time interval ¢ 15 t Rot P(a,p = ge (9.23) n! Fram Bg. (9.23) we recover the differential probability for anevent (n = 1) to occur in the differential interval dt. Since (dr ~ 0), Eq. (9.23) reads dP = P(1, dt) =r dt. (9.24a) 2) This is of course also tue for the decay of radioactive nuclet. 402 9 Scattering and Coincidence Experiments Similarly the probability that no events (1 = 0) occur in the interval 7 is. PO,t)=e™. (9.24b}. We can test this proposition by measuring the distribution of the time between the arrival of adjacent events, A time interval ¢ between events is: defined in this case by requiring no event for the interval ¢ and an event, at the time ¢ (in the differential dt). Thus the distribution is given by the: product of Eqs, (9.24a), (9.24b), which we write in the form”? It is interesting that the above distribution is exponential; namely, short time intervals ¢ between adjacent events are much more probable”* than: longer ones, comes The result for the case 7 = 1 can be generalized for the time interval2% between every second event (m = 2), every mth event, ctc. The derivation: Bee is given in Section 10.5.3 (see Eq. (10.75), and we obtain tate (ety lett (m — 1}! As m increases, the distribution tends to a gaussian centered at t = mr: Of course one could also test Eq. (9.23) directly by measuring how often: one, two, etc., events are found within a fixed interval t. However, mea=: suring the distribution of the time intervals between event arrivals, as done: here, is by far more practical and efficient. : Data are acquired by recording the time of arrival of every muon ina = computer file. Since the mean time between counts is ~40 ms, a precision, See of 0.1 ms is sufficient and can be easily provided by the computer clacks2244 The file can then be analyzed by sorting the time intervals between adjacent: : pulses (7 = 1) in time bins of 0.8 ms width. The same data are next and# 22225 lyzed by sorting the intervals for different values of m in correspondingly. oe longer time bins. on Results obtained by a student form = 1 areshownin Fig. 9.22, form = 3: in Fig. 9.23, and form = 100 in Fig. 9.24, One notes how the distribu= tion becomes narrower as m increases. Namely the interval between every: ete wt epupetn wale 1 mas Ser CHAS Ny eFEBSEEMSEAESHOMSEGT HME Eta E Reb SCRC REED SMCBLE EEE am(t) =r 9.26) 22We imply that the second count arrives after the first one with a delay between ¢ ann cee t+di. he "This justifies the old proverb that onc calamity is always followed by a second on See W. Bothe, Phys. Zeit. 37, 520 (1936). 4.4 Detection of Cosmic Rays 4 600 t/a = 35-00 ms Frequency o 0 0.02 0.04 0.06 0.06 0.1 0.12 Anival time (s) FIGURE 9.22 Distribution of the time between the arrival af two cosmic ray counts. The fit is the Poisson distribution for m = 1, Frequency Oo 0.05 0.1 0.15 a2 Time interval (s) FIGURE 9.23 As described in the legend to Fig. 9.22 but for m = 3. 4044 9 Scattering and Coincidence Experiments 1600 1400 1200 Frequency g 8 oa Time interval (5) FIGURE 9.24 As described in the legend to Fig. 9,22 bat for m = 100. Note that thi distribution is centered at a mean time? * 3.565, where? = (mm — I) /r 100/r. 100 events is much more “stable” (relative to its mean value) than betweeri every second event. As can be scen from Bq. (9.24) the distributions for m > | have a maximum (dq, /dt = 0) at mm — 1 i= . (9.27): r fos Thus, from the location of the peak in the distribution we can obtain the": average rate. We find that for the data shown in Figs. 9.23, and 9.24 rf m= 3, imax = 0.073 s, r= 275 /s m= 100, = 3.565, = 27.7 /s. Furthermore, a fit to the exponential for m = 1 (ee Fig. 9.22) yields . ije = 3.50 x {0-2 s, orr = 28.6/s in apreemicnt with the average rate. 9.4.3. Measurement of the Mean Life of the Muon The muon is not stable but decays into an electron, a neutrino, and an SE antineutrino: ee wt > ettyetvp ee wo > @ + Vet Vp. (9.28) 222 Pe ee . 9.4 Detection of Cosmic Rays 405 The mean life, or lifetime, G.e., the inverse of the decay rate) for this process is of order 2.2 js, and thus the decay is easily detectable for muons at rest. The neutrinos are nol observable bul the electron (or positron)*4 iS energetic enough to give a clear signal of the decay. The mass of the muon is my = 105.65 MeV/c’, approximately 200 times the electron mass. The maximum energy of the electron occurs when the two neutrinos recoil agamsi it as shown in Fig. 9.25a. This corresponds, in the rest frame of the muon, to l Ee(max) ~ 5 mc? = 53 MeV. The energy spectrum of the electrons from muon decay is shown in Fig. 9.25b. The long lifetime for muon decay indicates that the decay does not pro- ceed through the strong (nuclear) interaction but rather through the weak interaction responstble for the “f-decay” of nuclei, However, the process of Eq. (9.28) is very important because it involves only leptons (no strongly interacting particles participate) and thus can be used unambiguously to u (b) dN, de mc 2 =53 Mev End point lA 25 50 Mev E, FIGURE 9.25 (a) Configuration of the particles in yz-decay for obtaining the maximum electron energy. (b) The energy spectrum of the electrons from p-decay. 247 save words we will speak only of the electron even though we mean either — or et e~ ore’. 405 9 Scattering and Caincidence Experiments calculate the Fermi weak interaction constant Gr. The mean life of: the ie muon is given by 1 ot GE (mye?) t hi (Ac)® 19273 © Precise measurements of the muon mean life yield T, = (2.19703 + 0.00004) x 10° s and through Eq. (9.29) the value of the Fermi constant”? Gr (hey We will measure the decay of muons that have come to rest in the — 1.1664 x 1079 GeV~?. that muons entering the 35-cm-high liquid scintillator tank with enetg E,, S50 MeV will stop in the tank. The fraction of muons that do stop: eC of order 0.3% of the flux going through the tank. Thus, the stopping rate, & The experimental arrangement is shown in Fig. 9.26, When a mus ee enters the tank the PMT gives a signal, which is amplified and then: See discriminated. This pulse is used to start a “time-lo-amplitude converter’: ee functions. 9.4 Detection of Casmic Rays 40? u-mesan Electron 5-gal equiicd scintifatar tank Callbration FIGURE 9.26 Block diagram af the electronics for measuring muor decay. . mean lives, the TAC is reset and the start pulse ignored. To calibrate the = TAC one applies a fixed frequency (oscillator) signal to the discriminator : input. : [f the singles rate is too high, then the stop pulse may not be due to the : decay of the muon that started the TAC, but to a different muon entering : the tank. We call such cvents “accidental stops,” and we can estimate their « rates as follows. The singles rate is r = 25/s, so that using the Poisson : distribution of Eq. (9.23) for n = 2 and t = At = 25 ws we find for the : accidental rate _ P,(n = 2, Aft) ~ At : This is ten times smaller than the stopping rate Rs, and does not affect the determination of the mean life as discussed later. : Data obtained by a student-are shown in Fig. 9.27. The data were accu- “ mulated over five days and yielded Ns = 32,000 stops in 6921 min, The very early events, ¢ < 0.25 ws, were discarded, leaving a sample of 30,069 = events displayed in 100 bins each 0.25 1s wide. The data fort < 5 xs show "an exponential drop-off, as expected, and in this region are well fitted by Rs = 78x 107357. (9.31) N@)= Noe '’® 0.25 <1 < Sus. In contrast the data for late times, f > 15 ws, are flat and are well fitted by =a constant NiQ=C 5S <1 < 25 us. 406 9 Scattering and Coincidence Experiments Counts 10° tik including a an exponential decay and a constant background term are shown, A combined fit?’ of the form . N(t) = Noew!* + (0.32 yields t = 2.088 + 0.016 ys, No = 3410, and C = 28.8, and an excellent x? = 0.909 per degree of freedom. The contributions of the two terms 6! the fit are also indicated by the dashed lines in the graph. We bnefly discuss the background level. Since there are 100 chai nels, the total accidental count is V, = 2880, and thus the accidental rate is Ry = N,/6921 min = 6.9 x 107>/s in agreement with our estimate ot Eq. (9.31). One recognizes that the background does not affect ‘the mncasurement until “ Me™ ~ €, to delermiune t,,. CCAS Bo e Our value for the mean life is in close agreement with the accepte a , Se pe: value as given in Eg. (9.30a). The agreement is even closer because: the ee 27 ee Section 8.6.2. 9.5 y-pAngular Correlation Measurements 409 measured value for t, must be corrected for the following effect. When negative muons stop in matter, there is a finite probability that the »~ will be absorbed by a proton in the nucleus, leading to a “capture” reaction: mw +Z 3 (Z—1)* + uy. Thus the effective mean life is shortened and given by J I l . —=x=—4+-, tT Ty % where 1/t,, and 1/z, are the rates for decay and capture, respectively. As a result the observed mean Itfe is shorter; for mineral ol (the capture occurs mainly on carbon nuclei) and for the 4 /+ composition of cosmic rays this correction is approximately 4%. Therefore, the corrected measured value in this experiment is T = 2.172 + 0.017 ws. (9.33) The error shown in Eq. (9.33) is only statistical and does not include systematic effects, in particular any uncertainty in the TAC calibration. 9.5. y-y ANGULAR CORRELATION MEASUREMENTS 9.5.1. General Considerations We will now discuss the measurement of the correlation in angle between two gamma rays that were emitted simultancously from the same source. The origin of these gamma rays is frequently the cascaded decay of a nucleus, as in the case of ©°Ni Co) already discussed in Chapter 8. (See Fig. 8.20.) We reproduce in Fig. 9,28 the decay scheme of this nucleus and note that the 1.333-Me¥V gamma ray follows the 1.172-MeV gamma ray, the lifetime of the intermediate state being only about 10~!? s, so that for all practical purposes the two gamma rays are coincident. The fact that these two gamma mys are correlated in angle can be understood from the following general argument: the first gamma ray will have an angular distribution with respect to the spin axis of the nucleus; thus its observation at a fixed angle 9 = 0 conveys information about the probability of finding the spin at some angle y with respect to the 410 9 Scattering and Coincidence Expariments “Co,, ONiog 1.333 MeV 3 Ee FIGURE 9.28 Nuclear decay scheme of “Co by beta decay to Ni and subsequent’ 333 deexcitation of the “°Ni nucleus to its ground state by the emission of two cascaded gaming: ee Tays. Oe bution about the spin axis that now is known to be at y. Thus the probability. 224 that the second gamma ray will be emitted at an angle @ can be found}::22 this is called the angular correlation function C(@). The time coincidence ae signal assures us that the two gamma rays have indeed come from the: same nucleus and, therefore, are the two gamma rays of interest. A dis cussion of this correlation between cascaded gamma rays is presented iti: Section 9.5.4. me In Na, the angular correlation arises from a much simpler mechanism *2Na is a positron emitter as is shown from the decay scheme of Fig. 8.22 surround the source; the slow positrons are captured by the electrons Of: the copper to form positronium, which decays by the annihilation of th Thus the angular correlation theoretically is given by C(@) = da —@), correlation is so sharp, it is frequently used for calibration purposes. 9.5 »-y Angular Correlation Measuramants 411 e* a” elthar in 4S, or in 'S, A INS 8D ¥2 Fy FIGURES.29 Capture of a positron by an electron to form positronium and the subsequent annihilation of the positron—eleciron pair into two gamma rays. Preamplifier To HV Lead shiefcing (1480 kV) Z 5 EEE Z oe (radioactive source a L Photomultiplier ZB ‘ * + ay § . Stationary counter 4 5° . Intervals marked on circle FIGURE 9,30 Apparatus that can be used for angular correlation measurements. Two scintillation crystals mounted on photornultipliers are protected by appropriate lead shield- ing. One counter assembly ts fixed, whereas the other can be rotated about the position of ihe source. Angular correlations may also be observed between beta and gamma rays, alpha and gamma rays, etc. This technique has proved very fruitful for the analysis of nuclear decay schemes and the assignment of spin and parity to excited nuclear levels. We will descnbe a measuremeni of the gamma—gamma angular corte- lation of 2*Na and of ©°Co. The apparatus shown in Fig. 9.30 was used; it consists of two similar gamma-ray detectors placed at equal distances from the source; ove detector is fixed and the other is free to rotate around the source, varying the angle @ between the detected gamma rays. The detector outputs are fed to a coincidence circuit, and the rate of comcident counts C(6) is measured and compared with the theoretical correlation function. It is important to measure C(@) with the best possible resolution if the data are to be fitted with a polynomial in cos @ of high order. It can be shown that C(@) must be a polynomial in even powers of cos @, the highest power being 2k, where k < Jp, 1), fg where /, is the spin of the final nuclear state and /), /2 1s the angular momentum (multipole) of the ernitted gamma rays. Frequently the experimental measurement may be restricted to the 42 9 Scattering and Coincidence Experiments g Meavorennins. the nwisoarcony of the coincident gamma rays, that is, __ €(180°) ~ C(90°) 7 c(90r) a stronger source may be used, the solid angle may be increased, or the rate; again it depends on source strength and solid angle, but also o the source by detectors (1} and (2}, and let €; and e9 be their respectiv efficiencies. Then the “singles” counting rates are Ri = NAQ)€] : Ry = NAQz€2, . (9.34 mostly in nuclear decay), the coincidence rate** is Rc = NAQ AQ «1e5. (9.35 For most experimental arrangements AQ, = AQ: and €, = €2, so tha we find for the accidental rate Ra, Ra = Ry Ro Ai (9,36): = N72 AQ? At, (9.37): and for the ratio of the accidentals to true coincidences, The efficiency of the coincidence circuit has been sect to ¢. = 1 as it should be. = The limiting factors in these experiments are two: (a) The coincidence: = rate must be high enough to allow statistically significant data to be accu = mulated in a reasonable time interval. To increase the coincidence rate: - efficiency of the detector may be improved (if it has not been maximized: already). (b) The accidental rate must be kept well below the coincidence.:#4 ie ee gaan the resolving time. Let AS?) and AQ) be the solid angles subtended aps: where V’ is the number of disintegrations per unit time of the source. If thie: & oe two gamma rays are uncorrelated (or if the correlation is small, as happens. 9.5 y-~ Angular Correlation Measurements 413 since it enters Eq. (9.35) quadratically; however, the solid angle cannot be increased arbitrarily because this will destroy the angular resolution and wash out the correlation C(@). 9.5.2. The Apparatus The apparatus has been shown in Fig. 9.30, and we give here some addi- tional details. The reader should, however, refer to Section 8.4.2, in con- nection with the instrumentation and techniques of gamma-ray detection. The detectors were Nal crystals 1 in. in diameter and { in. thick, mounted on RCA 6655 photomulttpliers. Each was located 8 in. from the source. Both crystals are protected from scattered radiation with lead shielding, and the movable detector can be rotated about the center in 5° mtervals. The block diagram of the electronics is shown in Fig. 9.31, where the individual units are available from a oumber of vendors.”? The units are interconnected with 50-2 coaxial cable. Manuals accompanying the amplifier, discriminator, and coincidence modules should be con- sulted, especially to achieve the smallest possible resolution time. In the ensuing discussion we will assume that the circuits have been properly adjusted. , One of the outputs from each discriminator is fed to the coincidence module and a second output to a scaler capable of a peak rate of 10°/s. The coincidence output is also fed to a scaler. In this way the “singles” in each channel and the “doubles” are counted. The delay between the two inputs to the coincidence circuit may be easily adjusted by inserting appropriate cable lengths between the discriminator and coincidence in one or the other of the channels. One foot of typical 50-© coaxial cable corresponds to a transit ime of about 1.5 ns. Some care is required in order to properly set the discriminator bias levels and photomultiplier high voltage. First the system is checked out with a pulser, to adjust the setting and functioning of the scaler drivers and scalers. Next the actual signals are fed into the circuits and the discriminator outputs “looked ac’ on an oscilloscope to ascertain that the pulses are “clean” and uniform The high voltage is set by taking a plateau curve, which will not be completely flat but nevertheless should show a clear knee. If the system is 23 or example, Canberra Industries (http://www.canberra.com/) and Onec (http:// www.orteconline.com/) both give details of similar setups, including cross references to their own product line. 4144 9 Scattering and Coincidence Experiments (1) Scalar Discriminator driver noes Preamoptitier IH-51 fal Discrimmator IH-54 AGA : ga55 Preamplifier arate FIGURE 9.31 Block diagram of the electronics used for angular correlation measure ments. - 1 p+ & Ag=13.2 05 Colncidencas (Countess) 8) 5 10 15 20 20 30 45 . H} 1. merase Nhe 1 ‘ : a! Ss See eS SESE Sr Ru ce erate rate by a factor of at least 1000. The curve through the points is a simple spline interpolatiar and is only meant to puide the eyc. me “ wert a working properly, the “singles” rates A, and R2 in the two channels should! be (almost) equal. ae y It is possible to measure the resolving time of the coincidence..cit: it ss a cuit either by taking a “delay curve” (see Fig. 9.32) or by making. use 22 al ay eat 9.5 y-y Angular Corralation Measurements 415 TABLE 9.4 Determination of Resolving Time from Accidental Coincidences Counts/s Af (s) Channel (1) Channel(2) Comceidence (Al = C/R,R2) 2151 2056 0.061 13.8 x 107? 5920 6262 0.528 [4.2 x 1079 14,662 13,481 2,912 14.7 x 107? > 31,207 35,443 14.217 12.8 x 107? of Eq. (9.36). When the latter method is used, the two counters are sepa- rated by a very large distance and a separate source is placed in front of each. In view of the geometrical arrangement and the fact that an additional delay of 60 ft ts placed in one of the channels, all the coincidence counts are accidentals. By varying the distance between the source and the respective counter, the results given in Table 9.4 were obtained; the counting time was on the order of [0 min at each pomt. We note that the resolving time so obtained (column 4) is quite consistent despite the fact that the accidental coincidence rate increased by a factor of about 2000 between measure- ments; this resolving time is also consistent with the width of the two input signals (which were about 6 ns wide) and the data of Fig. 9.32. The above results as well as those to be presented in the following two sections were obtained by students. 9.5.3. The y—y Correlation of 7“Na A 100-.Ci 2*Na source, wrapped with a G.001-in. brass foil is placed at the center of the apparatus. The dimensions of the source are kept at a minimum, and it is positioned as accurately as possible. Since the solid angle is AQ = [mw x (0.5)71/(8)* % 42 x 1077 where the dimensions are in inches (see previous section). Assuming a detector efficiency €; ~ €2 ~ 0.3, the expected rate for “singles” is 10 y 49-4 Ry ~ Ro = are x (dx x 1073) x 0.3 = 1000 counts/s. a0 416 9 Scattering and Coincidence Experiments Since the two gamma rays are completely correlated when the two counters: are at @ = 180°, the expected coincidence rate at this angle is C(8) = nAGe? = Re = 300 connts/s. (9,39): The observed rates are on this order of magnitude. However, the 1.277-MeV gamma ray also contributes to the single rate; on the other hand, the finite size of the source and errors in geometrical alignment reduce the. coincidence rate from the calculated value. We first wish to check whether the coincidence eircuit is correctly ° “timed”—that is, whether the appropriate delay has been inserted so as * to make truly coincident signals arrive at the circuit at the same time. To °: this effect the movable counter is rotated to 180° and the counting rate is. obtained as a function of the variable delay introduced into channel (1); for: convenience, a fixed delay of 12 ft of cable has been introduced into channel: : (2). The data so obtained have already been given in Fig. 9.32 ona semilog * plot, which is the more appropriate representation for a delay curve. ; We note that (a) indeed, the peak counting rate occurs whena 16-ns delay © 224 is inserted in channel (1) as expected; (b) in the peak region, the delay curve a is flat over at least 6 ns; this indicates good efficiency and consequently that © small time jitters will not result in changes in the counting rate (provided * the delay is set at the center of the curve); (c) the width of the curve at ° half-maximum, which gives the resolving time of the circuit, is 13.2 ns, in excellent agreement with the values found in Section 9.5.2 and what is expected from the width of the input signals; (d) the accidental rate is very low; by inserting 40 ft of delay it is found to be 0.048 + 0.005 counts/s, yielding a ratio signal = 150 = ~ 3x 10°, 9.40 noise 0.05 * ( ) which is more than adequate. The considerable siope of the ascending and descending parts of the delay curve is due to the timte jitter of the input signals associated with their low peak amplitude. The stability of the system can he judged from : the fluctuations of the coincidence rate in the flat region as well as from the fluctuation of the singles rates given in Table 9.5, : We are now ready to obtain data on the angular correlation of 24Na.