The Evidence for the Photon

Transcription

1 The Evidence fr the Phtn The Atmic Nature f Matter and the Elementary Structure f the Atm The first evidence that nature is quantized arse frm the realizatin that there ere nly a small number f fundamental elements that cmbine in fixed ratis t prduce all the kn substances f nature. In the latter 19th century physicist applied this cncept f the atmic nature f matter t a gas cnfined in a vlume and ere able t derive the ideal gas la using nly a fe very simple assumptins. These and ther bservatins ere strng evidence that the rld in hich e live is quantized. As e shall see in hat flls, the quantum nature f things is nt restricted just t physical material, but manifests itself in many different areas - quantized energy levels, quantized spatial rientatins, etc. Many f the experiments perfrmed in the latter 19th and early 20 century demnstrated this quantum nature f ur rld. It is ur task t examine sme f the majr experiments frm that perid f time. J. J. Thmpsn' Experiments With Electrns

2 Evidence fr Phtns 2 Blackbdy Radiatin Max Planck's explanatin f blackbdy radiatin presented at the German Physical Sciety n December 14, 1900 may be cnsidered the beginning f quantum thery. In this sectin e ill briefly utline the thery f blackbdy radiatin and the events hich led up t Planck's histric paper. All bjects radiate and absrb energy in the frm f electrmagnetic radiatin. An bject in equilibrium ith its surrundings must necessarily emit and absrb equal amunts f energy. If this ere nt s, the bject uld either heat up as ther bjects cled ff, r vice versa! This is basically a statement f Kirchhff's la f absrptin and emissin. We define VX ( ), the radiated per (r radiant flux density) frm a heated surce, as the ttal amunt f energy radiated in all avelengths frm the surce per unit time per unit surface area f the surce. In 1879 Jseph Stephan determined experimentally that % V( X ) = -98=>+8> X here X is the abslute temperature. Tday e express this equatin in the frm VX œ%5 X % here % is the emissivity and depends upn the emitting surface, and here 5 is a universal cnstant (n knn as the Stephan-Bltzmann cnstant) given by: -8 % 5 = 5.67 x 10 Wm -K Objects ith an emissivity equal t unity are knn as blackbdies. This is because an bject in thermal equilibrium must emit and absrb equal amunts f radiatin. Thus, % is als the absrptin cefficient, and if the absrptin cefficient is equal t unity, then the bject must absrb all the radiatin incident upn it, s it is a "black" bdy! EXERCISE: Assume that the sun's surface temperature is 6000 K and determine the equilibrium temperature f the earth, assuming that bth the sun and the earth are ideal emitters and absrbers (i.e., % = 1). The avelength r frequency f the energy emitted depends upn the temperature f the bject. Thus VX ( ) = ' VÐ, X) d, here V(, X), the spectral radiant flux density, is the amunt f energy radiated per unit time per unit area in the frequency interval beteen and + d. The distributin functin fr radiated energy as a functin f frequency as first determined experimentally by Lummer and Pringsheim in 1899, and is shn in the flling figure.

3 Evidence fr Phtns 3 40 Blackbdy Radiatin Curves fr Different Temperatures. 35 Spectral Intensity (arbitrary units) K 3000K K frequency (arbitrary units) As this figure shs, hen an bject is heated up, the ttal amunt f energy radiated frm the bject increases (the area under the curve increases), and the peak in the radiatin curve shifts tard higher frequencies (r shrter avelengths, since - œ c " Ê - º ). Thus, an bject at rm temperature radiates in the far infra-red regin f the spectrum (invisible t the naked eye), hile a much htter bject might gl red-ht, and a still htter bject might gl hite-ht (i.e, the radiated light uld cntain much mre f the visible spectrum). [This simple bservatin explains hy red stars are cler then blue stars.] It as demnstrated by Wein that the frequency at hich V(, X) is a maximum is prprtinal t the abslute temperature, r 7+B ºX This is called Wein's displacement la and can als be ritten in the frm - 7+B º " X Based upn classical thermdynamics, Wein argued that V(, X) = J( X) $ V (,T) shuld have the frm here J( X) as sme functin dependent nly n the rati ÎX. Wein shed that the empirical frmula as f the frm $ V(, X ) = + exp[-, 5X],