- : The movable counter is rotated in appropriate steps to either side of 180°, and the doubjes and singles rates are recorded. The resulting data are shown in Fig. 9.33, and in Tabic 9.5 some representative points are listed. 9.5 y-y Angular Correlation Measurements TABLE 9.5 Representative Data on the y~y Correlation of 7*Na @ Stationary (°) counter 90 3011 150 2996 160 3013 170 2994 175 3011 178 2992 180 2995 182 3014 185 2991 190 3039 200 3005 210 3007 22Wa coincidences (Counts/s) 150 Counts/s Movable counter 3086 307] 3090 3064 3114 3189 3035 3178 30659 3127 3102 3136 160 170 Coincidence 1540.1 L524 0.2 17402 3,540.2 66.8 + 1.0 148.0 + 1.5 159.0 + 1.6 124.0 + 1.2 50.2 + 1.0 3,240.2 2.0+ 0,1 1840.1] 180 TAD Angles between detectors {dagrees) Coincidences (Counts/s-degree) 0,23 0,23 0223 0.49 9.2. 20.6 22.1 17.2 7.0 0.42 0.26 6.25 210 4i7 FIGURE 9.33 Angularcorrelation of the gamma rays froma **Na source. The coincidence Tate is plotted as a function of the angle between the two counters. Note that the full width of the correlation curve is 8.5°, which is entirely due to the angular resolution of the two counters, the isoopic background outside the peak is very small. The curve is a Gaussian fit to the peak region, with a fixed constant background, but only serves to guide the cye. 418 9 Scattering and Coincidence Experiments Columns 2 and 3 of Table 9.5 give the singles rates for the stationary and the movable counter, respectively; the coincidence rate* is given in col- umn 4, The counting time at each point was on the order of 1 min, which provides good statistics (about 1% in the peak region). Indeed we do notice a very pronounced correlation at @ = 180°, with an angular width of +4.25°. This width is on the order of the angular resolution of our system, which might be taken as the angle subtended at. the position of the source by one of the counters : 6 0.5 | an (*) ==> Ad =7.2°. (9.41). We therefore conclude that this correlation is compatible with La SM cot at Sal at Td pas A atk a REA Hs OTT CG) = 8(e — 8). (0.42) peti = Cer ie Rare The anisotropy as defined by Eq. (9.34) is _ €(180°)- C0°) _ 150-155 7 C(90°) “15 wernt pen a 100, (9.43): which is extremely large and compatible with a — oo as predicted by. Eq. (9.42). : The counts observed at large angles are still real coincidences, but due! mainly to the isotropic correlation of the 1.277-MeV gamma tray with one;: of the annihilation gamma rays; it should be on the order of the correlated: counts multiplied by the solid angle for one detector AQ ~ 10-7, as is: indeed the fact. Also, a small fraction of the background originates from: annihilation gamma rays that have scattered through a large angle in the: source or the converter. In column 5 of Table 9.5, the coincidence rate has been divided by the: angular acceptance A@ of the movable counter as given by Eq. (9.41): Indeed, since the correlation is a function of @, it is obvious that our systert, measures C (9) at @ within the differential range +AQ@. one : From the results presented we conclude that ?*Na provides a very good ae technique for aligning and adjusting the equipment, especially since the: strong correlation from the annihilation gamma rays is quite easy to detect: 30The rate for accidentals should have bcen subtracted from the results of column 4 however, itis so small (see Eq. (9.40)) that we neglect it. 9.5 y-y Angular Correlation Measurements 419 Also, the obtained correlation provides strong evidence for the annihilation of the positron-electron pair into two gamma rays; if a differential discrim- inator Is used after the detector, it is also possible to measure the energy of the coincident gamma rays, The angular resolution of the equipment may be easily improved by simply increasing the distance between the source and the counters. In fact, precise data on positron annihilation are quite sensitive to the momentum of the positronium just before it annihilates; this in turn provides information on the structure of the Fermi surface of the converter material. 9.5.4. The y~y Correlation of ©°Co Once the equipment has been adjusted and aligned (forexample, with Na) as described before, any correlation may be measured. A ®’Co source of (he same strength as the ?*Na source (100 Ci) was placed at the center of the apparatus, and data were taken every 15°, The discriminator levels could be readjusted, but it is usually preferable to leave everything as is. Since the ©°Co y—y correlation has a small anisotropy (as compared to 22Na, Eq. (9.43)) the expected coincidence rate is C(#) = N(AQ)*e? ~ 4 counts/s, (9,44) which is much smaller than that given by Eq. (9.39) for the same source strength, Consequently, also, the signal-to-noise ratio (Eq, (9.40)) will be only about 30, and the “accidental” rate, which was 0.070 counts/s, must be subtracted. Furthermore in view of the smaller correlation, better statistical accuracy is required. Representative data taken in one run are presented in Table 9.6 and plotted in Fig, 9.34. Incolumn 5 the coincidence rate after the subtraction of accidentals is given, while in column 6 the rate at each angle is normalized to the rate at 90°. At each point sufficient coincidence counts were taken to give 1% statistical accuracy (10,000 s = 3 h); these errors are indicated by the error bars shown in Fig. 9.34, where we plot a(@) = C(@)/C(90°) against angle. We see that the fractional errors on (@(@) — |) are now much larger, and on the order of 10%. It is known from theoretical considerations that the “Co correlation function is of the form wt) = © = 500°) = ] + a; cos? 8 + ay cos’ @, (9.45) 4 9 Scattering and Coincidence Experiments TABLE 9.6 Representative Data on the yy Correlation of Hoo Counts/s p Stationary Movable Corrected (3 counter counter Coincidences coincidences eae 60 2203 2129 0.880 0.810 1.080 50 2132 2157 0.820 0.750 1.000 105 2152 2127 0.857 0.787 1.049 120 2144 2130 0.864 0,794 1.059 135 2109 2175 0.886 0.816 1,088 150 2132 2136 0.933 0.863 L151] 165 2121 2123 0.931 0.861 1.148 180 2116 2124 0.944 0.874 1.165 210 2086 2134 0.889 0.819 1.087 i ‘ SEN hy, . oh, Lik ik ee 4.48 } —— Theory es --- Least squares fit # 1.16 2B 1,14 1.12 go & 1.1 O & 1.08 O 1.06 ee 100 120 140 160 180 Angle 4 between detectors (degrees) FIGURE 9.34 Data on the angular correlation of the two gamma rays fror correlation function C'(@}/C (90°) is plotted against the angle between the t Note, however, that the ordinates begin at the value 1.00. The experiment shown, and the dashed curve is a least-squares fit to the data. The solid lit theoretical curve, which is given by the function 1 + 0.125 cos? @ -+ 0,042. co ot Cat ata missin SESS snetatetate ‘ atte beb tata x metab Q : Satara to ut ‘“ : nee os SSeS SS woe 9.5 p-p Angular Correlation Measurements 421 A least-squares fit to Eq. (9.45) was made, using the entire set of expeti- mental data,*! and the following values were obtained for the coefficients a) and ao. a, = 0,190 + 0.08 a2 = —0.04 + 0.08 The theoretical values resulting from the spin assignments 7, = 47, J, = 2* and J. =O (see Fig. 9.28) are ay = 0,125 ay = 0.042, * The correlation function that results from the above coefficients is included in Fig. 9.34; the dashed line represents the least-squares fit, and the solid line the theoretical curve. From Fig. 9.34 we see clearly that an anisotropy in the angular distribution of the y—y coincidences from ©°Co exists; we obtain a = o(180°) — 1 = 0.165 + 0.016. (9.46) The error flags in Fig. 9.34 were set at 1.5%, but the data points scatter even more. This is not due to the “statistics,” but to random fluctuations and drifts of the equipment over the long counting intervals. 31 This included 21 more measurements in addition to those presented in Table 9.6. SESUNSEESES Se ee 7 q Selah a he SO ae ee i a hr ee CHAPTER 10 Elements from the Theory of Statistics 10.1. DEFINITIONS Statistics 1s the science that tries to draw mferences from a finite number of observations constituting only a sample, so as to postulate rules that apply to the entire population from which the sample was drawn. In the field of physics, statistics is needed (a) to fit data—that is, to est- mate the parameters of assumed frequency functions; (b) to treat random errors; and (c) to interpret phenomena that are inherently of a statistical nature. 10.1.1. Definition of Probability The probability of occurrence of an event can be axiomatically defined as egual to one (= 1) if the event occurred, or equal to zero (= 0) if the =: event did not take place. An alternative definition of probability is based on the frequency of occurrence of an event. Suppose that several trials of ' the same experiment have been made; then the probability of occurrence 473 474 =6©10 Elements from the Theory of Statistics of an event A, that is P(A), is given by the number of times event A wag obtained divided by the total number of trials (in the limit that the totak number of trials approaches infinity). This definition of probability retaing its ful value even in the case of nonrepetitive experiments, since the ofié trial can be considered as the first of a serics of trials. 10.1.2. Sample Space Any set of points that represents all possible outcomes of an experiment ig a sample space. For example, if a coin is tossed twice, the sample space consists of the 4 points indicated in Fig. 10.1. (Sample spaces can be finite: or infinite and discrete or continuous.) ws Once the sample space for a particular expernmentis constructed, we may assign (in the sense of Definition 10.1.1) a probability p; to each point of the space. From the definition of probability, we have o ae Bape pi > 0 p= Is ee all sample space points a aE thus 2 a pPixl Be and the probability of occurrence of an event A is : ya PilA) P(A) = ~S_— = )_ PilA), >. Pi d where >", indicates sunumation over all points that include event A. FIGURE 10.1 Simple example of a discrete and finite sample space. Here the samp space points correspond to all possible outcomes of “tossing a coin” twice. 5 10.1 Definitions 425 In most situations treated by statistics, equal probability is assigned to each sample-space point, a condition we will maintain throughout this discussion, Then ] i= Ts n n being the total number of sample-space points, and P(A) = Sta). » n where n(A) is the number of sample-space points containing event A. For example, in the case of the sample space of Fig. 10.1, the probability of obtaining heads at least once is P(heads) = n(heads at least once) a 3 n 4 while the probability of obtaining heads once and tails once (irrespective of order) can again be found by counting the appropriate points in the sample space of Fig. 10.1. We obtain acheads, tails) _ of P (heads, tails) = me 10.1.3. Probability for the Occurrence of a Complex Event _ The probability that both events A and B will occur is called the joint _ probability pag) = MAmd BY / where n = total number of sample-space points. The probability that either A or B will occur is called the either probability AorB PEA +P SES nh _ and the probability that A will occur when it is certain that B occurred is ; Called the conditional probability : n(A and B) P[A|B] = n(B) 476 10 Elements from the Theory of Statistics a region where both event A and event B can occur simultaneonsly. (b) No such region exisls; events A and # are mutually exclusive. ak All these probabilities are defined in the sense of Definition 10.1.2 as the number of sample-space points thal contain the stated condition divided by the total number of sample-space points allowed for by the statement. ©: Figures 10.2a and 10.2b illustrate two sample spaces. All points withit domain A include event A while all points within domain B include events:22423 B. The points contained in any intersection of the two domains A and & include both cvents A and B. Ls Lf such a common intersection does not exist in sample space, the tw events are mutually exclusive, and ° P{AB] = 0. It follows from consideration of Fig. 10.2 that Be P[A+ B] = P|A]+ P[B]— P[AB}. For the conditional probability ntA intersection B)} P[A|B]) = ———_—___——_, [A|B] n(B) since the condition that event 8 occurred restricts our sample within domain: “f B. However, OEE n(‘B) P[B| = 10.1 Definitions 427 and P[AB] = — = P[A|B]- P[B] = P[B|A)- PIAI. (10.1) If P{A|B] = P[A}, it means that the occurrence of B does not affect the probability of occurrence of A. We say that the two events A and # are independent. It then follows from Eg. (10.1) that P[AB] = P[A]- P[BI. (10.2) Equation (10.2) in tum implies (when combined with Eq. (10.1)) that for independent events P[B|A] = PLB. To illustrate some of the ideas we have just expressed, consider the fol- lowing. For the sample space of Fig. 10.1 we may define: event A = heads in first throw, and event B = heads in second throw. The domains are shown in Fig. 10.3, and it follows (assigning p = 1/4 to éach point) that ] ] PLA] = 2 PIBI= 5 I PIA +B) = P[A] + P[B] — PIAB]=—+~>-+=- [A+B] = P[Aj]+ [B] — =545747G FIGURE 10.3 The sample space of Fig. 10.] including the domain A (heads in the first ; throw) and the domain B (heads in the second throw). 428 10 Elements from the Theory of Statistics j | 1 PLA\B]= 5; PIBIAI= 5 11 1 P[AB] = P[A|B]- P(B] = 5-5 = 7 = PIAL: PLB) Thus events A and B are net mutually exclusive but are independent. Qa ae 10.1.4. Random Variable To study a sample space analytically (instead of geometrically), it is con venient to use a numerical variable that takes a definite value for each an every point of the sample space; however, the same value may be assignie to several points. Thus, a random variable used for the representation 6 a finite and discrete sample space will have a definite range and will take only discrete values. As an example, for the sample space of Fig. 10.1, w can assign to the random variable x the value 0 for points (b) and (c) (one each of heads and tails), the value —1 for point (a) (both tails), and th Bors ee valuc +1 for point (d) (both heads). Beas A ae Pata et SP a ba a SAN 16.1.5, Frequency Function A frequency function (of a random variable) is a function f(x) such tha f (xo) is the probability that the random variable x may take the specifi: value xp. By Definition 10.1.1, f(x} gives the number of points in the sample space that have been assigned the value of x of the random variable divided by the total number of sample-space points. The function f(x) i defined only within the range of x and need not have a definite analyti form. For the example considered above (the sample space of Fig. 10.1) f(x) is just a table, as shown in Table 10.1 (see also Fig. 10.4). : TABLE 10.1 Example of a Frequency Function *(*) of the Random Variable x Sample-space point x F(x) (a) -1 - (b,c) 0,0 i (a) +1 . 10.1 Definitions 429 f(x) Ni ee & aa a, ; FIGURE 10.4 The distnbution function of the discrete random variable x defined in Table 10,1. The summation of {(x) over the entire range of x must give 1: y.. Kbeaed all x The probability that the random variable may take any value smaller or equal to x Is given by F(x) = >> fit) f=x and is called the distribution function of x (or integral distibution function). It is sometimes convenient to describe a sample space in terms of two or more random variables, a frequency function existing for each of them. If these random vanables are independently distributed in the sense of Eq, (10.2), the joint frequency function Is Sf (41, *2,...) = fer) f(a) >> fn). If the random variable is continuously varying (for example, it describes the height of individuals), the probability of occurrence of the specitic value x when a measurement ts performed defines the frequency function J (x) dx of the random variable x. The random variable may now take any value within the range of its definition. Note, however, that the probability of occurrence of the exact value x is zero, while it is the probability of occurrence of some value in the infinitesimal interval dx about x that exists, For a continuously varying random variable, we have +00 f(s) = 0 and f(x)dx =1. “=o 430 10 Elements fram the Theory of Statistics Similarly B | f(xj)dx = Pla ~ |x; — m] f;, and so on. ee 10.2.3. Theoretical Frequency Functions As mentioned before, a theoretical frequency function f(x) might be of eee the discrete type—that is, the random variable x takes only integer values, ee rhe . * 10.2 Frequancy Functions of One Variabla 433 or of the “continuous” type. Most of the discrete random variables usually represent the number of successes, or of counts obtained, etc. In gomg from discrete frequency functions to continuous ones, obviously all summations are replaced by integrals. Moments are defined as in Eq. (10.3), but instead of the empiri- cal frequencics f;, the theoretical frequency function f(x) is used; the theoretical moments are designated by Greek letters, Latin letters being reserved for the empirical moments. Thus, the k" moment about the origin is r=+co.) up = >> x* f(x). r= 08 The first moment about the origin gives the mean; and ts denoted by = ws. The k moment of a theoretical frequency function about its mean is k=+05 He= > (x—p)F fF (2). x=—00 The square root of the second moment about the mean gives the standard deviation and is denoted by o = ./j12: x=+c0 2 = » (x — p)* f(x). atm—O) 10.2.4. The Bernoulli or Binomial Frequency Function This basic frequency function is applicable when there are anly fwo possible outcomes of an experiment, a3, for example, the occurrence of anevent A or ifs nonoccurrence (we designate this by B). If the experiment is repeated n times, the random variable x describes the number of times event A =. occurred. The frequency function—that is, the probability of obtaining a - certain x—is piven by nl x ae fid= lini? gq (10.6) “ Where pis the probability that event A will occur inthis experiment (defined = in the sense of Section 10.1.1); and g = ] — pis the probability that 8 will - happen, namely, that event A will not occur. 434 10 Elements from the Theory of Statistics To prove Eg. (10.6), consider the probability of obtaining event A, : times in a definite sequence AA+::A BB: B: ‘ne paar! ns i N—-X this joint probability of order n is according to Definition 10.1.3 pp --pqq::q= pg" x n—-x : since the outcome of consecutive experiments is independent. However: any other sequence, containing the same number x of occurrences, is also a satisfactory answer, since we are not interested im the order of occurrence: of event A. Thus we must sus over all sample-space points that give x occurrences; the number of all such sample-space points is given by the. permutations of 7 objects in groups of nm when x of them are alike (have probability p), which is n! x'(n — xj completing the proof of Eq. (10.6). i The frequency function fulfilis the normalization requirement as it, : should, since n! 3 f= 3 ree =(p+qy"=[pt+Q-p)=1 ee (10.7) 2 10.2.5. Moments of the Binomial Frequency Function From the definitions of SecUon 10.2.3, and since the range of x is from 0 ton, we have : nt n —1)! = .~ ta iy t= | | 4 Bete San Ra CInan ache CRON TET wink AM oantelniedeleteitettaty SASS 10.2 Frequency Functions of One Variable 435 If we let y = x — 1, it follows that n—I (n — 1} pri) ji =np = y 1 n= ly —yll I where now the sum is equal to (p +. q)"~'! = 1. Thus j= np. > (10.8) Next we wish to obtain the second moment about the mean, jz2 = o* We first calculate 45, given by t 2 x is Ba} (3° x=0 x(n— x)! We use x? =x(x—1) 42 so that y= Soxte=1 ree ail Z + He x=) = z pnt = Doe = <4 +p = 2 2)! —2 .n—s =n(n—1)p ieee soe q+ and letting y = x — 2, as before, the sum is equal to ( p + q)"~? = | and we obtain jt =n(n— 1)p* + w =n? p* —np* + np. Next we usé Eq. (10.5) to obtain = pig = 07 = 44 — pw” = —np* +np =np(i — p) =npq. : pa (10.9) = The binomial frequency function is applicable to many physical sit- = uations, but it is cumbersome to calculate with, When n becomes large, 43% i0 Elements from the Theory of Statistics Poisson distribution mn must be large, for example, n > 100, but 44 = np must be finite and small, for example, p < 0.05. | Gaussian distribution » must be large, for example, » > 30, and also p must be large, for example, p > 0.05. 10.2.6. The Poisson Frequency Function Be This is still a frequency function for the discrete random variable x, which describes, as in Section 10.2.4, the number of times event A will be obtained=24 if the experiment is repeated n times when n + oo for (large n). Contrary to Eq. (10.6), however, neither 7 nor p appears explicitly in the analyti¢: oe expression of the frequency function, but instead only their product y=np, (10.10) 22 which remains finite despite » - 00, since p > 0. The Poisson frequency function is given by Ta? f= and it is shown in the next section tbat y is the mean of the distribution: govemncd by Eq, (10.11). oe To prove Eq. (10.11), let us first note that since 7 is large, it (but not xy. may be treated as a continuous variable; second, we will assume that for a small (differential) number of trials dn, the probability of obtaining event A once is proportional to this number of trials: that is, P{l,dn) =ddn, ao: where 4 1s a constant. Note that Eq. (10.6) fulfills this requirement for ee = 1 in the limit that p > 0 org — 1. In terms of sample space our 3 assumption means that the density of sample-space points containing event os : A is uniform in the limit of a differential element of sample-space area, 227 ‘See, however, the detailed discussion in Section 10.2.9. 