4 Evidence fr Phtns 4 here + and, are arbitrary cnstants, and here 5 is the Bltzmann cnstant. This expressin fr the equatin indeed gives Wein's displacement la and seemed t fit the existing experimental data (fr l frequencies) reasnably ell. EXERCISE: Beginning ith Wein's expressin fr the spectral radiant flux density derive the Wein displacement la and the Stephan-Bltzmann equatin. An ideal blackbdy can be cnstructed using an enclsed cavity hich has nly a small hle pen t the utside. Any light hich falls n this hle ill enter the cavity and be cntinually reflected inside the cavity until it is eventually absrbed, because there is little likelihd that the light ill be reflected back ut thrugh the small pening. This is truly a blackbdy (a black hle). But if the bject cntaining the cavity is heated, the pening ill als act as an emitter f radiatin, and ill emit a blackbdy spectrum. N the amunt f energy hich exits the hle per unit area f hle per unit time in a particular frequency range is given by V(, X) and must be related t the ttal energy hich is cntained in the cavity. In fact, the energy density?(, X) times the speed f light (the rate at hich energy ill leave the pening) has units f energy per unit area per unit time s that e expect V(, X ) º -?(, X) - ÒIt is in fact given by V ßX œ %? ßX ÓÞ This means that the energy density ithin a cavity radiatr shuld fll Wein's la and be given by? ßX º + $ exp[, 5X] [Nte: The arbitrary cnstant + in this equatin is nt identical ith the ne in the previus equatin. The cnstant - and thers have been absrbed int this +.] II. THEORETICAL ATTEMPTS TO EXPLAIN BLACKBODY RADIATION In an attempt t derive theretically the energy density distributin functin?(, X), Rayleigh and Jeans assumed that the interir f a blackbdy cavity as made up f perfectly reflecting material (this pses n theretical prblem, since all blackbdies can be shn t be equivalent). Thus, the electrmagnetic energy reflected back and frth inside the cavity must be treated smething like standing aves (ith ndes at bth ends). The trick is t determine the number f different ays standing aves can be set up inside the cavity fr a given frequency interval (beteen and d ), and t determine the average energy density that such a ave can have. Classical arguments indicate tha the average energy stred in a cavity is equal t 5X times the number f pssible mdes f scillatin f the aves trapped in the cavity ØIÙ œ ØIÙÎ79. 8?7,< = N the number f mdes f scillatin depends upn the frequency f the ave trapped in the cavity. Classical arguments indicated tha the number f mdes f a transverse

5 Evidence fr Phtns 5 ave ith a frequency beteen and. is given by ) 1. Á transverse œ Z. -$ The average energy in a given frequency range, then, is given by ) 1 ØE( ) Ù d œ Z 5X. -$ and the average energy density (energy per unit vlume) in that frequency range is given by ) 1?. œ 5X. -$ A Cmparisn Of Experiment And Thery Wein's emperical la fr the energy density f a blackbdy radiatr? ßX. º + $ exp[, 5X ]. is nt the same as the energy density derived by Rayleigh and Jeans, based upn classical argurments. In fact, their equatin ) 1?, X. œ 5X. -$ appears t be quite different. Pltting these t functins ith respect t the frequency (see bel) e see that the t functins agree fr l frequencies, but at high frequencies, the Rayleigh-Jeans la fails badly. 5 Energy Density as a Functin f Frequency 4 Energy Density (arbitrary units) 3 2 Rayleigh-Jean's La 1 Wein's La Frequency (arbitrary units)

6 Evidence fr Phtns 6 Planck recgnized a cnnectin beteen the Rayleigh-Jeans la and Wein's la. If e rite Wein's la in the frm? ßX œ $ + expc, Î5X d and expand the expnential using the Taylr series, e btain? ßX œ + $, " ˆ, â 5X 5X In the limit as, Î5X becmes small enugh t ignre terms higher than first rder, e can see that this equatin is almst exactly the Rayleigh-Jeans la - except fr the factr f ne " in the denminatr. As a result, Planck prpsed that the crrect frm f the blackbdy radiatin equatin shuld be? ßX œ $ + expc, Î5Xd " hich has the frm f the Rayleigh-Jeans la fr l frequencies (lng avelengths) and the frm f Wein's la fr high frequencies (shrt avelengths). T make this equatin have the exact frm f the Rayleigh-Jeans la at l frequencies, e must chse the emperical cnstants + and, in Wein's equatin apprpriatelyþ In the limit f small, Î5X, e have hich means that $ $ + + +? ßX œ p œ 5X c, Î5Xd ",, exp + ) œ 1, -$ This gives as the prper frm f the Blackbdy radiatin equatin ) 2? ßX. 1 œ. -$ expc2 Î5X d " here e have replaced, ith 2 s that it is in the traditinal frm f Planck's equatin. We can determine the value f the cnstant, by integrating the energy density ver all frequecies. Yu shuld ntice that 2 must have units f energy (the same units as 5X) and that the term 2 expc2 Î5Xd " has taken the place f 5X in the Rayleigh-Jean's la. Yu ill recall that 5X as the classically predicted average energy f an scillatr. Thus, Planck's equatin implies that 5X

7 Evidence fr Phtns 7 the crrect frm fr the average energy f an scillatr is given by 2 ØIÙ œ expc2 Î5Xd " Ntice that this expressin fr the energy is identical t the classical expressin in the limit as 2 ges t zer! In fact, Planck fund that the classical calculatin f the average energy culd be frmulated t give the result abve, ith 2 being the smallest increment f energy. Classically, this smallest increment is zer (i.e., e let 2p! in the expressin fr the energy, 2 ), and e btain the classical value fr the average energy, 5X. The experimentally measured blackbdy spectrum, hever, demands that 2 cannt g t zer! Thus, e find that nature has a ler limit n energy increments: energy cannt be subdivided int units smaller than 2! Planck's cnstant is bviusly related t the Stephan-Bltzmann cnstant. If e rite V ßX œ -? ßX œ 1 2 % - 2 Î5X " e can find the relatinship beteen the Stephan-Bltzman cnstant 5 and Planck's cnstant 2, by integrating ver all frequencies, giving 1 2 VX ( ) = ( VÐ, X) d œ (. œ5x - 2 Î5X " The value f Planck's cnstant turns ut t be 2 œ 'Þ''!(' "! J s "& œ %Þ"$&''* "! ev s $% %

8 Evidence fr Phtns 8 The Phtelectric Effect The phtelectric effect as first bserved in 1887 by Hertz as he as experimenting ith radi aves. Hertz discvered that a spark culd be mre easily initiated at a spark gap in the presence f ther sparks. As he studied this phenmena further, he fund that the enhancement f the sparks as due t ultravilet light, that the effect as mre prnunced if the terminals ere clean and smth, and that the negative terminals ere the mre sensitive t the incident light. Sn after Hertz' initial bservatins, thers began t study this phenmenn. A schematic f a typical vacuum tube used t study the phtelectric effect is shn in the diagram abve, and the results f sme f these experimental studies are listed bel. [Ntice the cntinuity f the names f persns h ere invlved in the research and the dates hen the research as reprted.] 1. The phtelectric effect invlves negatively charged particles. (Hallachs, 1889) 2. These charged particles are emitted frcibly by the light. (Hallachs, Elster, and Geitel, 1889) 3. A clse relatinship exists beteen the cntact ptential (r electrnegativity) f the metal and its sensitivity t light (phtsensitivity). The mre electrpsitive a metal the mre sensitive it is t lnger avelengths. (Elster and Geitel, 1889) 4. The phtcurrent is prprtinal t the intensity f the light. (Elster and Geitel, 1891) 5. The emitted particle as determined t be the electrn. (Thmpsn and Lenard, 1899) 6. The kinetic energy f an emitted electrn is independent f the intensity f the incident light, but the number f emitted electrns is prprtinal t the light intensity. (Lenard, 1902) 7. The maximum kinetic energy f the emitted electrns is greater hen the light incident upn the metal has a shrter avelength. N electrns hatsever are emitted if the avelength is lnger than sme maximum threshld value. (Lenard, 1902) 8. The phtelectrns ere bserved t be emitted frm the metal surfaces ith n appreciable time delay after the surce as turned n. G