10.2 Fraquency Functions af One Variable 437 S The Poisson frequency function then follows for all populations for which = assumption (10.12) is valid. = Let P{x,n} be the probability of obtaining event A, x times in n trials, = go that P{0, n} is the probability of obtaining no events A in » trials. Then the probability of obtaining no events in nm + dn trials is P{O,n + dn} = P{O,n}- fl — Pl, dn}] = since the events are independent.” Using Eg. (10.12) we obtain = P{0,n + dn} — P(0,n) _ = of =—P{0,n}-d dn (0.7) dP (0, SEO) _ Pin) a, " dn = which has the solution In P{0, n} = —nd P{0,n}=e™ (10.13) "and use has been made of the initial condition that for n = 0 P{0,0} = 1. In a similar manner we obtain Pfi,n+dn} = Pf{l, nm} P{0,dn} + P{0, x2} FP{i, dr), : where the two possible either probabilities are sunumed. Making use again of Eg. (10.12), we may write the above result as Pil,atdn} = P{l,n}-(l—Aadn|]+ P{0,n}-Aadn by further transforming and ustng Eq. (10.13) as well, aP(1,n} dn The solution of this linear first-order equation is straightforward, leading to +AP{L,n}—Aeg™ =0. P{i,n}=e™ | ene "dn + c| = (ndje™, (10.14) making use of the initial condition P{1,0} = 0. Since the increase in the number of trials dn is differential, the possibility of obtaining more than one event in dn is excluded. 438 10 Elements from the Theory of Statistics In general the following recurston formula holds a ane 7) Pl n}—- API — Dn} = which is satisfied by Ln —Ak | f@x)= Pix, n)= a (10.15): as can be verified by substitution. : Thus &q. (10.11) has been proven, and we can identify the proportion ality constant 4 as the probability that event A will occur in one trial. As: pointed out before, however, it is only the product y = An = pn that may be properly defined: it is the theoretical mean of the discrete random, variable x when the same (large) number of 7 trials is repeated many times. Equation (10.11) correctly fulfills the normalization requirement A=0o >» f@) Le =e %e7 = 1. x=0 It is shown in Section 10.2.9 that Bq. (10.11) is the limiting form of : Eq. (10.6) when p > Oandn —> o., : 10.2.7. Moments of the Poisson Frequency Function Following the approach used in Section 10.2.5, the moments of the Poisson : frequency function will be obtained by direct evaluation of the defining : equations; note that as n — oo the upper limit of x is also co: ‘ i x=0 ~@—D! 1); oo yor 1) -y Yup? — =e Lea J eye =. Thus way (10. 1 3P{l,l}=de7* @ Awhend large and |np — x} << np. Consider Eq. (10.4): | f(x )= pt ght x!(n —x)! If —> co but np —> ys remains finite, we may write rr yn n° xf fey = MAM = Din) ory’ ey _ py () — py ! However, (1 — py = [(1 — p)~O/P)] 8? —» eH since from the definition of 2, lim(1 +2)! =e z+ and in the present case we have p — Q. Puriher fen WL U/mL = (= Din] nro (I= py because p —> Oand x 1s finite; by substituting the last two expressions intg Eq. (10.23) we obtain the Poisson frequency function, Eq. (10.11): : wee f(Qx)= We now use the further condition that x be a continuous variable and: ae np - x] < np, namely, its deviations from the mean jz be small; then the: Hee following approximate expression is valid: gee ~ — i —~x\4 nk =n(i+4 *\=(4 *\-3(¢ =) 4 x x x 2 x 10.2 Fraquency Functions of One Variable 443 Hence and 1(u—x) je” ® x” exp(yt — x) exp 5 ee *) . 3 From Stirling's formula we have xl V2nxxte? and by substituting (j2)* and x! into Eq. (10.11) we obtain wre eo bxte exp{ —31(u — x)*/x]] f@Q)= xl fla xxte—* ] l fp-x ; ~ Ving 5 ( vx | oe Thus the binomial frequency function in its limil approaches a Gaussian frequency function with mean ji = np standard deviation o = ./x © ./npq, (10.25) where x *¥ npg follows from | — x| < yz and p -> 0. From Eq. (10.25) we see that the moments of the limiting Gaussian frequency function are the limits of the moments of the original binomial frequency function. 10.2.10. Praperties of the Gaussian Frequency Function | Let us now interpret the frequency function given by Eg. (10.18). We ' could refer to our original example of obtaining event A, x times when a choice between A or 2 1s made m times; x then can vary from 0 ton in integer values. It 1s easier, however, to consider the measurement with a ruler of the length of a rod; we Jet the continuous random variable x represent the result of one measurement, If the true length of the rod is xo, Raq. (10.18) specifies that a result between x and x + dx will be obtained eee 444 10 Elements from the Theory of Statistics Gs = with a frequency (xjdx = exp | —= ( dx. (10.26 oe y of ln PL aN a : z One may also say that the probability that the measurement will “yield & result x” between x and x + dx is given by Eq, (10.26). In simpler words if N measurements are performed, a result between x; and x3 is likely to be obtained in n(x1, x2) of these measurements, where 7 SESS a ke al otal Nf Lfxm—x\2] A(x}, 42) = N+ F(x, x2) = exp | —- (* *) dx. Of Zit X] as shown in Fig. 10.5. Note that in Eqs. (10.26) and (10.27) the standard deviation o is deter minced by the conditions of the measurement. The applicability of th ae Gaussian distribution to the results obtained from such measurements lies: 4427 in the fact that: (a) n, the number of (least) divisions of the ruler, is large: and (b) the errors in measurement |xg — x| are small as compared to x. ~: In Table 10.3 are given the values of f(x) and its integral, F (ce), for th normalized Gaussian function (Eq. (10.22)). 7 From Table 10.3, for example, we see that half of the measurements du yield a result x between 4 Xo — 0.690 < x < x9 + 0.690 or that only 2.23% of the results may yield x, such that x >XxXy4+ 20. TABLE 10.3. Some Numerical Values of the Normalized Gaussian Function f@)= vo exp(—x*/2) Fl(-e,e}= pre f(@jdx f@Q) == 0.3989 F(—1, 1} = 0.6826 f(l) = fC-D = 0.2420 F(—2, 2) = 0.9554 FQ = f(—D = 0.0540 F(—3, 3) = 0.9974 F(—0.69, 0.69) = 0.5000 10.3 Estimation of Parameters and Fitting of Data 445 As another example we see thaf a result x in the small interval Ax about xg, will be obtamed (0.3989) /(0.0540) = 7.4 times more frequently than a result in the same small mterval Ax about xg + 2c. 19.3, ESTIMATION OF PARAMETERS AND FITTING OF DATA In Section 10.1 the basic definitions were given; in Section 10.2, analytic expressions for some frequency functions were obtained. We will now see how statistics can be applied to the interpretation of a measurement or an experiment. We can consider one of more measurements to form a sample of a pop- ulation that obeys a certain frequency function; we are then faced with one of two estimation problems: {a} Given the frequency function and its parameters, what is the probability of obtaining from a measurement the result x? (b) Given the result x of a measurement, what are the parameters of the frequency function (or the frequency function itself)? In physics we are usually faced with estimation of type (b), since a set of experimental data are obtained, and it 1s then desired to reduce them fo afew parameters that should describe the whole population and therefore, also any new measurement that may be perfonned. There are several methods for obtaining “estimators” to an unknown parameter. Some of these methods are almost subconsciously applied, but most of them can be derived from the principle of “maximum likelihood” introduced by R. A. Fisher in 1920. 10.3.1, Maximum Likelihood To apply this principle we must have knowledge of the normalized frequency functions of the variables x; that form the data, f(x 0), where @ is the parameter to be estimated and upon which the frequency function depends. We may then form the product of the frequency functions for all observed variables, L(xi, £2, --- nO) = 1, 8) F (x2, 8) Fn, 8), (10.28) 445 10 Elements from the Theory of Statistics which is called the likelhood function for the parameter @ (note that £. is not a frequency function for the parameter @). The theorem of maximum: likelihood then states that the value of @, @*, that maximizes © (for the. set of observed data) is the best estimator of @: OL(x), 42,-.-Xn, 8) ae a—9* 4, since when W = log & is maximum, so will also be L. As an example, we consider a set of m data x; that obey a normal frex quency function about a, with a standard deviation o; let us seek the best value for the paramcter a: V2 Ij,a) = + exp _* (<—*) oC Then n \2 W =logl = —n log (6 V2z) _ >) (< =) r=] aw Pax Qa 72a ge setting (9W)/(da) = 0 leads to the estimator a*; OT Thus if a sct of measurements is distributed normally, the best estimator : e for the true value of the parameter is the mean of the measurements (first 22 moment). “i er aa 10.3 Estimation of Parameters and Fitting of Data A47 Similarly we may obtain the estimator, o*, for o, by differentiating Eq. (10.30) with respect to a ee O(( AS) and setting AW /do = 0 gives ] 7 aye =. — yj? 1 (a”") = : (a —x;)*, (10.32) where, in Eq. (10.32), a should be replaced by its estimator a* given by Eq. (10.31). Again we obtain the familiar result that the best estimator for the standard deviation of the theoretical frequency function is given by the second moment (about the mean) of the observed measurements. The principle of maximum likelihood can be further extended to give the variance S* of the estimator 0"; that is, if the determination of estimators @* is repeated, the values so obtained will have a standard deviation S, where 1 aw Se ag We may apply Eq, (10.33) to our sample of measurements that obeys a normal frequency function, where W was given by Eq. (10.30). We obtain (10.33) l ew Hil it 32 ages Se get Thus the standard deviation of the estimator will be a sS= ie (10.34) where n is the number of measurements used for obtaining each estimator, Equation (10.34) is a well-known result that we will obtain again when we discuss the combination of errors in Section 10.4. 10.3.2. The Least-Squares Method Until now we have discussed the case where all measurements are made on the same physical quantity whose true value is a, for example, the data of 448 10 Elaments from the Theory of Statistics FIGURE 10.6 Least-squares fit of a two-dimensional curve to a set of data points obtained: for different values of x. Note that each data point has associated with it a different error as indicated by the flags; this is taken into account when forming the least-squares sum. ° Eg. (10.29). However, consider now a set of measurements yielding values Yi, ¥2,---, ¥x depending on another variable x; the corresponding true: values of y, which we designate by j, are assumed to be a function of x and: of one or more parameters w, common to the whole sample. Thus we write: Fi = YRS dys... dy). (10.35): Further, each measurement y; has associated with it a standard deviation’ a;, which is not the same for each point. This situation is shown in Fig. 10.6::: It is possible that the form of Eq. (10.35) is known or may be correctly: inferred from the physics of the process under investigation, in which case: the estimation is reduced to finding the best estimators for the parameters’ ay. If, however, the form of Eq. (10.35) is not known, various functional. relationships must be assumed, for example, a polynomial of order k. We: then speuk of fitting a curve to the data. Even though special techniques are’ developed in Scction 10.3.4 to ascertain which curve fits hest, the following | discussion is generally applicable. | The method of least squares follows directly from the assumption that: each individual measurement y; is a member of a Gaussian population with. a Inean given by the true value of y,;, p(x;; @,); for the standard deviation. of this Gaussian we use the experimental error o; of each measurement. - Then in analogy to Eq. (10.29) we write for the frequency function of yj". L| yi — ¥O%GS a) exp y—-xz me f 2 l 2 : is Xz: = — ; 10.36): J (vis 47: ay) o, Jin ( , 10.3 Estimation of Parameters and Fitting of Data 449 and in analogy with Eq. (10.28) we form the likelihood function "i Evi Yai Xt a) = | | SO%5 25m). i=] We seek the estimators aj that maximize this function, or its logarithm W | W = logl " n —s., 2 =a 2,108 (0:27) — 5 > feed ; (10.37) i= Since the values of o; are fixed by the measurement, the estimators af are those values of a), that minimize the sum i =a x [yw — P03 aa) ]? i=] ! that is, those that give the “least-squares sum." They are obtained by solving the simultaneous equations —$ = X= ltov. 10.3.3. Application of the Least-Squares Methad to a Linear Functional Dependence The simplest case of functional dependence y(x} ts the Linear one: yp=ax+8. If we assume that every measurement y; has the same standard deviation (statistical weight), we may obtain the estimators a" and 5* that minimize Eq. (10.38) in closed form. Since a) = o2 = -+- = oy, = a, instead of Eq. (10.38) we need only minimize R=) Ly - @t+ ox]. (10,39) i=] er 450 {0 Elements from the Theory of Statistics Hence 3, =? Lo- (a + bx;)] =0 ~ = 23" {hy — (a+ bux} = 0, i=l which after some manipulation* leads to a= > x? 2 >> xi SGry) n> x? xP — Sox Do xi pe = ML) -— Vw ny x? — Sox; Vox; The standard deviations for the above estimators may be obtained by ait extension of Eq. (10.33), which now yields a symmetric square matrix a2w 1 @°M ey H,, = —-———- = —-———-. 10, Aw 04, da, 2a% da, da, C 0 tay The clements of the inverse matrix give the variance of the estimators a*. A complete discussion of this error matrix is given in Section 10.4: suffice it to say here that the usually given expressions (Eqs. (10.43)) for the standard deviation of the estimators (Eqs. (10.41) are the square roots of the diagonal elements of B—! (see Eo. (10.63)). We then obtain yx? nox? — oxi x In casé 0) 4 02 4 -:+ A op, it is M and not R that must be minimized.” Bee Clearly, such calculations are best done using computer programs, ff fact, many packages and self-contained programs that are designed to see handle these kinds of problems are available (both commercially anich igs 4Note that the second of the above eyuations is by no means equal to the first orig ee multiplied by x;. ae 10.3 Estimation of Parameters and Fitting of Data 451 through “shareware”). Io this book, we default to MATLAB (see Appendix B), which is in fact well suited for dealing with problems formu- lated in terms of matrices. For the problem of linear (or, more generally, polynomial) function fitting with equally weighted data ports, MATLAB provides the polyfit utility for exactly this purpose. For more general problems, the reader is referred to ‘other textbooks on the subject of data analysis. For example, the problem of linear fitting with unequally weighted data,points is discussed in Chapter 5’of Nume- rical Methods for Physics, 2nd ed., by Alejandro Garcia (Prentice-Hall, Englewood Cliffs, NJ, 2000). A program linreg for this task, is described and the code is available online from the publisher as a MATLAB m-file, as well as in the languages C++ and FORTRAN. 10.3.4. Goodness of Fit; the x~ Distribution We have seen how the least-squares method, as a consequence of the prin- ciple of maximum likelihood, may be used to fit a curve to a set of data. Once the curve has been found, however, the necessity to ascertain quao- titatively how good the fil is arises. This is important especially if the functional dependence is not known, a poor fit might indicate the neces- sity for fitting with a curve of higher order, or a poor fit might indicate inconsistencies in the data. Similarly, we may wish to test whether a certain hypothesis is supported by the data, in which case the goodness of the fit may establish the level of confidence with which the hypothesis should he accepted. Let us first suppose that we know the tre functional relationship of y to x, that is, yx) = f(x); we may then form the least-squares sum i __3 2 M = yb yee (10.38) The range of Mis 0 < MM < +00 but we would be surprised if MI = 0 and would be equally surprised if © was extremely large. Thus we have already a quantitative indication as to how well the data fit the known (or assumed) curve y = f(x). if a new set of data pertaining to the same experimental situation is obtained, and Eq. (10.38) is again formed, a new value M will result. : Clearly, if enough such measurements are repeated, each time yielding : a value for {, we will obtain the frequency function for M. Once the 452 10 Elaments from the Theory of Statistics frequency function is known, it is then easy to tell what the probability of: obtaining a specific is. We may, for example, calculate that in 95% of the cases MU < Mp; if then a specific set of data yields M; > Mo, we know : that such data should be obtained only in 5% of the experiments and can : therefore be rejected. of Obtaining the frequency function for the least-squares sum in this way - is obviously impractical. Nevertheless, it is true that the distribution of.’ ‘M is independent of the curve y = f(x) and of 9;, and can therefore be: calculated theoretically; it depends only on the number n of points that are:: compared, and is called the x7 distribution (pronounced “chi-sqnared”) = M/2)- | exp(—M/2) 2/20 (y/2) where v is the number of “degrees of freedom” of M. In the present case: we set SO) dM = dM = fx2%dx?, (10. 44) : v=n because this is the number of truly independent points being compared. In Eg. (10. 44) T(x) is the “gamma function,” which for positive integer: arguments” is simply a Tin} = (nx — De Consider next that y = f(x) is not known, but that a two-parameter curve is fitted to n data points, yielding estimators a* and b*. Then one::: forms again the least-squares sum M using » = f(x; a*, b*) but now the:: frequency function for the © values is given by Eq. (10.44) with the n: degrees of freedom reduced by the number of estimators obtained from the: data, that is, 7 vo=n—2. The x? distribution may also be used for comparing the frequency of: occurrence of aclass of events with the theoretical frequency (function). Let:: us consider, for example, 100 measurements of a radioactive sample, and: divide the sample into seven classes, with mean value N = 85 counts/mity *The general definition of the gamma function is oo ['{z) =f peal exp{—z) di: 0 for more details see any text on advanced calculus, 10.3 Estimation of Parameters and Fitting of Data 453 TABLE 10.4 Observed and Expected Frequenciés of the Results of 100 Measurements of a Radipacive Sample Class 0-75 75-79 79-83 83-87 87-91 91-95 %5-co Cannts/min O; 15 133 15 15 18 12 14 Observed freq ej 13 12 15 16 16 13 [5 Expected freq (ej —9;)*/e? 0.307 0.083 OG 0.062 0.25 0.077 0.067 x? a and approximately equal expected frequencies; the resulting frequency of the experimental observations o; in each class is given in Table 10.4. Next we obtain from the data the estimators for the parameters of a Gaussian () p* = N, 2) 0* = VN, and (3) the overall normalization, namely, > 0; = >_ e;; thus the degrees of freedom of y7 are four, corresponding to seven classes less three estimators. From the Gaussian distribution we calculate the expected frequencies e; for each class; they are also given in Table 10.4. In complete analogy with the least-squares sum, Eq, (10.38), we form the x7 sum i: (e — 04)" r=) > 2 / i=l Note that y* is now a discrete variable, since frequencies of classes are compared; however, Eq. (10.44), which holds for a continuously van- able y?, is valid provided the number of classes n > 5 and the expected frequencies ¢; > 4. For this experiment we obtain x? = 0.846, and we explained before that vy = 4. From a table of the x? distribution we find that in 93% of the cases the y* distribution would be larger than the result obtained here. Thus one may suspect that the data are “too good” a fit to the estimated Gaussian. The x? distribution of Eq. (10.44) for different degrees of freedom is shown in Fig. 10.7. Tables of this distribution may be found in refer- ence manuals, or easily calculated in any number of computer programs. It should not be surprising that when the number of degrees of freedom 454 10 Elements from the Theory of Statistics FIGURE 16.7 The frequency function for the distribution of x, for different degrees of freedom. All curves afc normalized to the same unit area. Note that for large v the x? distribution approaches a Gaussian. increases p > 30, the x* distribution approaches a Gaussian® with mean po=v— 1/2. 10.4. ERRORS AND THEIR PROPAGATION 10.4.1. Introduction When we perform a measurement of a physical quantity x, it can be expected that the result obtained, x,, will differ from x; this difference is the error of the measurement and consists of a systematic and a randem contribution. Suppose, now, that the measurement is repeated under the: same conditions n times; then the results x, will be distributed (in most cases) nonmally aboul a mean x with a standard deviation o. The difference. betwcen x and the true value x is then the systematic error, and the standard: deviation o of the Gaussian is a measure of the dispersion of the results due to the random error, - The object of the measurement, however, is the dctermination of the unknown true value x; since this is not possible, we scek to find whether x lies between certain limits, or whether the true value x is distributed hr is Teally the distribution of J2yx* that approaches the Gaussian with mean we /(2v — 1) and unit standard deviation (R, A. Fisher’s approximation). 10.4 Errors and Their Propagation 455 about some mean x* with a standard deviation o*. Note that im a rig- orous sense, this statement is mcorrect, since the unknown true value x is not distributed, but is fixed; what we mean is that the probabiliry, x = x*,x > x", etc., is given by the nommal frequency function with mean x and o = j22, the second moment of the measured data about their mean x. Thus, by repeating the measurement several times, it is possible in prin- ciple to circumvent the random errors because (a) a knowledge*of x and ¢ contains all possible mformation about the unknown true value x, and (b) as n increases, the second moment should decrease as 1/./n and may be made arbitrarily small. On the other hand, the systematic errors can- not be extracted from a set of identical measurements. They can either be estimated by the observer or be judged from a performance of the same measurement with a different technique. Therefore, it is unadvisabie to reduce the random errors much below the expected limits of the systematic errors. In what follows we will discuss only the treatment of random errors and work under the assumption that the results of the measurements follow a normal distmbution. Until now we have considered the simple case where the unknown value x is directly measured and an error o, can be associated with the Measurement; that is, the frequency function of x depends only on one variable: fix) = qx | G2) 4 /2xa;y PI 3 a Most frequently, however, the unknown value x is not directly measured, and we distinguish two cases: (a) x 1s an explicit function of the quantities y;, yo,..., ¥, that are measured and have with them associated errors o|, 79, ..., 0,. Namely, x = (1, Yas +s Ynds (10.45) and it is desired to find the estimator x™ and its standard deviation o,. (b) x is an implicit function of other unknown variables #1, 42,.-., Um, and of the quantities 1, ya,..-, ¥, that are measured and have with them associated errors a], 02,...,0,,- Namely, Ox, e1, H2,..., 4p; ¥is ¥2,-+-. ¥n) = 0, (10.46) 455 10 Elements from the Theory of Statistics and it is desired to find the estimators x*, uf, u3,..., 45, and the symmetric error matrix ojj(i, } = 1,...,# + 1). Such an example was treated in Section 10.3.3, and we know that at least m + 1 sets of measurcments are required to obtain the mm + | cstimators. The techniques for obtaining the best estimators were discussed in Section 10,3. In this section we will discuss how the random error of x may be determined from knowledge of the errors of the independent vari- ables y,; this procedure is frequently referred to as the combination or the propagation of the errors of the measured values y,,. 10.4.2. Propagation of Errors Let us first assume x to be an explicit function of the measured y, as discussed previously (Section (10.4.1)): x= (1, Yase-65 Yn) (10.45) By applying the maximum likelihood method, it can be shown that the estimator x* is obtained by using the mean values, ji,, of the measured y, (provided! the y,, are distributed normally). Here the mean values ji, are obtained from r different measurements Im. | yn = r DOny. f=] Thus x" = (v1, Y2,-++5 Va) = P41, 9, 12-5 Ma). (10.47) Next we make a Taylor expansion of Eq. (10.45) about x*, through first order a X= OCU, Ha, ---5 Ha) + | (4) — ya) Yidu oe] yy) bees Fa _ +|5 we yo) + + Dyn fe Yn)- ? Clearly if x is variable, all measurements yt, are made so as to correspond to the same point x. hae i es ee are) ae ee ee a ee vaaa er ra) 10.4 Errors and Their Propagation 457 where [d¢/dy,], means evaluation of the derivative at the point about which we expand—that is, (447, 442,.-., in). We can now form the sec- ond moment of the distribution of the x! values as they result from the observed y,' values. The superscript i here refers to the r different sets of measurements: $50 r BBB) Hiei i=] ad d@\ le (2) (G8) Ese ls t= (5°)? Hla jato tar) (ge) eat” (10.48) Equation (10.48) is the most general expression for the propagation of errors. [f we assume that the errors are uncorrelated, namely, 01; = 0 when i 4 J, we can obtain the results for the simplest functional relationships: (a) Addition X=yl t+ yate- + Yn oy = op tazt---+a2, (10.49) (b) Subtraction x = yl) — y2 _ 2 2 Oy =o, + os. (10.50) (c) Multiplication X= Vt * V2 X--- XK Yn 458 10 Elements from the Theory of Statistics (sr) ay, = 2 X+-* Mn Yisp oF X (Mass fn)? +++ +2 x (uipea-++)* = (10.51) 2 2 2 o O OC, (=) +(2) peng (2). HY H2 Ly (d) Division yi x= — y2 (#) mS (==) sae (10.52): dy iu 2 dy. 7 (42)? | | oe ad (yt1)? a ) ; 2 Oy = ,{—s + —— = x* J (—} +1 —]. (0.53) ee : (2)? (424 (= & ae From the above examples we see that in general the errors are combined in quadrature—that is, it is their squares that are added. Consequently, if the © error in one of the variables o; is large, it will dominate all other terms and the error of x, 0, will be almost equal to o;, despite good measurements made on the other independent variables. Our simple rule for the case of addition, Eq. (10.49), may be used to obtain in a different way the result derived in Eq. (10.34). Let a variable x be measured and let the mean of a set of measurements be x;, with a standard deviation o;; if this set of measurements is repeated under identical conditions, anew mean result x; 3 x; will be obtained, but let the standard deviations be equal, that is, 0; = o;. If n such sets of measurements are performed, the new estimator for x will be : oe es x* = mace + Xo ++++Xn), a¢) Ge Tn and thus 10.4 Errors and Their Propagation 459 Hence, from Eq. (10.48) or (10.49), (=) fcc (=) =yraaae (1054) Namely, the standard deviation of the mean of n measurements of a Gaussian distribution is o/./n, where o is the standard deviation of the individual measurements. 10.4.3. Example of Calculation of Error Propagation As an example, let us consider an experiment to determine Stefan’s constant &, from the relation E = bT", where the following values of E and T were obtained with the indicated standard deviations: T (K) E (Wim?) 800(1 £0.02) (3.0 + 0.3) x 104 1000(1 + 0.02) (8.00.8) x 104 1200(1 + 0.02} (15.6 £0.6) x 104 We wish to calculate the estimator b* and its standard deviation oy. There are two ways to proceed in this case. We either may calculate b* from each of the three sets of measurements and then combine these values to obtain b* = b*, but weighing each b* according to its standard deviation, or we may use least squares in the observed variables E and T+, Note that a mean of T or E of the three listed measurements makes no sense whatsoever since each measurement is made for a different T. We will follow the first procedure, and we first obtain the error on T* from the known error on 7. For this we should use the general expression, Eq. (10.48), but since @ = T* is a function of only one variable,® simple differentiation gives the desired result directly —=4T3 = =4—., (10.55) 81f we choose to write @ = T x T x T x T, we may not apply Eg. (10.51), since these variables are correlated; use of Eq. (10.48) and apy = a7 gives back the result of Eg. (10.55). 460 10 Elements from the Theory of Statistics TABLE 10.5 An Example of a Calculation of Propagation of Errors Set of data T* E/T*= bt a(T*)/T4 o(b;)/b% 1 0.41 x 10/2 7.3 x 1078 0.08 0.13 2 1.0 x 19!2 8.0 x 1078 0.08 0.13 3 2.0 x 19! 78x 10-8 0.04 0.06 We note from Eq. (10.54) that it is easier to work with relative errors, and . we thus fonn Table !0.5, where un o(b) _ er] + [22] b T* E since the errors in 7 and E are uncorrelated. Bbees For the best estimator of b, we will use the mean of the three measure- 2455 ments but weighed in inverse proportion to the square of their standard : deviation (see Section 10.3.3). Thus : - 4 b= (7.348.044 x78) x 1078 = 7.75 x 1078; for o (b} we used Eg. (10.49), a(b) = av a*(b1) + obo) + 404(b3) or the convenient approximation a(b) 1 [Tetp) , fobs] won) = 2 | +| . +4] 2) 0.08, so that the final result is b* = 7.75(1 + 0.043) x 1078 W/°K*-m?. mnie My Sue MSE ROR 10.4.4. Evaluation of the Error Matrix In the two previous sections we have discussed the case where only 22% one unknown variable x was sought. We will now consider the random 224: 10.4 Errors and Their Propagation 461 errors when several unknown variables are stmultaneously estimated or measured. When only one variable is measured, we know how to obtain from the data the second moment about the mean 1 A 2 = 2 aq” = — x —x;)”. Lk Hi i=] + If now p variables are simultaneously measured in an experiment, we must form the p( p + 1)/2 second moments about the mean; for example, if we measure x, y, and z, we must calculate the six expressions le. 7 en = DE MDG — ai) Tyy =e Tez =" Loa. 7 Oxy = ae — x1)(9 — yi) = yx; (10.56) i= Oy, S++ ' = Ozy Tyz = + = Ozy. (In this notation, the dimensionality of a quantity o pg is that of the product pq. Hence, o2 has the same dimensions as o,,. We avoid the notation a2, etc., because it misleads one to think that o',,, for example, 1s positive definite.) If the distribution of the variables x, y, and z is normal, then these six moments form the symmetric error matrix; if the variables are uncorrelated, the matrix is diagonal. Clearly, the error matrix must be known if it is desired to apply Kq. (10.48). Consider, for example, that from the measured variables x, y, and z we wish to obtain a new unknown u and its standard deviations o (1), where Hu = O(x, y, 2). (10.37) Then the values of a? that were obtained from the data with the help of Eq. (10.56) are substituted in Eq. (10.48) along with the partial derivatives of uz, which are obtained from Eq. (10.57). Conversely, if the frequency function of the three variables x, y, and z, and thus of wu, is known, flu) = flbtx, y. x9) 2 eee 452 10 Elements from the Theory of Statistics ae it is possible to calculate theoretically the elements of the error matrix 22 through the usual expression ee Pie p(t, y) = II f&, y,axydx dy dz (10.58) of H2(%, y= fff FO, YY, (its — x) ty — y) dx dy dz, where Oxy = o2(X, y), ete. In most practical applications, however, it is difficult to use Eq. (10.56) or (10.58). Equation (10.56) may not be usable because the unknown vari- ables may not be measuted directly (although they are measuredimplicitly); also, extensive data are required to yield meaningful results, and the cal- culation is cumbersome. Equation (10.58) may not be usable becausc the 22 multidimensional integrals are frequently too difficult to calculate. Instead, %& the method of maximum likelihood provides an easy way for obtaining the = = CITor malrix. As already discussed in Section 10.3, if the set of data xj, yj,..., 2% has been measured, and the estimators for the » unknown variables G,,05,.-., Om aré sought, we may fonm the likelihood function wae Ce Be ee i eae Ls te ae or ad A eo ee ee ae eat et td See ee a ad a POPU a tat a Se Li(1, X25 0004 Xn, Vis Vos ees Vane eee 210225 ees Zn Go, Gh, v0, An) = FURY 1s 0005 215 Fas Obs 06s Om) F (x2, V2, 06252050, Gby ees Onde % fF Ons Vay ses Zits Gas Gh «+s Onn) where f is the frequency fonction of the measured variables and is usually assumed to be a product of Gaussians. Then the estimators @7, re are given by the values that simudtaneously maximize ©, namely, at, at al more | = 0, (10.59) a8, 6% .6%,....0% 80m 6*.6F requiring the solution of m coupled equations. Equation (10.41) is a simple example of such a solution of Eq. (10.59). We note that the number of independent data points taken, n, must be larger than or equal to mm. 10.4 Errors and Their Propagation 4683 The elements of the error matrix can be obtained from the inverse of the matrix ew Hi; = od , (10.60) JG; a8 ét, oF es where the second-order partial derivatives must be calculated at the values of the estimators, and W = log ©. We have —i oki = (H),, ’ where the rule for matrix inversion is pases 1 Det ( jé minor of H) Det H and the minor ts the matrix resulting from AH when the jth row and ith column are removed; obviously, the inverse matrix does not exist unless Det H # 0. We will now apply this method of obtaining the error matrix to the simple example treated in Section 10.3.3. The measured variables are x and y, and estimators are sought for the variables a and 6; we assume that x is known exactly and that y is distributed normally for each measurement, and related to x through (H™');; = (-1**/ (10.61) y =f + bx. Using Eq. (10.37), we have 7. el | 1 exp { —-—= [Wi ~ YOY; 2, ort | It oO 2a? and = log = — log(2m) — Yoga _t as pene i O; . To simplify the calculations we assume o) = a2 = --- = Gy, SO that a2 Ww r a2 Ww _ Lei acw _ yx? da2 a?’ “9aab at bbe gt Hence | |# > xi H=G *. x =) (10.62) 464 140 Elements from the Theory of Statistics and Thus = a? > (37) — Di avady—-(> xi)” — px n which gives the results stated in Eq. (10.43); the indices v, stand for. a or b, : un = (10.63) | 10.4.5. The Monte Carlo Method It is clear that the calculation of the propagation of errors may become extremely involved, especially when the frequency functions of the van- : ables cannot be expressed analytically and when intermediate processes of | a statistical nature take place. It is then preferable to use computer programs “2225 based on the so-called “Monte Carlo” method. ee By this technique, we follow a particular event through the sequence = of processes it may undergo. For each process, all possible outcomes are . weighed according to the frequency function and divided into x classes of equal probability. Then, from a table of these classes, one class is selected at random: for example, by looking up a table of x random numbers. The outcome of this proccss is incorporated in the progress of the event until a new decision point is reached, when again random selection is made. Thus, at the end of the sequence of all processes, certain final con- ditions will be reached from the initial conditions with which we started and through the intermediary of the random choices made at each decision point. We follow in this fashion several events, always starting with the same -: inital conditions, but because of the random choices, the final conditions © : will be spread over some range. If enough events have been followed =: through, we are able to find the frequency function of the combined process ~ i and of its parameters, namely, the mean and the standard deviation for the . = final conditions that result from a given set of initial conditions. : For more discussion, including examples with accompanying com- chapter. 10.5 The Statistics of Nuclear Counting 465 10.5. THE STATISTICS OF NUCLEAR COUNTING In many experiments related to nuclear physics, we count the particles or photons emitted in the decay of a nucleus. Usually only a very small fraction of the total sample undergoes such decay. The decay of one nucleus is a completely random phenomenon, yet from the number of counts in a given time interval, we may determine the decay probability of this species of nuclei or unstable particles. We have already made use of these’concepts in Chapters 8 and 9, 10.5.1. The Frequency Function for the Number of Decays We start with the assurnption that the decay of one nucleus is purely ran- dom and the probability (unnormalized) for decay in a time interval At is proportional to Af and some constant A with dimensions of inverse time’: Da = AAT. (10.64) If we have a sample of N nuclei, since the presence of one nucleus does not affect the decay of another, the probability that one nucleus out of the sample of N nuclei will decay, in time Ar, is P(l, At) = ANAT. (10.65) Equation (10.65) is completely analogous to Eq. (10.12) of Section 10.2.6, which leads to the Poisson distribution; the only difference is that the product Nt of Eq. (10.65) is the equivalent of the number of trials n of Eq. (10.12). Consequently the probability (frequency function) for obtaining n decays in a time interval ¢ is ew ANt ON sy" n! Pin, th= (10.66) The first moment of Eq. (10.66) (in the discrete unknown variable n), as we know from Ea. (10.16), is A=ANt?. (10.67) °K. Schweidler, 1905; this assumption bas been proven absolutely correct from the agreement of experiment with the deductions following from Eq. (10.64) as developed in the following paragraphs. 466 10 Elements from the Theory of Statistics Since n/t is the average number of decays per unit time (the average decay 4 ratc), we find the physical significance of the constant parameter A. That -: is, VA gives the average decay rate of the sample; N is the total number of : nucle: in the sample. Similarly, the second moment about the mean of Eq. (10.66), as we . know from Eq. (10.17), is oF =ANt=7. Hence the very frequently used expression, a= Jn. (10.68) Note, however, that n/t = NA is, the theoretical average rate, which is usually unknown (unless 4 and N are precisciy known for the sample under consideration). The average rate that we measure, R = n/t (counts 228 per unit time), will, in general, differ from the true rate NA = n/t, but “2228 if n is large, R will be distributed normally about NA, (See Eq. (10.66a) below.) From the considerations of Section 10.2.9, it is clear that when the total | number of observed counts 7 is large, Eq. (10.66) is well approximated by | a Gaussian with mean yp = Naz and standard deviation o = / NAr: (Nat — ny? P(a,t) = -——- = 10.664 (1, t) » 2m Nit 2NAL ( | I (A — a = exp | -————_ ] . 10.66b XP an ( ) Thus, unless we are dealing with very few counts, Gaussian statistics may be safely applicd. Finally, we summarize here some simple consequences of Eq. (10.64) (or a single nucleus: (a) Uf the probability for decay in di is pa{dt) = dAdt, (b) then the probability for not decaying (survival) in the time interval fromt =Qtor =/fis ‘ p(t) = e7* | (lor proof see Eq. (10.13)). rr ee eee rr eee a ee a ey Pre ee ee ae a Par ae Peary ar al ea nie a oo a] rr ee 10.5 The Statistics of Nuclear Counting 467 (c) The probability for decay in df at time f£ is pa(t,dt)=e™')dt. (d) The probability for decay im the ume interval from ¢ = Otof = 1 is pat) =1—p,(t)=t—-e™. Note that only (c) is properly normalized, so that on Oo [ pg{t)dt = { e“adt=I. 0 0 Expressions (b) and (d) are, correctly, always <1 and reduce to Q and 1, respectively, as ¢ approaches infinity. As to expression (a), we must keep in mind that it holds only for Af such thatAArt < 1, 10.5.2. Behavior of Large Samples Having obtained the frequency functions, we may flow examine the behav- ior of the total sample.!° Fram Eq. (10.67) we see that given a sample of N nuclei, on the average, in a time interval At there will be n=ANAt decays; that is, the total sample will be decreased by an amount —AN = N2A?. (10.69) Equation (10.69) then leads to the differential equation for the number of nuclei in the sample dN — = —Ad! N with solution N(t) = Noe™, (10.70) where No is the number of nuclei at time ¢ = 0. Frequently r = 1/2 is used for the exponent in Eq, (10.70); 1 is called the /ifetime of that particular species of nuclei and is the time in whicb the population of the sample [s OT he pnnciples and formulas in this section have already been used in Section 8.6. 468 10 Elements from the Theory af Statistics reduced to 37% (1/e) of its original value. The haif-life 1 T1j2 =T in | = 0.693r gives the time in which the population of the sample is reduced to half its original value. Using Eq. (10.70) we find, for the decay rate as a function of time, that dN 7 R(t) = -AN(t) = —-ANoe*, (10.71) © {) | which has the same time dependence as Eg. (10.70). Experimentally we - usually measure R(t) and obtain acurve as shown in Fig. (10.8); fromsucha ; plot A may be obtained. If the sample contains two or more different species | of nuclei with different decay constants 4;,A2,..., the time dependence - of the decay rate is no longer the simple exponential of Eq. (10.71); instead - aN ar = RO) = —ANge “Al — AgNge* —--. If, however, A] > Ao, then for small ¢ (that is, t ~ 1/41) R(Z) is dominated : by the first term; for large t (for example, ¢ ~ !/42), R(t) is dominated by | 0.75 0.50 Relative activity W/Ny Te T 2a Itz Aria Elapsed time FIGURE 10.8 Exponential decay of a sample of radioactive nuclei. The abscissa is F calibrated in units of the half-life of the sample; the lifetime is also indicated. . 