9 Evidence fr Phtns 9 Althugh Netn had first pstulated that light as crpuscular in nature (i.e., that light cnsisted f small 'particles'), Yung's duble slit experiment had firmly established that light as a ave: it clearly exhibited interference phenmena. By the the early 1900's, the mathematical mdels used t describe aves and ave interference had been successful in explaining many prblems invlving light. Hever, many f the experimental bservatins cncerning the phtelectric effect culd in n ay be explained by the classical ave mdel f light. One particular aspect f classical ave thery is rth nting here. The intensity f light (a measure f the amunt f energy hich is incident upn a surface per unit time) is calculated by taking the square f the abslute magnitude f the amplitude f the electrmagnetic ave at a pint in space. This ave amplitude might cnsist f the summatin f several different aves cming frm different surces (hich is h ne explains the phenmena f interference), but the intensity f light at the pint f interest can nly be determined by taking the square f the abslute magnitude f the resultant electrmagnetic ave. As a simple example, cnsider a sine ave as a mdel fr light. We kn that the time average f the sine ave is zer if the average is taken ver a cmplete perid. Hever, the time average f the square f the sine ave ill nt be zer, and the classical mdel f light uld imply that if light is incident upn a surface fr a lng time, then the energy frm the light ave hich is imparted t the surface ill increase ith time. Hever, the bservatins f the phtelectric effect (in particular 6 and 8) uld seem t disagree ith this picture. One uld think that if the light surce ere turned n, the energy frm the surce uld cntinually be added t the metal surface until enugh energy as added t eject an electrn. This uld imply sme time delay beteen turning n the surce and the ejectin f an electrn. But there as n measurable time delay if an electrn ere ejected at all. And the energy f the ejected electrn as dependent, nt n h lng the light surce had been n, but rather n the frequency f the incident light, a cnditin hich as in n ay indicated by the classical thery. Rather classical thery uld predict that the phtelectric effect shuld ccur fr any frequency as lng as the surce ere n lng enugh. The pertinent bservatins cncerning the phtelectric effect are summarized by t diagrams bel. The first diagram plts the phtcurrent (the current f ejected electrns) as a functin f "retarding ptential" fr t different intensities f the surce (the surces bth being mnchrmatic surces f the same avelength). Each electrn " hich reaches the cllectr plate has an energy f 7@, the mst energetic being the nes hich reach the plate at. Thus, e have Z 9 " 7@ 7+B œz9 hich is independent f the intensity f the surce, but hich des depend upn the incident radiatin frequency.

10 Evidence fr Phtns 10 I I a b V 0 If e plt the cut-ff ptential fr a given metal as a functin f the frequency f the incident light, e btain the flling diagram, here the results fr t different metals are pltted tgether. Here e see that bth curves exhibit the same slpe, but different intercepts, depending upn the phtsensitive metal. Stpping Ptential (V) Metal 1 Metal 2 0 Light Frequency When Planck's paper n the quantum explanatin f blackbdy radiatin came ut in 1903, Einstein immediately sa that this idea f quantized energies culd be used t explain the phenmena f the phtelectric effect. He pstulated that light as made up f "phtns" each f hich has a "quantized" energy, % œ2, and that the intensity f light as given by 82, here 8 is the number f phtns present. This hypthesis uld immediately indicate that the energy f a phtn is directly related t the frequency f the incident light, and that the phtcurrent hich is bserved shuld be a functin f the number f phtns (the intensity f the surce) hich arrive at the phtsensitive metal. In this hypthesis, the energy f a single phtn, 2 uld be given t a single electrn, s that the electrn uld acquire a kinetic energy hich culd be determined by the equatin " 2 œ 7+B [ here [ is the rk functin fr a particular phtsensitive substance. N, since the maximum kinetic energy is equal t the electrstatic ptential energy Z 9, e can rite 2 œ Z9 [

11 Evidence fr Phtns 11 r, slving fr, e have Z 9 2 [ Z9 œ Œ and e see that this mdel fr the energy in an electrmagnetic ave fits the phtelectric effect data very ell. The fact that the slpe f the data fits the numerical value f 2Î as first demnstrated by Millikan hen he made an attempt t disprve Einstein's hypthesis. Einstein's theretical explanatin f the phtelectric effect prbably did mre t establish Planck's quantum thery than any ther phenmena. An interesting prblem has n been intrduced, hever. A prper descriptin f light evidently requires us t smeh blend the idea that light can be described as a ave but als as a discrete "particle". The reslutin f this "prblem" ill take up a cnsiderable part f ur study.