10.5 The Statistics of Nuclear Counting 469 Time (hr) FIGURE 10.9 The decay curve for a sample containing two species of radioactive nuclei, each decaying with a different lifetime, Note that the composite decay curve a is the sum of curves 5 and c. the second term. This 1s shown im Fig. 10.9, which gives the decay curves on 4 semilogarithmic plot. See also Section 8.6.3, im particular Fig. 3.37. Another situation of interest arises when nuclei of species A decay into species B with a constant 4,4; nuclei B, however, decay in turn into species C with a constant Ag. Let, at ime f = 0, the number of nuclei of species A be No and that of species B be 0. Then the number of nuclei of species A as a function of time is still given by Eq. (10.70), N4 = Noe *4!. However, for the number of nuclei of species B, the following differential equation holds: ans atiaNa —ApNg. The solution of this first-order linear differential equation is straight- forward, and with the initial conditon Ng(i = 0) = 0, we have dA —\ —A Ne = Noy [eat — en *8t] (10.72) 10 Elements from the Thaary of Statistics e that Eq. (10.72) always gives Nz > 0, as it must be, irrespective © vhether 44 > Ap orAg > Ag. Equation (10.72) correctly reduces to | = 0 for = Oandt = ow. The two limiting cases for the decay rate - n B ta C can also be obtained from Eq. (10.72) if we take into account - ’Raec(t) = Npap. Thos for rg > Aa Rractt) = Nokae forka Ag Reclt) * Nodge *#E6° 5.3. Testing of the Distribution of Radioactive Decay; the Distribution of the Time Intervals between Counts s frequently desirable to test whether a sample of counting data does =: eed come trom the decay of radioactive nuclei, that is, thatitfollows the quency function of Eq. (10.66). A very sensitive test can be devised if plot the distribution of the time intervals between successive decays, or ~ sry second, third, etc., decay. This method was applied to the distribution the arrival times of cosmic rays in Section 9.4.2, First we obtain the distribution of the time intervals between two succcs- e decays. Let ¢ = 0 when a decay occurs; we then seek the probability itno decay occurs until ¢ = , but a decay occurs within dt at = t. This ybability is piven by Eq. (10.66) with a = 0, multiplied by Eq. (10.65); nely, P(t, dt) = q(t) dt =e" Nidt. (10.73) uation (10.73) indicates that the shortest time intervals between two ints are much more frequent than the longer ones; this is true for any rdom events, since they obey Eq. (10.64) and is shown in Fig. 9.22. Next we consider the distribution of the time intervals between every ‘ond, third, etc., #:th count. In practice this arises when the counts from Output of a “scaling circuit” are recorded, Consider, therefore, a circuit ing one oulput count for every ym input count. If the true rate is r, then Output rate #& is related to r by 'Compare this equation with the probability for the decay of a single nucleus, a given | 2 ection 10.5,1(c). tee a ‘= e SRE ASRS ENS ANSE TEE SA NtCert AES Se nS EN SESS > =e ee Le ee 10.5 Tha Statistics of Nuclear Counting 471 = 0 when an output pulse arrives, and let Q,,(f) be the probability other output pulse arnives in the time interval ft, gp,(t) dr will then é probability that this other output pulse amives at? (between ¢ and t). other output pulse will arrive if the input counts n > m, so that Oo co _ (rt ne rr Ont) = S> Pint) = 5s PE . n> n> Ht me n=m—l (rt)"e =]- » ——. (10.74) fe the last equality follows from the normalization of Eq. (10.66) oO > Pa,i)=1. n=O by considering the sample space of Fig. 10.10 we see that the set of S$ On(t) is a subset of O,,(¢ + dt), so that any sample-space point ging to O,,(f + dt) but not to @,,(f) represents an output count een f and? + dt. Thus Gm(t) dt = Qm{t + dt) ~— Qn{t) nk oer at) tE 10.10 Sample space indicaring the domain Q,,(1), which contains all paints )ooding to the arrival of an output count in the time interval from O to ¢ after the us copnt, This domain forms a subset of Q(t + dt), which contains all points yonding to the arrival of the output count in the time interval from 0 to ¢ + di. The of the ontpat count alt is ga (2} = Om(t +d —- Om(r). 472. 10 Elements from the Theory of Statistics or 2 Om (t) Gn{t) = it Taking the derivative of Eq. (10.74) a=nt—l1 _ y rni(rty’ te! (rt Pe! = n! ni n=rn— 1] —_ i=ri— _, 3 ae rr vs (rt) te —r! — _ (nT ~ 7 By replacing in the second sum n by / = nm — 1, we see that only the last . term of the first sum survives, so that (rty"-1e-74 2 tj} r-————_.. 10.75) Une G@m(t}=r (m — 1)! ( J : ; Equation (10.75) correctly reduces to Eq. (10.73) form = 1 (sincer = * NA). For me > 2, Eq. (10.75) has a maximum at dg, (t)/dt = 5 [r2(m _ Lerty" 7a") _ (rztrt) te 7 —Q. Hence t = Gn — 1)/r and for large m, f — m/r = 1/R. Thus we see that the most probable time interval is not the shortest one, but instead approa- ches the mean time interval between oufpui counts 1/; that is, the scaling circuit regularizes the counts. Equation (10.75) is shown in Fig. 10.11 for Frnfé} nh Go ay FIGURE 10.11 The probability g,,(¢) that the mth count will follow any original count at the time interval ¢, Note that the abscissa is calibrated in units of rt where r is the: unscaled rate of events; for nz large the curves approach a Gaussian with mean {rf} = me: or (4) = mfr, sites AA Saat Daa os SNAG 10.6 Refarances..:.. 473 different values of m. Comparison of these curves with experimental data has been presented in Section 9.4.2. 10.6. REFERENCES There are many texts, both elementary and advanced, on the subject of statistics, data fitting, treatment of errors, and computational modeling. The teferences given below were consulted for the preparation of this chapter. L. Lyons, A Practical Guide to Data Analysis for Physical Science Students, Cambridge Univ, Press, Cambridge, UK, 1994, A succint guide with plenty of examples. J. R. Taylor, An infroduction to Error Analysis, secood ed,, University Science Books, Sausalito, CA, 1997. A thorough treatment with applications to the physical sciences. B. P, Roe, Probability and Statistics in Experimental Physics, Springer-Verlag, Berlin, 1992. A slightly more advanced and mathematical text. P. G Hoel, Introduction 9 Mathematical Sratisitcs, Wiley, New York, 1958. The presentation of Sections 10.1 and 10.2 follows Hoel closely. A. L. Gareia, Numerical Methods for Physics, second ed., Prentice-Hall, Englewood Cliffs, NJ. 2000. A genera] text including chapters on data analysis and Monte Carlo techniques, with plenty of coding examples in MATLAB, FORTRAN, and C++. H. Gould and J. Tobochnik, An fniroduction to Computer Simulation Methods: Applications ta Pitysical Systems, secood ed., Addison-Wesley, Reading, MA, 1996. A text devoted to simulations, with extensive use of Monte Carlo methods, with programming examples in BASIC, FORTRAN, C, and PASCAL. APPENDIX A Students We gratefully acknowledge the many students who have contributed the data used to illustrate these experiments. Students from the University of Rochester: « R. Annstrong, class of 1994 e W. Lama, class of 1966 ¢ D. Boyd, class of 1963 e T. Londergan, class of 1965 * C. Border, class of 1994 e E. May, class of 1962 * M. Dobbins, class of 1994 e S. McColl, class of 1962 © R. Dockerty, class of 1962 e T. Middleton, class of 1994 « P. D’Onomo, class of 1962 « R. Nebel, class of 1962 ® K. Douglass, class of 1964 ¢ P. Nichols, class of 1963 e E. Glover, class of 1961 e D. Owen, class of 1963 « R. Harris, class of 1963 ® D. Peters, class of 1962 ¢ EF. Holroyd, class of 1966 « S§. Pieper, class of 1965 « M. Klein, class of 1962 e W. Rakreungdet, class of 2000 ¢ D. Kohler, class of 1962 e J. Reed, class of 1961 474 476 A Students « A. Rosen, class of 1962 e M. Thomas, class of 1994 ¢ T, Safford, class of 1994 e J. Traer, class of 1994 e D. Sawyer, class of 1963 e T. Wagner, class of 1961 « P. Schreiber, class of 1962 e T. Walters, class of 1962 e D. Stanchfield, class of 1995 e J. $. Weaver, class of 1962 e D. Statt, class of 1963 « J, Witkowski, class of 2001 ° R. Stevens, class of 1963 e E, Yadlowski, class of 1962 Students from Rensselaer Polytechnic Institute: « Daniel Bentz, class of 1996 e Kristen Ryby, class of 2003 ¢ Jeff Fedison, class of 1994 ¢ Jeffrey Schneider, class of e Adam Grossman, class of 2002 2003 a ¢ Jackie Krajewski, classof 2005 — * Joseph Schreier, class of 2003: e Jane Krenkel, class of 2003 ¢ Peter Thies, class of 1996 : * Katie Newhall, class of 2005 e Tristan Ursell, class of 2003 « John Orrell, class of 1997 ¢ Jeff Wereszczynski, class of e Ryan Quiller, class of 2003 2004 « Herman Riese, class of 2003 « Jeff Yu, class of 1997 ot eae i ae ee car en) a) a aT er seas rer tab ae vee es ae ar Car en) ee a ae ee Pare Par ee a ee ar) a eae en at V2 A Short Guide to MATLAB The experiments described in this book can be analyzed with any of a wide number of computer programs. All that is needed is the ability to sort and plot data, and basic statistical analysis. We have chosen to illustrate the analyses using MATLAB. Although it is a very sophisticated package, a felatively inexpensive student edition that is more than adequate for all of our illustrations is available. This appendix collects some information that should help you navi- gate your way through MATLAB. The MATLAB User’s Guide is a very useful reference, but there is much more in there than you will need for these experiments. Also remember that you can get help online from http://www.mathworks.com, This site includes a long, searchable list of frequently asked questions, and it is likely that yours is among them. This site also offers you access to programs donated by other users, which you can download and use or modify yourself. aT? 478 = B A Short Guide to MATLAB pnt at a fae a’ ate B.1. A MATLAB REVIEW The following is a brief summary of key MATLAB commands and: procedures. Input Medes, Commands can be executed one by one in the command- line mode in MATLAB of you can write a program consisting of the. appropriate command lines in a convenient word processor such as notes: in Windows or emacs on a Unix system, and store it as a file with the “.m” extension such as programname.m. Daia Input. Lists of data points are usually input as one-dimensional: matrices (vectors). You can do this in a command line within MATLAB: | SR ee NS SNS x=[l 23 4 5 6]; y=[0.1 0.2 0.3 0.4); aerate thts (The semicolon at the end of the line is not necessary, but if you do not: include it, then MATLAB will echo values.) You can also store data in: ASCII columas in a file with the “.dat® extension, such as mydata.dat. If the x data are in the first column and the y data are in the second column: of your ASCII file, then you would usc the following commands to load it into your MATLAB session: a aa load mydata.dat x=mydata{:,1)}; yemydata(:,2); Simple Arithmetic. To get an online list of simple functions, type help elfun. Formatting for simple calculations with numbers is straightforward? Addition is a+b, subtraction is a-b, multiplication is a*b, division is alts and raising to a power is a b. Scientific functions include: « abs{x) for absolute value ° round(x) to round to the nearest integer * reallx) to take the real part of a complex number / « sign{x} to find the sign (it returns +1, —1, or 0) < * tog(x) for the natural logarithm e lag10{x) for the logarithm to base 10 © sqrt(x} to find the square root B.1 A MATLAB Review 479 as well as the familiar tigonometric and hyperbolic functions and their inverses, sin(x}, cos(x}, tan(x}, asin(x), acos(x}, atan(x), sinh{x}, cosh{x}, tanh(x), and $0 on. Vector Construction. The easiest way to create a vector with regularly spaced elements is with the command X = (start:increment: last) where start is the first element of a vector, last is the last element, and increment is the step size between the elements. For example, x=(0:0.1:1) creates the vector x = (00.1 0.20.3 0.40.5 0.6 0.7 0.8 0.9 1.0) (The parentheses “{)" are optional, or they could be replaced with brackets “[]’.}) This is also equivalent to using the faction linspace(start/ast.number), where number is the number of entries in the vector. If you would like to define a vector where the increments are loga- rithmic, i.c., separated by a constant factor instead of a constant difference, use logspace(start,last, number}. Array Arithmetic. To get an online list of matix functions, type help elmat, For operations between a scalar and an array, addition, subtraction, multiplication, and division of an array by a scalar Jook just like sunple anthmetic, and the operation applies to every member of the array. For operations between two arrays of the same length, addition, subtrac- tion, multiplication, and division apply on an element-by-element basis, but the syntax for multiplication and division is different than that for sim- ple arithmetic. Multiplication is written a.*b and division is a/b, where a and b are vectors of the same length. (Multiplication and division without the dot correspond to normal mainx muldplication and division.) Data Analysis, There are some simple MATLAB functions for calcu- lating often-used quantities for analyzing a vector x of data values: « length(x) returns the number of elements in the vector ¢ sum(x) adds all the elements in the vector * mean(x) averages all the elements in the vector ¢ std{x) finds the standard deviation of the elements. Note that std(x) is equivalent to sqrt(sum((x-mean(x)).2}{length{x}-1)}. The command [n,x]=hist(y,nb} takes a vector y of data values, calculates a histogram with nb equally spaced bins, and returns vectors n and x, which pive the frequencies and midpoints, respectively, of the binned data. 460 B A Short Guide to MATLAB Least-Squares Fitting. The theory of the least-squares method is'dis- 2 cussed, along with reference to MATLAB, in Section 10.3.3. 5 When the data points are equally weighted, all of the operations nec- =: essary to fit a polynomial to a set of (x,y) data points are included in the ~ command p=polyfit{x,y,m), where m is the order of the polynomial. A fitto -: a straight line is therefore p=polyfit(x,y,1). The vector p holds the best-fit °: values in order of decreasing polynomial order. For example, if m=2, then =: you are fitting to a quadratic function ax* + bx + ¢ and polyfit returns p=ia,,cl]. The values of the fitted function can be computed for a set of x values x1 using the command yl=polyval(p,x1). (if you want to compute the fitted function at the data points, just use something like yfit=polyval(p,x).) If the data points are not equally weighted, then you can use Garcia’s function linreg to fit to a line. Note that you can retrieve this code from the MATLAB Web site. Nenlinear Least-Squares Fitting. If you cannot express the function you want to fit as a polynomial, then you cannot use poilyfit or linreg. ff the function is still linear in the fitting parameters, though, you can use matrix - techniques to solve the equations. However, it may be simpler just to resort .°: to numerical techniques to minimize x? directly. You are forced into this - : situation if the function is nonlinear in the fitting parameters anyway. For «: example, if you want to fit some decay data to y = Ae~*/4, then youcan = instead fit a straight line to log y = log A—x/A, but ifthereisabackground = term, as in y = Ae—*/* + B, then you must use numerical techniques. Defining the x* function in MATLAB is quite straightforward, and there |= is a MATLAB function called fminsearch, which will do all the hard work = of finding the values of the parameters that minimize the x* function. (See, Z for example, Section 8.6.2.) cs Simple Plots. There are several simple variations on the plot command Ee 2 that will give you everything you need for these experiments. If you really 22: want to do more, see the next section of this appendix. * glot(y) plots the column values of y versus index. It autoscales the axes. Points ane connected by solid lines. ° plot(x,y) plots vector y (vertical) versus vector x (horizontal!) on an autoscaled plot. Points are connected by solid lines. ¢ plot(x,y, finetype’) allows you to specify the type of line that connects the points of the type of symbol that is printed on a data point. for “linetype” use “-,” “:,” “- -,” or “-.” for solid, dotted, B.2 Making Fancy Plots in MATLAB 481 ft 44 dé aa as 14 be dashed, or dot-dash lines, respectively, or use “.,” “o,” “x,” “+, or “*” for the corresponding plot symbol. © barly) draws a bar graph of the elements of y versus index. « bar(x,y) draws a bar graph of y at the locations specified by vector X. » stairs(y) and stairs(x,y) draw “stairstep” histogram plots. You can plot more than one set of data, or data and a fit, hy specify- ing more than one set of vectors in plot. For example, plot(x,y,'0',x,yfit,~'} plots “data” vector y versus xX as little circles, and then overplots the “fir” vector yfit as a solid line through the points. Another way to over- lay plots is to hold a plot and then just repeat the plot command with new vectors. When you are finished collecting overlays, use the command hoid off. Simple labels are put on the graph using the commands xlabel’fabel on the x-axis’) ylabel(‘/abel on the y-axis’) title('title for your plot’) text{x,y, some text’) puts some text at point (x,y) lagandl ‘string?’, 'string2’,...) labels different sets of data added to the same plot To print your plot on the default printer, use print. Printing to files or to other printers will depend on which system you are using to nin MATLAB. Consult the online help or the User’s Manual for details. B.2. MAKING FANCY PLOTS IN MATLAB It is simple to make MATLAB plots with the default gharacteristics. Some- times, however, that is not quite what you want,/especially if you are preparing a formal lab report. You can also, of course, consult the Mathworks Web page help directly for some hints. For example, if you want to know how to add Greek char- acters to your plot, click “Tech Support Solution Search” on the Web page, and search for keywords “Greek AND plot.” You will find “492 Haw can | place Greek characters in my plat?” in the search results list. Clicking on this solution tells you not only how to do it, but also tells you how to get an m-file, which will make a chart for you that shows the mappings for all the various Greek letters and symbols. 492 8B A Short Guide to MATLAB You can dress up plots quite a bit in MATLAB using what is called “handie graphics.” Every plot, and plot element, has a “handle” that you can access in order to change properties of the corresponding element. Most plotting commands retum the value of the handle if you ask for it. For example, h=plot(x,y}; will return a value for the handle h that can be used with the command set for modifying properties of the plot. } Refer to the on-line documentation for more information. Sa ; Laser Safety Laser radiation can be dangerous, and in particular it can result in serious and permanent damage to the eye. Thus it is important to be aware of the hazards involved and to follow the rules for safe use-and operation of lasers. Explicit rules and standards are given in publication ANSI Z136.4-1986 of the American National Standards Institute (1430 Broadway, New York, NY 10018). The damage a laser can cause depends on the level of the emitted power for CW lasers and on a combination of power and energy for pulsed lasers. The energy per unit area is a better measure of the hazard from direct wradiation. The most serious danger, however, from laboratory lasers in the visible and near infrared G.e., Nd: YAG) 1s that they can be focused by the eyeball onto the retina where they will create a permanent blind spot. This is particularly serious for infrared lasers where the beam is invisible. Thus protective IR absorbing glasses (typically of optical density 4) must always be worn in rooms where IR lasers or beams are present. Lasers with power below 1 mW are classified as Class 1 lasers. At this power level the exposure in the time it takes for the eye to “blink,” 483 464 € Laser Safety approximately 0.25 s, is considered safe. The HeNe laser used in this laboratory 1s a Class 1 laser. Still one should never stare directly into the beam, or let a specularly reflected ray enter the eye. No eyeglasses are needed but one must use common sense and remain alert. The lasers installed in commercial scanners to which the public is exposed are Class 1 devices. One advantage of the HeNe is that the beam is clearly visible SO One is aware of stray beams. Stray beams result from reflection off the various optical elements and other smooth surfaces; they should be blocked or minimized. Lasers with more than 1-mW powcr are generally classified as Class 4 devices, as are most pulsed lasers. Nd:YAG and argon-ion Jasers can easily deliver several watts of power. Such lasers wili cause permanent eye damage instantaneously before one is aware of it. In the case of Class 4 lasers only qualified trained personnel can enter the laser room, which must be kept locked with appropriate signs indicatmg laser operation. The nitrogen pulsed laser emits in the ultraviolet at A. = 337 nm. UV is invisible but can be absorbed by plexiglass, so that ordinary safety glasses are not effective; certain materials (i.e., a business card} wilt fuoresce and can be used to locate the beam. Similarly, (R beams are located with special fluorescent cards and/or with IR viewers. The need for obeying safety rules and procedures around lasers is a real one, and not a “bureaucratic whim.” Never look into a laser beam, be aware of the stray beams, and wear glasses when required. Do not lIct others be exposed to your laser, ‘1 a = aa = "a a ‘ 7 . = et ear ee as at, = sisitattrniarsiet SSSERSENSS Pee er ri a ee 2 ea APPENDIX D Radioactivity and Radiation Safety In a series of experiments on quantum physics, the student comes in contact with radioactive sources, either while studying the properties of the nucleus itself or when using the sources to obtain energetic beams of alpha or beta particles or gamma radiation. As is well known, radi- ation can be harmful to humans, and therefore precautions must be taken against undue exposure to il, and in the handling of radioactive materials. In addition to the naturally occurring radioisotopes (which have long lifetimes), a great variety of isotopes have been produced artificially and many of them can be purchased. A convenient table of radioisotopes, many of which, like ©°Co, ?*Na, and '?’Cs, are quite standard for training, testing, and calibration purposes, is available online from the Particle Data Group (PDG) at Lawrence Berkeley National Laboratory: http://pdq.|bl.gov/2000/sourcesrppbook.pdf 485 48 0D Radioactivity and Radiation Safaty The table gives the type and energy of the radiation, as well as the half-life, with separate information for the different decay schemes, of each radio isotope. Much more detailed information is available from the National Nuclear Data Center (NNDC) at Brookhaven National Laboratory. This information includes level and decay schemes, radiations emitted, and thorough documentation on using the various online programs made available to the user: v —t http://www.nnde.bni.gov/nnde/nudat/ In the handling of radioactive materials the following regulations should always be observed: 1. Wear a film badge when using radioisotopes. 2. Refrain from eating and smoking while using radioisotopes. 3. Check haods for activity after completing work with radivisotopes (use the appropriate detector, that is, for alphas, betas, etc.). 4. Use gloves when danger of contamination exists. 5. Use tongs for handling strong samples {but only if you can do so safely). 6. In case of a spill, wash it off immediately. 7. Report all accidents and mishaps connected with radiotsotopes. 8. Do not take radioactive sources out of the laboratory. Radiation is harmful to living organisms because by ionization it desiroys individual cells, and also because it may induce genetic changes. Tt seems established that low levels of radiation do not produce permanent injury, but the effect is assumed to be cumulative. A genetic change, on the other hand, can be produced by low-level radiation as weil as by high-level Tadiation, but it should not be forgotten that human bemgs have always been exposed to cosmic rays and natural radioisotopes. In all estahlishments where some potential radiation hazard might pre- vail there must exist an agency (the health physics group) that is responsible for personnel and area monitoring, and for source custody. The health physics groups keeps a record of radioactive sources and other hazards, and of radiation accidents, and im general helps in the enforcement of safe _ procedures. It should be clear, however, that the sole responsibility for ° enforcement of proper practices rests with the individual who has been granted the privilege to work with a radioactive source. The aversion of {45 many scientists to observe strict rules ts a common phenomenon, but it | must not be imitated by the student. - D Radioactivity and Radiation Safety 48 Two peculiar aspects of harm from radiation need special mention and warming: (a) radiation is neither visible nor painful; hence one may not be aware of having been exposed unless proper detectors are used; and (b) in general it is too late to do anything after one has been exposed. Excluding nuclear reactars and particle accelerators, the most serious radiation hazards come from X-ray machines and from taking intemally a small amount of radioactive material from a source used in a laboratory. The PDG publishes online an excellent summary of the units and con- version factors for radiation and radiation doses, as well as recommended exposure limits and radiation protection procedures: http://pdq.lbl.gov/2000/radiorppbook.pdf Finally, we conclude wit / some remarks about radiation shielding. This is important not only for parsonnel protection, but also to reduce back- grounds in an experiment in Which the primary radiation from a source Is not meant to be detected. The purpose of shielding is to attenuate the radiation beam. If the beam consists of charged particles, they do lose energy as they cross matter, and if the shield is sufficiently thick the beam will be completely stopped. Since the energy loss is proportional to the number of atomic electrons Z of the shielding matenal, low-Z materials have a larger stopping power per (nucleon) gram. On the other hand, tbe higher the density, the higher the stopping power per unit length of shielding. The attenuation of a gamma-ray beam, however, is different; no grad- ual energy loss occurs, but there exists a finite probability (cross section) for an interaction. Interactions (electromagnetic) of a gamma-ray beam with matter are either the photoelectric effect, Compton scattenng, or pair production, depending on the energy of the beam. As explained in detail in Chapter 8 through a series of such processes a fraction of the beam becomes completely absorbed in the material used for shielding. Since the interaction probability is proportional to the amount of material present, we have dl Pan Ik, hence f=Ipe **, where x is the length of the shield, « = 1/£ = a, Npo is the absorption coefficient, and Z is the radiation length (£ = 0.51 cm for lead). 488 0D Radioactivity and Radiation Safety If the beam consists of particles with strong interactions, suchas neutrons -° or protons, the formalism is similar, but now « = 1/4, where 2 is the mean free path, where 4, can be roughly taken as 60 g/cm’. Despite these considerations, still the best shielding against a radioactive source is distance; since the inverse square law holds, keeping ata 10-m “= distance dilutes the flux over the valuc it had at contact with the source, 225 (assuming an extent of 5 cm) by a factor of 40,000; for gamma rays such | 22: attenuation is equivalent to shielding by 7 cm of lead. APPENOIX Ey Optical Detection - Techniques ( , If we are going to do experiments with light, we have to learn how to measure it, There are several propertes of light that can be measured, for example, its intensity, wavelength, or degree of polarization. In this section we discuss ways to measure the intensity, either as energy per unit time or number of photons per unit time. In order to work with intensity quantitatively, we need to convert it to a voltage level that can be recorded or digitized or so on. However, the simplest option, namely photographic film, still lets you distinguish “dark” from “light and has some advantages. We discuss it first. E.1. PHOTOGRAPHIC FILM Photographic film uses light and chemical reactions to record light intensity. It of course has some obvious drawbacks. For example, itis hard 490 E Optica! Detection Techniques to convert this record into a voltage, although film-scanning machines are built for this purpose. Another disadvantage is that it is inconvenient to record large amounts of data this way, unless some fast and efficient scan- ning method is available. On the other hand, film has some preat advantages “: as well. " First of all, film is economical. You can record light intensity“over : quite a large area for very little money, Astronomers, for example, photo- -: eraph large sections of star fields on a single photographic plate, giving an accurate and reliable record, all for only a few dollars (in film) per picture. Secondly, film gives you data that you can easily relate to. Distances between images are true, at least to the extent of your focusing device, and. you can remeasure or check then easily. There can be an abundance of data on a single photograph, and you can always go back to the same picture if. you want to recheck things, | Most importantly, however, film has outstanding position resolution, especially for its price. This resolution is limited by the grain size of the film, and 10 y.m is simple to achieve while 1 im is routine with a little care. What is more, this resolution can be achieved simultaneously over many centimeters of distance. This is almost impossible to achieve with direct electronic means, and can be quite important to astronomers measuring star naps and to optical spectroscopists measuring precise wavelengths. An important trade-off is between resolution and speed. A film like ™: Kodak Tech-Pan can be used routinely for 1-m resolution or smaller, but ‘: it takes a lot of photons to convert a prain. Thus, such a film is limited to cases of rather large light intensity or where you can afford long exposure -:. times. Somewhat faster films, like Kodak Pan-X, are much faster, and still .:: give resolutions perfectly suitable for most applications. K.2. PHOTOMULTIPLIER TUBES The photomultiplier tube (sometime shortened to “photctube” or PMT} ir is probably the oldest device for converting optical photons directly into. electrical signals. It does this with very high efficiency and is very reliable, 2-2 Some can detect single photons and easily distinguish the signal from back- © 425 ground noise. Others are made to measure beams of light. Photomultiplier::: tubes have been in development for more than 50 years, and have evolved into lots of varieties, some of which are quite sophisticated, The basic operation, though, is simple. E.2 Photomultiplier Tubes 491 The photomultiplier tube is based on two:effects, both of which involve the emission of electrons from the surface of materials. The first is the photoelectric effect, where a photop is’ absorbed by an electron on the material surface. The electron then emerges with some small kinetic energy; thus a photon is “converted” intd-an electron. The second effect is that when an electron of some moderate:energy:strikes a surface, a number of electrons are emitted. (This process is called “secondary emiission.”’) Secondary emission is used to multiply::the ‘initial electron into a large number of secondary electrons. All of this takes place on surfaces enclosed within an evacuated glass tube, hence, the name phatomultiplier tube. Aschematic photomultiplier tube is shown in Fig. B.1. The photoelectric effect acts at the front surface, or face, of the PMT, and there one photon is converted into one electron (with a certain efficiency less than 1). There is a potentia! difference of ~ 100-300 V between the face and the first “stage” of strikes the first stage, it emits more eléctfons; which’ are accelerated to the next stage, and so on. These materials that act as stages are called emitters of electrons (i.e., cathodes). After several (usually between 6 and 14) stages, a significant number of electrons emerge in place of the incident photon. Electrical connections are made with the outside world by pins that penetrate the glass envelope on the end. The front window of the PMT is made of glass or some other transparent material. A thin layer of some optically active material is evaporated on the inner surface of the window. This layer, called the photocathode, is sermitransparent and is usually brownish in color. If the tube breaks and air fills the inside, the photocathode oxidizes away ahd the brownish color disappears. In this case, the photornultiplier tube will never work again. Window Glass envelope ( y _ ; Anode out 3 F f i i] Proton Photocathode Connection pins FIGURE E.1 How a photomultiplier tube works. The connection pins are used to supply high voltage to the individual dynodes, and to extract the anode output. eee ee 492 E Optical Detection Techniques A photon incident on the window penetrates it if it can. In fact, glass window tubes become very inefficient in the near UV because photons with wavelengths below 350 nm are quickly absorbed in ordinary glass. Special UV transmitting glass is available on some photornultiplier tubes, and this can extend the range down to 250 nm or so. To get further into the UV, special windows made of quartz or CaF are necessary, and the devices become very expensive. : If the photon penetrates the window, it reaches the photocathode and has: « ®: a chance to eject an electron through the photoelectric effect. Recall that in the photoelectric effect, a photon of energy Av gives rise to an electron of kinetic energy T given by T =hv — ¢, where @ is called the “work function” and represents the energy needed | to remove the electron from the surface. Several different materials are — used for photocathodes, but all are designed to have work functions small enough so that optical photons can eject electrons. It is in fact hard to find materials for which ¢ is less than +2 cV, so photomultipliers become quite — insensitive at the red end of the visible spectrum. The probability that an incident photon ejects an electron from the = photocathode is called the “quantum efficiency” or QE. It is clearly a function of wavelength 4, tending to zero for both A < UV and A => red. lt is also a function of window and photocathode material for the same — reasons. Figure B.2, taken from the Burle photomultiplier tube handbook, shows the “spectral sensitivity” 5 (in mA/W) for various combinations of windows and photocathodes, Manufacturers tend to quote 5 rather than QE since it is closer to what the PMTs actually measure. By shin- ing so much light energy per unit time (P) on the face of the PMT, and | measuring the current (J) of electrons coming off the photocathode, they determine s= i _ Nelectran xeft _ electron x A = QE x where S is written in mA/W and 4 is in nanometers. Curves of constant * QE are drawn in on Fig. E.2. Typical quantum efficiencies are maximum :' jn the blue region and range upward of 25% or so. S Now let’s return to Fig. E.1 and see how the photomultiplier tube ampli-...: fles the signal. The incident photon has ejected an electron with something 2 ee eee ey - ‘a E.2 Photomultiplier Tubes 493 Absolute responsivity-mA/Watt 300 400 500 600 700 Wavelength-Nanometers FIGUREE.2 Spectral sensitivity (“absolute responsibility”) and quantum efficieacy (QE) for some photomultiplier tube windows and photocathodes, From the Burle photomultiplier tube handbook, available online at http//www.burle.com/. like an electronvolt of kinetic energy. This electron is accelerated to the first dynode and sirnkes it. The dynodes are constructed out of materials that give « significant mean number of electrons out for each that strikes the surface. This muluplication factor 5 is a strong function of the incident electron energy, and is roughly linear with energy up to a few hundred electronvolts or so for most materials used in PMTs. There is clearly some randomness associated with the operation of a photomultiplier. The quantum efficiency, for example, only represents the probability that a photon will actually eject an electron. The result is that the output voltage pulse corresponding to an input light signal will have random fluctuations about a mean value. We therefore frequently talk in terms of the “mean number of photoelectrons” Nps that correspond to a particular signal. Assuming that Poisson statistics domunate, this number will dominate the size of the fluctuations, since the number of electrons ejected in subsequent 44% § Optical Detection Techniques stages will be larger. That is, the fractional rms width of the signal fluctu- | ations should be given by ./Npe/Npe = 1./Npp. This can be particularly ° important if the signal corresponds to a very low light level, i.e., a srnall : value of Npg. In this case, there is a probability e—*?? that there will be no photoelectrons ejected and the signal will go unobserved. : The gain g of a photomultiplier tube ts the number of electrons aut the : back (i.e., at the anode) for a single incident photon. So, for an n-stage = tube, : == 5y X 82-++ Kb, eS", where we tacitly assume that 6 is the same at each stage; i.e., all dynodes F are identical and the potential difference across each stage is the same. If 8 : is proportional to V, then these assumptions! predict that g is proportional = to V". Thus if you want to keep the gain constant to 1% in a \0-stage 22 z photomultiplier tube, you must keep the voltage constant to 0.1%. This is © not particularly easy to do. 2 The accelerating voltage is usually applied to the individual stages by ~: a single external high-voltage DC power supply, and a multilevel voltage’ : divider. The voltage divider has output taps connected to each stage through : the pins into the tube. This is connected to the circuit that extracts the - signal from the anode. The extraction circuit and voltage divider string are = housed together in the photomultiplier tube “base,” and their design will vary depending on the application. The base is usually some sort of closed | box with a socket that attaches to the tube pins. Two examples of base ~ circuits, taken from the Philips photomultiplier tube handbook, are shown — in Fig. E.3. Lf the signal is more or less continuous, and, for example, a | meter reads the current off the anode to ground, you must use the negative high-voltage configuration so that the anode is at (or near) ground. If the - output is pulsc-like, such as when “flashes” of light, or perhaps individual photons, are detected intermittently, then itis usually best to use the positive high-voltage configuration since that leaves the photocathode at ground. ». In this case, an RC voltage divider at the anode output allows fast pulses ~ to reach the counter, but the capacitor protects the downstream electronics — from the high DC voltage. a I These assumptions are almost always wrong. We are using them just to illustrate the — general performance of the PMT. For actual gain calculations, you must know the specific _ characteristics of the PMT. E.2 Photomultiplier Tubes 495 FIGURE E.3_ Typical photomultiplier base circuits. The upper figure shows connections for a positive high-voltage configuration, while the lower shows negative high voltage. No maiter what circuit is used, either those in Fig. E.3 or otherwise, you must choose the resistor values carefully. Although the stage voltages only depend on the relative resistor values, you must make sure the average current passing through the divider string is much larger than the signals passing through the PMT. Otherwise, the electrons in the multiplier will draw current throngh the resistors and change the voltage drop across the stage. Even if this is a small change, it can affect the gain by a Jot sincc the gain depends on voltage to a large power. On the other hand, you cannot make the resistors arbitrarily small so the divider current gets very large, because this would require a large and expensive high-currcnt, high-voltage DC power supply. What is more, the power dissipated in the divider string, i.e., /7R, gets to be enormous, making things very hot. Trade-offs must be made, and always keep your eye on the gain. 49% EF Optical Detection Techniques E.3, PHOTODIODES Photodiodes are an alternative to photomultipliers. Both turn light directly into electrical signals, but there are distinct differences. First, let’s leam how photodiodes work. Recall our discussion about diades in Section 3.1.4. A piece of bulk silicon is essentially an insulator. Only thermally excited electron$can move to the upper, empty energy band to conduct electricity, and there are few of them atroom temperature. By adding n- or p-type dopants, lots more charge carriers can be created, and it is a much better conductor. A piece of silicon doped m on one end and p on the other, a pn junction, only conducts | in One direction. If a “reverse” voltage 1s applied, only atiny current flows, . due to the small number of thermally excited electrons. A photodiode uses light (photons) to excite more electrons than those excited thermally. This 1s possible if the photon energy is larger than the band gap. Thus, the “reverse” voltage current would increase if you shine — light on the diode. This is the principle of the photodiode. The actual mechanism is a bit more complicated, because of how excited electrons actually conduct. So, for example, for a given applied voltage, the output current is not very linear with intensity. That is, if you double the light intensity, the output current does not change by quite a factor of 2 (over the “noise” from the thermal electrons). Furthermore, a photodiode can work if there is no applied voltage, reverse or otherwise. This all means that you must calibrate your photodiode response if you want a quantitative measure of the light intensity. A popular form of photodiode puts a large region of pure, or “intrinsic,” silicon in between the p and n ends. This increases the active area and decreases the thermal noise current. These photodiodes are cailed p—i—n or “pin” diodes. Now let’s look at a clear advantage that photodiodes have over photo- tubes. The energy gap in silicon is 1.1 eV, so photons with wavelengths up to 1.1 wm can be detected. This is well past red and into the IR. Photomultiplier tubes become inefficient at around 600 nm (see Fig. E.2) or so because of the work function of the photocathode. The band gap of germanium (another popwlar semiconductor) is 0.72 eV, so germanium photodiodes reach 2 * 2 pm. So, if you need to detect red light, you - ee probably want to use a photodiode, and not a photomultiplier tube. Another big advantage of photodiodes over photomultiplier tubes is cost. A photomultiplier tube with voltage divider circuitry, high-voltage supply, | E.3 Photadiodes 497 and mechanical assemblies can easily cost upward of $2000. A photodiode costs around $1, and is very easy and cheap to instrument. Photodiodes can also be made with very small active areas (say 50 p.m across). This along with their low cost makes “photodiode arrays” practi- cal, These are lines of photodiodes, separately instrumented, that measure photon position along the array. Such things are frequently used in spec- trographic instruments. A typical example might be 1024 25 pm x 2.5 mm photodiodes arranged linearly in a single housing with readout capability, as discussed in Section 5.5. The cost for such a device is typically less than a thousand dollars. Of course, photomultipliers have some advantages over photodiodes. The biggest is the relative signal-to-noise ratio. A microwatt of incident light power gives around a 1-4 signal in a photodiode, but around 1 A in a photomultiplier tube. This big enhancement in signal is due to the large gain (~10° or more). Thermally excited electrons are plentiful in a photodiode, but rarely does such an electron spontaneously jump off the photocathode in a photomultiplier. Therefore, the noise is a lot larger in a photodiode. Thus, the signal-to-noise ratio is much worse in a photodiode. So, if you need to detect very low light intensities (“photon counting” for example), you probably want to use a photomultiplier tube, and not a photodiode. Photomultipliers also give a more linear response, particularly if care is given to the base design. Some of these relative advantages and dis- advantages are shown in Table E.1. Another advantage of photodiodes is that they work in high magnetic fields. Photomuitiplier tubes rely on electrons with 100-300 eV energy to follow the electric field lines to the dynodes. A few-gauss magmetic field disturbs the trajectories enough to render the PMT useless. In most cases, magnetic shielding solves the problem, but sometimes this is impractical and photodiodes are used instead. TABLE E.1 Photomulbplier Tubes Versus Photodiodes If you are Then your choice should likely be interested in... Photomulaplier Phoiodiode Low cost af Red sensitivity af Low intensity J Lineanty af 498 E— Optical Datection Techniquas Finally, we mention that photosensitive transistors, or phototransistors,, :: are also available. They use the natural amplification features of the tran- sistor to get a ~100 times larger signal than the photodiode. Of course, °: the transistor also amplifies the noise, so there is no improvement in the : sensistivity at low intensities. r ey rom . F; oo oe ao a. we ae at _ i ‘- v- = Lor teal ta bate at Par te Ce CeCe ee a ee Saat aa ta het ht a SC ot Ota APPENDIX F Constants Table F.t of fundamental constants is taken from the “Review of particle properties,” published in Phys. Rev. D 50 (1994). The uncertainties in the values are very small and can be neglected for the experiments in this book. TABLEF1 Fundamental Constants Quantity Symbol Value Speed of light in vacuum c 299792458 m/s Planck’s constant h 6.6260755 x 10-44 Js Rife 6.5821220x 10-22 MeV s Electron charge é 1.60217733x 10-9 ¢ he 197327053 x 107 !3 MeV m Vacoum permittivity ég 8.854187817 x 10—'2 F/m Vacuum permeability LO 4 x 1077 N/A? Electron mass Me 0,51099906 MeVic* Proton mass mp 938,27231 MeVic? Deuteron mass ma 1875.61339 MeV/c? Atomic mass unit rF 931.49432 MeV/c? Rydberg energy he Rog 13.605698 1 eV Bohr magneton fig = eh/ 2m, 5.78838263 x 107 |! MeV/T Nuclear magneton Ln = ehf{2my 3.15245166x« 107 !* MeV/T Avogadro conslant No 6.0221367x 10%) atoms/mole Boltzmann constant k 1.380658 x 10723 /K 499 APPENDIX G Exercises The following exercises may be used. 1. The foilowing table lists data points for the decay rate (in caunts/s) of a radioactive source: Time Rate Time Rate Time Rate (i) @) () @w (7) 0.6 18.4 2.0 4.92 43.6 1.72 0.8 10.6 2.4 2.61 4.0 1.61 1.2 8.04 2.8 2,08 4.2 1.57 1.6 6.10 3.0 150 0 4.3 1.85 a. Plot the data using an appropriate set of axes, and determine over what range of times the rate obeys the decay law R = RoeW’/. . Estimate the value of Ro from the plot. . Estimate the value of t from the plot . Estimate the value of the rate you expect at f = 6s. an 2. An experiment determines the gravitational acceleration g by measuring the period T of a pendulum. The pendulum has an adjustable 501 2 G Exercises length L. These quantities are related as iL Yt =2x [-. & A rcsearcher measures the following data poinis in some arbitrary units. Data ‘a point -L T “ey ] 06 14 2 15) 619 4 20 2.6 4 2.6 2.9 5 3.5 3.4 One of these data points is obviously wrong. Which one? 3. Consider the following simple circuit: — Vion Let the input voltage Vj, be a sinusoidally varying function with amplitude Vg and angular frequency w. a. Calculate the gain g and phase shift ¢ for the output voltage Telative to the input voltage. b. Piot g and @ as a function of w/wp where wo = 1/RC. For each of these functions, use the combination of linear or logarithmic axes for g and for @ that you think are most appropriate. 4, Consider the following simple circuit: Ye Ai Viva eee ag wae G Exercises 503 Let the input voltage Vig be a sinusoidally varymg function with amplitude Vo and angular frequency w. a. Calculate the gain g and phase shift # for the output voltage relative to the input voltage. b. Plot g and ¢ as a function of w/wp9 where w) = R/L. For cach of these functions, use the combination of linear or logarithmic axes for g and for @ that you think are most appropmiate. » 5. Consider the following not-so-simple circuit: Vin R Vout a. What is the gain g for very low frequencies «2? What is the gain for very high frequencies? Remember that capacitors act like dead shorts and open circuits at high and low frequencies, respectively, and inductors behave in just the opposite way. b. At what frequency do you suppose the gain of this circuit is maximized? c. Using the miles for impedance and the generalized voitage divider, determine the gain ¢(w) for this circuit and show that your answers to (a) and (b) are correct. 6. Suppose that you wish to detect a rapidly varying voltage signal, However, the signal ig superimposed on a large DC voltage level that would damage your voltmeter if tt were in contact with it. You would like to build a simple passive circuit-that allows only the high-frequency signal to pass through. a, Sketch a circuit using only a resistor R and a capacitor C that would do the job for you. Indicate the points at which you measure the input and output voltages. 504 G Exercises b. Show that the magnitude of the output voltage equals the magnitude of the input voltage, multiplied by 1 Jl + 1/@2R2C2’ where « is the (angular) frequency of the signal. c. Suppose that R = 1 k& and the signal frequency is 1 MHz =“. 10°/s. Suggest a value for the capacitor C. 7. Anclectromagnet is designed so that a 5-V potential difference drives |. 100 A through the coils. The magnet is an effective inductor with an induc- ..° tance £ of 10 MHz. Your laboratory is short on space, so you put the DC. power supply across the room with the power cables along the wall. You notice that the meter on the power supply must be set to 6 V in order to get 5 V at the magnet. On the other hand, you are nowhere near the Limit of . the supply, so it is happy to give you the power you need. Is there any reason for you to be concerned? Where did that volt go, and what are the implications? If there is something to be concerned about,. suggest a solution. 8. You are given a low-voltage, high-current power supply to use for an |: experiment. The manual switch on the power supply is broken. (The power supply is kind of old, and it looks like someone accidently hit the switch with a hammer and broke it off.) You replace the switch with something you found around the lab, and it works the first time, but never again. = When you Lake it apart, the contacts seem to be welded together, and you know it wasn’t that way when you put iLin, What happened? (Hint: Recall that the voltage drop across an inductor is L di/dt, and assume the switch disconnects the circuit over 1 ms or so.) 9. The following table is from the Tektronix Corp. 1994 catalog » sclection guide for some of their oscilloscopes: Model Bandwidth Sample rate Resolution Time bases 2232 100 MHz 100 M5/s 8 bits Dual 22214 100MHz 100MSi/s 8 bits Single 2212 60 MHz 20 MS/s 8 bits Single 2201 20 MHz 10 MS/s 8 bits Single You are looking at the output of a waveform generator on one of these: oscilloscopes. The generator is set to give a +2-V sine wave output. [f the.: sine-wave period is set at 1 jis, the scope indeed shows a 2-V amplitude. :: G Exercises 505 However, if the period is 20 ns, the amplitude is [ V. Assuming the oscilloscope is not broken, which one are you using? 10. You want to measure the energies of various photons emitted in a nuclear decay. The energies vary from 80 keV to 2.5 MeV, but you want to measure two particular [ines that are separated by I keV. Lf you do this by digitizing the output of your energy detector, at least how many bits does your ADC need to have? Il. Pulses emitted randomly by a detector ate studied on an oscillo- scope: The vertical sensitivity is 100 mV/div and the sweep rate 1s 20 ns/div., The bandwidth of the scope is 400 MHz. The start of the sweep precedes the trigger point by 10 ns, and the mput impedence is 50Q. a. Estimate the pulse nsetime. What could you say about the risetime if the bandwidth were 40 MHz? b. Estimate the trigger level. c. These pulses are fed into a charge-integrating ADC, also with 50 2 input impedence. The integration gate into the ADC is 100 ns long and precedes the pulses by 10 ns. Sketch the spectrum shape digitized by the ADC. Label the horizontal axis, assuming 5 pC of integrated charge corresponds to one channel, d. The ADC can digitize, be read out by the computer, and reset in 100 js. Estimate the number of counts in the spectrum after 100 s if the average pulse rate is | kHz. What is the number of counts if the rate is | MHz? 12. Adetector system measures the photon emission rate of a weak light source. The photons are emitted randomly. The system measures a rate of 10 kHz, but the associated electronics requires 10 1s to régister a photon, and the system will not respond during that time. What is the true rate at which the detector observes photons? 506 G Exercises L 2) . 13. You measure the following voltages across some resistor with a - three-digit DMM. As far as you know, nothing is changing so all the - measurements are supposed to be of the same quantity Vp. 231 235 2.26 2.22 2.30 Zof 2.29 2.33 2.25 2,29 a. Determine the best value of Vp from the mean of the measurements. b. What systematic uncertainty would you assign to the measurements? c. Assuming the fluctuations are random, determine the random uncertainty from the standard deviation. d. Somebody comes along and tells you that the true value of Vp is 2.23, What can you conclude? 14. (From G L. Squires, Practical Physics, third ed., Cambridge -: (1985).) In the following examples, g is a given function of the independent. : measured quantities x and y. Calculate the value of g and its uncertainty © 8q, assuming the uncertainties are al! independent and random, from the. : given values and uncertainties for x and y. ; a. q=x*forx =25+1. b. gq =x — 2y forx = 10043 and y = 45 +2. c. g = E: 2n./L/g as 69 — 0.) The pendulum length is L = 87.2 + 0.6 cm. The = period is determined by measuring the total time for 100 (round trip) swings. * a. Atotal time of 192 s is measured, but the clock cannot be read to better than +100 ms. What is the period and its uncertainty? b. Neglecting the effect of a finite value of 6, determine g and its uncertainty from these data. Assume uncorrelated, random uncertainties. c. You are told that the pendulum is released from an angle less than 10°. What is the systematic uncertainty in g from this information? d. Which entity (the timing clock, the length measurement, or the unknown release angle) limits the precision of the measurement? 22. The f-decay asymmetry, A, of the neutron has been measured by : Bopp et al. Phys. Rev. Lett. 56, 919 (1986) who found that : 2a(1 — A) A = ———— = -0.1146 + 0.0019. 1 + 3A2 my This value is perfectly consistent with, but more precise than, earlier results. : The neutron lifetime, t, has also been measured by several groups, and the results are not entirely consistent with each other. The lifetime is given by .: _ 3163.7 ~~ 14342 and has been measured to be 918 + 14s by Christenson et al., Phys. Rev. D 5, 1628 (1972), G Exercises 509 881 + 8 s by Bondarenko ef al., JETP Lett. 28, 303 (1978), 937 + 18s by Byme e al., Phys. Lett. B 92,274 (1980), and 887.6 + 3.0 s by Mampe et al., Phys. Rev. Lert. 63, 593 (1989), Which, if any, of the measurements of t are consistent with the result for A? Which, if any, of the measurements of t are inconsistent with the result for A? Explain your answers. A plot may help. 23. The “weighted average” of a set of numbers is > _ ray Wi Xj iw = Diet Wits — (7.1) where the “weights” w,; = i/o?. a. Prove that this definition for the weighted average is the value that minimizes x°. b, Use propagation of errors to derive the uncertainty m the weighted average. 24. Let’s suppose you have some peculiar dice which each have 10 faces. The faces are numbered from 0 to 9, You throw etghr of these dice at a time and record which numbers land face down on the table. You repeat this procedure (1e., throwing the dice) 50 times. a. For how many throws do you expect there to be exactly three dice landing with either face 1 or face 5 landing face down? b. What is the average number of dice you expect to land with either face 1 or face 5 down, for any particular throw? What is the standard deviation uncertainty tn this number? c. Use the Poisson approximation to calculate the same number as in (a), d. Use the Gaussian approximation to caiculate the same number as in (a). 25. Aradioactive source emits equally in all directions, so that the inten- sity falls off like 1/r? wherer is the distance tothe source. You are equipped with a detector that counts only radioactivity fron the source, and nothing else. Atr = 1! m, the detector measures 100 counts in 10s. a. Whatis the count rale, and its uncertainty, in counts per second? b. What do you expect for the fractional uncertainty in the count rate if you count for 100 s instead of 10? S10) 386G Exercises c. Based on the original 10-s measurement, predict the number of counts you should observe, and its uncertainty, if the detectot-is, moved to a distance of 2 m and you count for 1 min. 26. Suppose you are using a Geiger counter to measure the decay rate ©225 of a radioactive source. With the source near the detector, you detect 100 7°35: counts in 25 s. To measure the background count rate, you take the source - “=: very far away and observe 25 counts in 25 s. Random counting uncertainties | dominate. a. What is the count rate (in counts/s) and its uncertainty when the EE source is ncar the Gciger counter? ae b. What is the count rate (in counts/s) and its uncertainty when the 9 225 source is far away? Ee c. What is the net count rate (in counts/s) and its uncertainty dueto 24 the source alone? d. Suppose you want to reduce the uncertainties by a factor of 10. How long must you run the experiment? es or a ieee 27. An experimenter is trying to deiermine the value of “absolute zero” 25: in degrees Celsius using a pressure bulb and a Celsius thermometer. She “2 assumes that the pressure in the bulb is proportional to the absolute tem- 222 perature. That is, the pressure is zero at absolute zero. She makes five measurements of the temperature at five different pressures: Pressure (nm of Hg} 65 75 85 95 105 cE Temperature (°C) —21 1 41 93 129 Ee Use a straight line fit to determine the value of absolute zero, and its = uncertainty, from these data. 28. Fit the following (x, y) values <= 235 63 89 132 147 y= 496.6 507.2 551.5 625.5 651.7 to a straight line and plot the data points and the fitted line. a. Does it look like a straight line describes the data well? 2 b. Study this further by plotting the deviations of the fit from the = data points. What does this plot suggest? c. Try fitting the points to a quadratic form, i.e., a polynomial of degree 2. Is this fit significantly better than the straight tine? 29. The following results come from a study of the relationship between & high school averages and the students’ overall average at the end of the first “+ G Exersises 4511 year of college. In each case, the first number of the pair 1s the high school average, and the second is the college average. 78,65 80,60 85,64 77,59 80,56 82,67 81,66 89,78 87,71 80,66 85,66 87,76 @4,.73 87,63 74,58 91,78 B1,72 91.74 86,66 90,68 a. Draw a scatterplot of the college average against the high school average. b. Evaluate the correlation coefficient. Would you conclude there is a strong correlation between the grades students get in high school and the grades they get in their first year of college? 30. Using the data in Table 2.1, draw a scatterplot of electrical conductivity versus thermal conductivity for various metals. (Electrical conductivity is the inverse of electrical resistivity.) Calculate the linear correlation coefficient. 3]. Graph the ratio of the Poisson distribution to the Gaussian distribu- tion for mean values jt = 2 and for 4 = 20. Use this to discuss where the Gaussian approximation to the Poisson distribution is applicable. Repeat the exercise, but cormmpare the Gaussian approximation directly to the binomial distribution with p = 4. 32. Consider blackbody radiation. a. Show that the wavelength at which the intensity of a blackbody radiator is the greatest is given by “Wien's displacement law”’: 29x 1074 Amax (1) = T (K) Hint: You will need to solve an equation like xe* /(e* —1) = A for some value A. If A > | then this ts trivial to solve, but you can be more exaet using MATLAB. In MATLAB you would use the “function” fZero to find the place where f(x) = Ale — 1) —xe* crosses zero. b. Stars are essentially blackbody radiators. Our sun is a “yellow” star because its spectrum péaks in the yellow portion of the visible light. Estimate the surface temperature of the sun. 33. A particular transition in atomic neon emits a photon with wave- length A = 632.8 nm. 412 G Exercises a. Calculate the energy £ of this photon. LAS b. Calculate the frequency v of this photon. c. An optical physicist tells you the “line width” of this transition is Av = 2 GHz. What is the linc width AF in terms of energy? d. Use the Heisenberg uncertainty principle to estimate the lifetime At of the state that emilted the photon. e. How far would a photon travel during this lifetime? f. Suppose the neon is contained in a narrow tube 50 cm long, with Mirrors at cach cnd to reficct the light back and forth and “trap” it in the tube. What is the nominal “mode number” for 632.8-um photons, that is, the number of half-wavelengths that fit in the tube? g. What is the spacing in frequency between the nominal mode number m, and the wavelength corresponding to the mode m + 1? h. Compare the mode spacing 4v (part G)} with the line width Av. i. What is this problem describing? 34. Estimate the “transit time” for a typical photomultiplier tube. That . is, how much time elapses between the photon ejecting an electron from . the photocathode and the pulse emerging from the anode? Assume the - photomultiplier has 10 stages and 2000 V between cathode and anode, divided equally among al! stages, and that the dynodes are each separated by 1 cm. 35. Some high-quality photomuitiplicrs can detect the signal from a . single photoelectron, and cleanly scparate it from the background noise. . Such a PMT is located some distance away from a pulsed light source, so - that on the average, the PMT detects (Npe) photoelectrons. If {Npp) « 1 : and Ng pulses are delivered, show that the number of pulses detected by the photomultiplier is given by {Npg) No. 36. A photomultiplier mbe observes a flash of green light from an Ar* : laser. (Assume the photons have wavelength 4 = 500 nm.} The photomul- - tiplier is a 10-stage tube, with a RbCsSb photocathode, The voltages are » set so that the first stage has a secondary emission factor 8, = 5, while the © other 9 stages each have 6 = 2.5. The laser delivers some huge number of. . photons to a diffusing system, which isotropically radiates the light, and _ only a small fraction of them randomly reach the photomultiplier. On the 5 average, 250 photons impinge on the window for each flash of the laser. «2328 a. What is the average number of electrons delivered at the anode output of the photomultiplier tube, per laser flash? G Exercises 513 b, Assume these electrons come cut in a rectangular pulse 20 ns wide. What is the height of the vof/tage pulse as measured across a 50-2 resistor? c. You make a histogram of these pulse heights. What is the standard deviation of the distribution displayed in the histogram? d. Suppose the photomultiplier tube is moved four times farther away from the source. For any given pulse of the laser, what is the probability that no photons are detected? ° 37. A Geiger counter is a device that counts radioactive decays, typically used to find out whether something is radioactive. A particular Geiger counter measures 8.173 background counts per second; i.e., this is the rate when there are no known radioactive sources near it. Your lab partner hands you a piece of material and asks you whether it is radioactive. You place it next to the Geiger counter for 30 s and it registers a total of 253 counts. a. What do you tell your lab partner? b. What do you do next? 38. The Tortoise and the Hare have a signal-to-noise problem. A very weak signal sits on top of an enormous background. They are told to deter- mine the signal rate with a fractional uncertainty of 25%, and they decide to solve the problem independently. The Tortoise dives into it and takes data with the setup, and he determines the answer after running the appa- Talus for a weck. The Hare figures she is not only faster than the Tortoise, bul smarter (00, so she spends two days reducing the background in the apparatus to zero, without affecting the signal. She then gets the answer after running the improved setup for one hour. (The Hare really is a lot Smarter than the Tortoise, at least this time.) Assuming Poisson statistics, a. What is the signal rate? b, What is the Tortoise’s background rate? 39, Consider the passive filters shown in Fig. 3.11. a. Determine the gain as a function of w = 27 v for each filter. b. Plot the gain as a function of w/c for the three low-pass filters. Define the critical frequency wc using the simplest combination of the two components in the circuit, that is, ac = 1/RC, ac =1//LC, or ac = R/L. Ibis probably best to plot all three on the same set of log-log axes. 514 G Exarcises a =a c. Do the same as (b) for the high-pass filters. d. Can you identify relative advantages and disadvantages for the different combinations of low-pass and high-pass filters? 40. Consider the following variation on the circuit shown In Fig. 3.12: Vin R Veurt Lt a. How does this circuit behave at high frequency? b. How does this circuit behave at low frequency? c. Calculate the gain g = | Vour/ Vin! as a function of frequency. What is the behavior for intermediate frequencies? d. Give an example of where this sort of filter would be useful. 41. Aparticle detector gives pulses that are 50 mV high when measured as a voltage drop across a 50-{2 resistor. The pulse rises and falls in a time span of 100 ns ot less. Unfortunately, there are lots of noisy motors in the laboratory and the pround is not well isolated. The result is that a 10-mV, 60-Hz sine wave is also present across the resistor, and adds linearly with the pulses. a. Draw a simple circuit, including the 50-£2 resistor and a single capacitor, that allows the pulses to pass, but blocks out the 60-Hz noise. b. Determine a suitable capacitance value for the capacitor. 42. You are measuring a quantity Q that is proportional to some small voltage. In order to make the measurement, you amplify the voltage using a negative feedback amplifier, as discussed m Section 3.5. a. Show that the gain g of the full amplifier circuit can be wnitten as 1 1 e=00[1- 35 +0 (aes). ae ae we hare voy ete eee ray Cr a age ae eee eT a ed wa pe ea Pree erie ee) ae ee a ie eT ee re ay sie a oe ae Chee a ars ae wal a rele a, ae Pee ha ae pea Pree ae ee ey ea ate ee aa na te wan witatiale vera ie ner eae re a a ar eee eee ate ttt ot ae ea at oe Pe ial eee ee oe) ad rr ad eee Par ie rer eae ee ae a ee bel eae ee oe rae eal eae er eer aay or Me eee et ate ee ee ee a ae ee aire a ea Pa eee a eee ees Poe eee aera a ta oat oe te ee Par eeieal ee area ea eT aaa at ate eee Cae cate ee Drea se ee ate a aie ee ve ee ed cn a a ee G Exercises 515 where 29 = 1/8 and a >> | is the internal amplifier gain, £ is the feedback fraction, and af > I. b. You measure @ with such an amplifier, with 8 = 0.01. The temperature in the lab fluctuates by 5°F while you make the measureraent, and the specification sheet for the opamp tells you that its gain varies between 2.2 x 10* and 2.7 x 104 over this temperature range. What is the fractional uncertainty in Q due to this temperature fluctuation? 43, A**Na radioactive source emits 0.511- and 1.27-Me¥ y-rays. You have a detector placed some distance away. You observe a rate of 0.511- MeV photons to be 2.5 x 103/s, and of 1.27-MeV photons to be 103/s, with just air between the source and the detector. Calculate the rate you expect for each y-ray if a 1/2-1n.-thick piece of iron is placed between the source and the detector. Repeat the calculation for a 2-in.-thick lead brick. 44. Aracioactive source is situated near a particle detector. The detector counts at a rate of 104/s, completely dominated by the source. A 2-cm-thick slab of aluminum (density 2.7 gm/cm?) is then placed between the source and the detector. The radiation from the source must pass through the slab to be detected. a. Assuming the source emits only 1-MeV photons, estimate the count rate after the slab is inserted. b, Assuming the source emits only 1-MeV clectrons, estimate the count rate after the slab 1s inserted. 45. Consider a small rectangular surface far away from a source. The surface ts normal to the direction to the source, and subtends an angle a horizontally and £ vertically. Show that the solid angle subtended is given by af. 46. A photomultiplier tube with a 2-in. active diameter photocathode is Jocated 1 m away from a blue light source. The face of the PMT is normal to the direction of light. The light source isotropically emits 10° photons/s. Assuming a quantum efficiency of 20%, what is the count rate observed by the photomultiplier? 47. Two scintillation detectors separated by 3 m can measure the “time- of-flight” for a particle crossing both of them to a precision of +0.20 ns. Each detector can also measure the differential energy loss dE/dx = constant/ 8”, 8 = u/c, to 410%. For a particle with a velocity of 80% the speed of light (i.e., 8 = 0.8), how many individual detectors are needed 516 G Exercisas along the particle path to determine the velocity v using d £ /dx to the same precision as is possible with time-of-flight? 48. A Cerenkov detector is sensitive to particles that move faster than the speed of light in some medium, i.¢., particles with 8 > 1/n, where n is the index of refraction of the medium. When a particle crosses such a detector, it produces an average number of detected photons piven by 1 = K]1-—-——]|. . ( wna) The actual number of detected photons for any particular event obeys a Poisson distribution, so the probability of detecting no photons when the mean is 44 is given by e ”. When 1-GeV electrons (8 = 1) pass through the detector, no photons are observed for 31 out of 19,761 events. When 523-MeV/c pions (8 = 0.9662) pass through, no photons are observed for 646 out of 4944 events. What is the best value of the index of refraction n as determined from these data? What is peculiar about this value? (You might want to look up the indices of refraction of various solids, liquids, and gases.) ee ae roe rar ae Absorption coefficient, 300-301 AC circuits, 93-96 ADC, See Analog-to-digital converter Airy equation, 190 Alpha particles, 306, 324, 325f, 351-354 Amplifiers, operational, 119-120 Analog-to-digital converter (ADC), 113 Angular momentum, addition of, AQ, 226f, 228 Atomic structure. See specific particles, effects Atomic vapors, 1-13 Autocollimation, 27 Avalanche detectors, 347 Avogadro's number, 300 Babinet’s principle, 184 Balmer series, 2, 25, 29-33, 235, 235f Band theory, 49-54, 72 Bandpass filters, 103, 122, 133 Barnum, 355 Bamer-layer detector, 345 Beams, atomic. See specific types, techniques Beams, laser. See Lasers Bernoulli distribution, 433-434 Berry’s phase, 210-213, 213f Bessel functions, 58, 190 Beta decay, 20 Bifurcations, 133, 137-138 Binouual distribution, 433-436, 443 Birefringent materials, 203 Bismuth, 66, 68 Blackhody radiator, 511 Bloch magnetic susceptibility, 267 Bohm-Aharonov effect, 211 Bohr magneton, 216 Bohr model, 10, 21, 22 Bohr, N., 20- Boltzmann constant, 47, 124, 125, 131 Boltzmann distribution, 48, 73, 154 Boltzmann, L., 45 Boron, LO1 517 518s Index Bose-Einstein statistics, 45 Bragg curve, 305, 353, 354f Bremsstrahlung, 304, 316-319 Brewster angle, 161, 161f, 162 Bridge circuit, 276-278, 289 Builouin surfaces, 52 Brownian motion, 5 Capacitance, 93, 95 Capacitors, 93-98 Cavity, 151. See Lasers Cesium, 360 Chaos, 133-143 Charged particles, 10. See specific particles Chi-square distribution, 451-454 Cherenkov detector, 516 Circuit theory, 89-104, 116-119 Circular apertures, 191f, 188-!91 Coaxial cables, 104-107 Coincidence experiments, 367-418 Combinatorial analysis, 430-431 Compton, A., 370 Compton scattering, 313-314, 369-385 experimenial design, 375-378 K-N formula and, 313 shifts in, 371 wavelength and, 371 Computer interfaces, 147-149 Conduction bands, 54, 72-74 Confocal resonator, 158 Conservation laws, 20 Constant deviation instruments, 33 Cosmic rays, 399-409 Coulomb-bartier effects, 298 Coulomb force, 20, 21 Coulomb potential, 34n15, 218 waa ee Crimping, 107 Cross section, defined, 298-299 Crystal efficiency, 378 Crystals. See Semiconductors, 52-56 Currentdensity,55 aE Current, electric, 90 Cyclotron, 64 DAC, See Digital-to-analog 2 converter aS Darlington pair, 59n7 USE Data analysis, 149, 445-453 [SE DC power supplies, 108-109 oe dE/dx curve, 349, 353 ES Dead time, 115, 332, 333 Te Decay rate, 354, 405, 466,468,501 2: Degrees of freedom, 452, 453 REE Delay curve, 414 ES Delta rays, 305 SS Deuterium, 235f no Diffraction, 164, 179 EE calculation of, 185 ES circular aperture and, 188-191 ES gratings and, 180,192-198,217 =: prism and, 30 te resolving power and, 217 ES specroscopy. See Spectroscopy AEE slit and, 180-184 OSS See also specific effects, ES equipment Digital multimeters, 110 Bae Digital oscilloscope, 116 ES Digital-to-analog converter (DAC), 5 114 ES Digitizers, 113-115 BE Diodes, 99-102 oe bifurcationsand,142 0 ii iE ae eae rr ee * we Pare Par chaps and, 143 circuits with, 139-144 current through, 80 I-V characteristic of, 139 oscilloscope traces, 130 p-n junctions, 78, 100, 345 properties of, 78 recombination regime, 79 reverse bias on, 101 semiconductors and, 78, 100 symbol for, 100 Dirac theory, 39, 224n Direction cosines, 186 Dopants, 101 Doppler effects, 16], 245n, 387 Eddy currents, 57 Finstein, A., 153 Flectric current, 90 Flectric-dipole transition, 221, 222n9 Electric field, 2 Electric potential, 90 Electrical conductivity, 511 Electrical resistance, 55 eddy current technique, 57 metals and, 54 physics of, 54 temperature and, 63 Electromagnetic cascade, 320 Electromagnetic radtation, 312 Electron spin resonance (ESR) spectrometry, 254, 283-290 Electrons, 40—43, 254-292, 322f angular momentum of, 220 bremsstrahlung, 316-320 charge on, 1, 4, 10 coupling of, 40-43 Index 515 current and, 90 drift velocity, 55 energies of. See Energy levels, atomic excited states, 20 fractional charge, 10 pround state, 20 holes, 54, 76, 347-348 ions and, 319, 320f magnetic moment of, 224—228 matter and, 298-319 mean free path, 63 one-dimensional problem, 50 orbits of, 218, 367 positrons and, 319, 320f radiation length, 319 reduced mass, 233 scattenng angle of, 316 semuconductors and, 72 solids and, 45-88 thermal motion, 123 wells and, 50 Energy levels, atomic, 20, 49, 152-154, 203, 227, 254, 337, 353. See also specific particles Error analysis, 454-464 ESR. See Electron spin resonance Estimation of parameters, 445-453 Exponential growth, 134 Extrinsic carriers, 72 f-number, 190 Fabry-Perot apparatus, 172-177, 217, 239, 239f, 241 Far-field amplitude, 188 Farad, unit, 93 Paraday effect, 201, 203, 205f, 207f, 210 i290 8=6oindex Faraday’s law, 57 Feather’s rule, 378 Feedback, negative, 119 Feigenbaum, M., 137 Feigenbaum number, 138, 143, [44 Fermi constant, 406 Fermi-Dirac statistics, 45-49 Fermi distribution, 73 Fermi energy, 47, 53, 73-75 Fermi particies, 47 Fermi weak interaction, 406 Fermi’s golden rule, 258 Filter circuit, 104 Fine structure, 36-39 Floating terminals, 108 Fluorescence, 247, 248f Foucault pendulum, 211 Fourier analysis, 133, 316 Fourier optics, 180, 198 Fourier transform, 95, 188, 198 Frank—Hertz cxpenment, 1, 10-19 apparatus for, 2, 12, 14, 15 beam current, 17f excitation potential, 16 ion current, 19 ascilloscope display of, 17 preferred elements for, 13 wiring diagram for, 15 Fraunhofer diffraction, 180 Frec-electron gas, 72 Free induction decay, 271 Free radicals, 284 Free spectral range, 157 Frequency bifurcations, 133-138 Frequency filters, 102-104 Frequency functions, 431-445 Fresnel diffraction, 180 Gage number, 105n Gain curve, 157 Gain function, 125 Gamma function, 452 Gamma rays, 328F, 336, 337, 339, 409-42 1 angular correlation of, 413-413, 417f, 419-421 amisotropy, 412, 418 attenuation of beam, 487 coincidence rate, 412 coincidence circuit, 416 decay scheme, 338, 339 electron-positron pairs, 298 pamma-camma correlation, 409-411, 415-419 low-cnergy, 371 pulse-height spectrum of, 375 recoil effects, 387 spectra, 378 Gaseous ionization detectors, 320-333 Gaussian approximation, 509 Gaussian distribution, 132, 436, 439 as limiting case, 439-442 binomial frequency function, 433 moments of, 434-438 normal distribution and, 435 properties of, 443 Gauss’s law, 302 Geiger counter, 320-333, 510 cylindrical, 321f dead time of, 332, 333 plateau region, 329, 33] Germanium, 54, 74 Goodness of fit, 451-454 Gratings, 25, 180. See also Diffraction Ground, electric, 90 Ground state, 11 Gyromagnetic ratio, 255 Hadronic particles, 10 Half-life, 354 Hall coefficient, 65, 66, 70 Hall effect, 63-70 Hamiltonian operator, 5] Helium, 20, 160, 160f Helmoltz coils, 275, 286 Henmite polynomials, 158 hfs, See Hyperfine structure High-field magnets, 85 High-pass filter, 122 High-resolution filters, 177 Holes, electron, 54, 72, 75 Huygens—Fresnel principte, 185 Hydrogen, 20, 235f Balmer serics of, 29 hydrogen-deuterium shift, 234 orbits in, 22 spectra of, 20 Hyperfine structure (his), 215-216, 228-238 Doppler effect, 238 isotope shift, 228, 232 of mercury, 238 of rubidium, 246-247 Image-forming detectors, 296 Image plane, 199 Impedance, 95 charactenstic, 106 coaxial cable, 104 Index ‘521 Impurities, 72, 74, 10] Indium, 356—359 Inductance, defined, 99 Inductors, 98-100, 141 Input registers, 115 Insulators, 53 Interferometry, 167, 172. See specific types ° Intrinsic carriers, 72, 74 Inverse matrix, 450 Ion current, 16 Ionization chamber, 321, 323-326, 344 Ionization potential, 13, 14, 18 Isotope shift, 215n, 228, 232, 234f Jarrell-Ash spectrometer, 235f Johnson notse, 122, 125, 126f, 129 Junctions, 75-78, See Semiconductors Kimball-Slater spacing, 52 Ktein-Nishina formula, 313, 374, 384, 385 Ktystron, 286 Knock-on electrons, 305 Lande g factor, 225 Larmor precession, 260 Lasers, 151-177 beam profile, 165-167 cavity, 155 collimation of beam, 164 defined, 170 Fabry-Perot interferometer, 172-177 HeNe laser, 159-162 522 Index Lasers (continued ) interferometers and, 172 lasing medium, 154 Michelson interferometer, 168 principle of operation, 152-156 properties of beams, 156 safety, 483-484 spatial filtering, 201 telescope, 164 See also specific parameters, effects Latches, 115 Lattices, 52, 389. See also Crystals LCR circuit, 99 Least-squares method, 29, 447-451, 480 Lifetime, of nuclei, 467 Light, 201-210, See aiso specific effects, instruments Linear devices, 99 Linear functional dependence, A49_-451 Load buffering, 122 Lock-in amplifier, 144-146, 208 Logistic map, 133-138 Longitudinal modes, 156 Lorentz transformation, 374, 390 Low temperature approximation, 74 Lyman series, 24 Magic tee circuit, 286, 289f Magnetic-dipole transitions, 252, 255-261 Magnetic fields anomalous effects, 229n light, 201 magnetic moment, 219f, 261f magnetic resonance, 251-293 optics and, 200 refractive indices, 203 spectral lines and, 221f spin and, 256f Malthus theory, 134 Marginal oscillator circuit, 276, 281-282 MATLAB programs, 132, 149, 342, 358, 451, 477, 482, 511 Maximum likehhood methods, 445-447 MaxwWell-Boltzmann distribution, 237 . Maxwell’s equations, 58 Mean, defined, 432 Mean free path, 7, 70 Meissner effect, 83 Mercury, 42f, 33-43, 43f, 232f, 238 Metals, resistivity, 54, 56, 72 Meters, types of, 109 Michelson interferometer, 167, 168f, 171 Microscope, parallax errors, 5 Microwave cavity, 289 Millikan oil drop experiment, 2-10 Minority carriers, 76 Monte Carlo method, 464 Missbauer effect, 385-399 Multiple-bearn interferometer, 217 Multiple scattering, 310-311 Muons, 404-409 Negative feedback, 119, 120f Neon, 160, 160f Neutrinos, 404, 405 ) dee Pore ees errs Pare a) oe eae a oer ae) vane aan ee aa ae oa ree ee a ae) or rey wae ay ea a ae) eee ee rae ee aa cay aa ela ee) a aaa vee ds oe a wee Para er) era ea wo et er a) Parke he ty ker er) Pee) tee +4 ata ae ae a ee ne aa Par a he ha ae ee! vga ee rps aes eae eae ae enn ee eer) al ere ae va eae ay aoe eel ee ee ee er oe oe) eae aan a oe a eee wae ea a ree a ee Park ee ee aye roe ae ae a ae he har er er ed ee a eT aa ae a eee Pak rer ee aaa ee ee Pari eae) ee Peet mr ee ae ee La as ae aa ee ie ead or i at a) | ee ei a ri nts ae vate ae Ce a a a ee er ed eae es ee ar ee a eee cere ee ee eee a al Neutrons, 296, 355, 508 n-p junctions, 76, 345, 346£ Noise, 102, 119, 146 Johnson noise, 122 Nyquist noise, 122 rejection, 146 spatial filtering, 201 temperature and, 124 Nonlinear components, 133 Nonlinear methods, 480 Normal distribution, See Gaussian distribution Nuclear magnetic resonance (NMR), 146, 267, 283 bridge circuit, 277-279 detection of, 277-279 ESR. See Electron spin resonance free induction decay, 270-273 line width, 266-267 marginal oscillator circuit, 281-282 protons and, 278f, 280f, 273-282 pulsed, 270-273, 279-281 Rabi experiments, 254 spin and. See BSR; Spin transverse relaxation time, 267 Nuclear magneton, 229 Nuclear resonance radiation, 389, 390 Nucleus, atomic decay of, 409, 410f, 465-467 electron—positron pairs, 298 half-life, 354-363 mean free path, 298 moments, 230, 230f, 262-273 NMR. See Nuclear magnetic resonance nucleons, 229n ao "a*a"s ‘a? spin and, 229-232, 2568 statistics for, 465-473. ses: See also specific particles, ay effects ok shyt Null methods, 264 Nyquist noise, 122 Occupied states, 73 Ohm's law, 54, 55, 64, 104 Oil drop method, See Millikan oil drop experiment Operational amplifiers, 121f, 119-12] Optical detection techniques, 489-498 Optical experiments, 179-213 Optical fiber, 211f Optical spectroscopy, 20 Organic free radicals, 284 Organic scintillators, 334 Orthogonal] triads, 211 Oscilloscopes, 110-113, 117 bandwidth, 113 digital, 116 Fourier analysis, 133 ion current, 16 lock-in amplifier, 208 Output coupler, 159 p—n junction, 75, 79, 100-101, 345 p-n—p junction, 77 p-n-p transistor, 77, 102 Pair production, 312, 314 Paraelectric matenals, 268 Parallel circuitry, 91 Paramagnetism, 268, 284 Parameter estimation, 445453 524—S ss Indax Particles, 50, 295-365. See specific types Paschen series, 24 Pashen-Back effect, 228 Pauli principle, 4 peak-to-total ratio, 378 Period doubling, 137, 142n Permutations, 430 Phase space, 460 Phase transitions, 98, 144-146, 210-213 Phonons, emission of, 389 Photodiodes, 166f, 165-167, 496-498 Photoelectric effect, 312 Photofraction, 378 Photographic film, 489 Photomultiplier tube, 490-497 quantum efficiency, 492 spectral sensitivity, 492 transit time for, 512 Photons, 152, 245, 295, 312, 343 Pinhole, 190, 191 Pianck’s constant, 21, 152 Plastic scintillators, 334 Pockels effect, 203 Poincare, H., 135 Poisson approximation, 509 Poisson distribution, 337, 436 Poisson statistics, 493 Polarization, 153, 179, 180, 201, 2062, 205, 210 Polonium, 298 Population growth, 134 Population inversion, 154, 160 Positrontum, 419 Positrons, 312 Power supplies, 108-109 Poynting vector, 373 Prism spectrometers, 25 Probability theory, 423-427 Proportional counter, 327 Protons, 273-283, 278f, 280f, 283f Pulse-height spectrum, 337 Pulse transmission, 105 Quadrupole transitions, 36 Quality factor, defined, 277 Quantization, defined, 1 Quantum efficiency, 492 Quantum electrodynamics, 21] Quantum number, 22, 203 Quarks, charge on, 10 Rabi frequency, 273 Radiation, 36 absorption of, 10, 153 blackbody, 511 diffraction of. See Diffraction discrete, 10 electromagnetic, 312 electrons and, 318 energy of. See Energy levels, atomic photons and, 312 quanta of, 20 radioactivity and, 323-363, 485-488 safety, 485-488 spectral analysis. See Spectroscopy Standing waves, 156 use of, 296 waves, See Waves See also specific effects, types, equipment Radiofrequency field, 260-262 Random events, 401-409 Random vanables, 428 Raudom walks, 123 Range, of particle, 308 Rayleigh range, 158 Recombination regime, 79 Recursion methods, 438 Reflection coefficient, 106 Reflection grating spectrometer, 25, 26 Refractive indices, 25, 203 Relative phase, 97 Relativistic particles, 304 Relaxation, of moments, 262-267 Resistivity. See Electncal resistance Resonance, 99, 141, 264, 284, 390 Resonant frequency, 141 Rotation, of fields, 259-262 Rubidium, 218, 243-246 Russell—Saunders coupling, 40 Rutherford cross-section, 310 Rutherford experiments, 367 Rydberg constant, 22, 29 Sample space, 424-426 Saturation spectroscopy, 243, 245, 262-265 Scanning spectrometers, 177 Scattering experiments, 367-421, Sée Specific types, effects Schrodinger equation, 20, 34, 233 energy eigenvalues, 50-51 hydrogen-like atom, 34 stationary states, 218 Scintillation counter, 333-344 index 525 Selection rules, 35—36, 222, 226, 252 Self-absorption, 237 Semiconductors, 71-81 bulk detectors, 345 diodes. See Diodes dopants, 72, 74, 101 electrons and, 72 energy bands and, 72 extimsic carriers, 72 Ferm level, 75 Hall effect, 63—70 holes, 54, 72, 76 impurities, 72, 74, 10% junctions, 75-78 properties of, 71-78 valence band, 53, 72 See also specific types Sensitive volume, 349 Series circuits, 91 Shockley array, 52 Signal analyzer, 142 Signal-to-noise ratio, 419 Silver, 363-364 Single-mode fiber, 212 Slater-Kimbal] spacing, 52 Shts, diffraction in, 180-184 Snell's Jaw, 162 Sodium, 33-43, 53, 54 Solder, use of, 107 Solenoid, 59 Solid angle, defined, 368 Solid-state detectors, 344-353 Solid-state materials, 46 Spatial filtering, 201 Spectroscopy, 2, 146, 147f, 177 crossover lines, 245 diffraction, See Diffraction gratings, 25—28, 198 . 526 Index Spectroscopy (continued) hfs. See Hyperfine structure high-resolution, 215-250 line width, 237f, 236-238 magnetic fields, 221f photomultipher tube, 493f rubidium, 243-250 selection rules, 35-36 self-absorption in, 237f sensitivity, 493 spectral lines, 215, 221f, 228, 236, 237f See also specific types, elements Spherical wavelets, 185 Spin, 39 ESR. See Electron spin resonance magnetic field and, 256f NMR. See Nuclear magnetic resonance nucleus, 256f spin-lattice effects, 265 spinning sample technique, 267 statistics and, 45 Stability analysis, 135n Standing waves, 156, 289, 432 Stationary states, 218 Statistical mechanics, 45 Statistics, theory of, 423-473 Stefan constant, 459 Stellar spectra, 36 Sterm—Gerlach experiment, 220n Stirline’s formula, 443 Stokes equation, 3, 7, [0n8 Superconductors, 81-88 Sweep generator, 111] ‘Temperature conductivity and, 51] intrinsic carriers,. 74 low temperature approximation, 74 noise and, 125, 512 tesistivity and, 63, 105 viscosity and, 7 Thermocouples, 16 Thermodynamic properties, 45 Thomas precession, 224n Thomson cross-section, 313, 314f, 372 Time-dependent perturbation, 257 | Time-to-amplitude converter, 406 ‘Time-to-analog converter, 114 Transform plane, 199 Transistors, 99-102 Transmission grating, 199 Transverse modes, 158 Turbulence, 133 Uncertainty principle, 267, 387011 Yalence band, 72 Vapors, atomic, 1 Variance, defined, 432 Verdet constant, 204, 204f, 207, 208 Viscosity, 7 Voltage divider, 92, 94f, 96f Waves, 22 antisymmetric, 45 diffraction of. See Diffraction generation of, 109-111, 116, 128 seer eo em ee a ei) ree) a eek a at ee ar] ay Cae ey ee cas ae ee Tae he aoe ee Carat) ie a ied weet Ce ay ea ge ae ae aaa ge ree iy ra phase, 187 radiation and. See Radiation recording of, 114 wave function, 211 See also specific parameters, types Wien displacement law, 51] Work function, 13, 492 X-rays, 372 Xenon-methane counter, 392 index : GS-88 Se es Young two-slit experiment, 193f Zeeman effect, 203, 215-228 magnetic resonance, 254 mercury and, 238—242f Miéssbauer effect, 396 norma), 223