12 Evidence fr Phtns 12 X-rays X-rays ere first discvered in 1895 by Röntgen. In his first attempts t identify these mysterius rays Röntgen discvered: 1) they culd nt be deflected by a magnet, and s it seemed they ere nt charged particles; and 2) they exhibited n apparent diffractin r refractin effects, and s it seemed they ere nt electrmagnetic aves. Because he had n idea hat these rays ere, Röntgen cined the name x-ray. Althugh he did nt initially detect any deflectin f these x-rays by diffractin r refractin effects, Röntgen reasned that these rays ere electrmagnetic in rigin, since they ere prduced hen electrns ere accelerated tard a metal electrde. He reasned that the charged electrns ere decelerated ithin the target metal, and the decelerating charge uld radiate energy. Indeed, further study f these x-rays indicated that an x-ray beam as indeed slightly bradened hen passed thrugh an extremely narr slit. This seemed t indicate that the avelength f these x-rays must be extremely shrt. Electrmagnetic aves are diffracted mre hen they pass thrugh penings hich are cmparable in size t their avelength. Since the avelength f x-rays appeared t be very shrt, ne needed a diffractin grating ith very small spacing. In 1912 Laue prpsed using a crystal as a three-dimensinal grating since the grating spacing uld be n the rder f the atmic dimensin. Sn after this, Friedrich and Knipping ere successful in btaining a x-ray diffractin pattern. Their experiment cnfirmed Röntgen's belief that the x-rays ere indeed electrmagnetic aves and als cnfirmed the idea that the atms in a crystal ere arranged in a regular array. Using crystal gratings, the spectral intensity f x-rays culd be studied. These studies indicated that there ere t distinct patterns t x-ray emissins as seen in the diagram bel. The first type f x-ray emissin are the relatively sharp x-ray lines hich depend upn the type f target used in the x-ray tube. These lines typically ccur in pairs as indicated in the figure. Here a Oα and O" pair are indicated. Secnd, a cntinuus spectrum is bserved hich appears t have a shrt avelength limit, -. This part f the x-ray spectrum is knn as Bremsstrahlung radiatin.

13 Evidence fr Phtns 13 I K α λ ο K β Bremsstrahlung λ It as determined experimentally that the l avelength limit t Bremsstrahlung radiatin varied inversely ith the accelerating ptential applied t the x- ray tube. Assuming, as did Röntgen, that x-radiatin is prduced hen the bmbarding electrns decelerate near charged particles ithin the target, and that the radiatin is in the frm f phtns, e can argue that the energy f the phtn is the energy given up by the electrn as it decelerates. The maximum energy hich the electrn has is the energy given t it by the accelerating ptential applied t the x-ray tube. That energy is given by the electrn charge multiplied by the accelerating ptential. Thus the maximum energy phtn is given by giving B œ œ Z œ 2- Z in agreement ith the bservatins. Thus, the bserved spectrum f Bremsstrahlung radiatin als indicates that electrmagnetic aves can best be described in terms f phtns f energy 2. Frm his study f the x-ray emissins f different atmic species, Mseley fund a relatinship beteen the avelength r frequency f the O α line and the atmic number f the frm " ^œe WœFÈ α W È- α here ^ is the atmic number, - α and α are the avelength and frequency, respectively, f the emitted Oα line, and here E, F, and Ware cnstants. This relatinship is immediately derivable frm Bhr's mdel fr a hydrgenic atm.

14 Evidence fr Phtns 14 Accrding t Bhr's mdel, the frequency f an emitted phtn is given by ^I ^I " " 2 œ œ ^ I Fr fixed values f 8 and 8, e have ^œfèα hich is pretty clse t Mseley's equatin. If e realize that the transitins hich e bserve are arrising frm electrns is uter levels falling int the innermst level f the atm, e see that the nucleus f the atm is smehat shielded by electrns hich may surrund the nucleus. This shielding f the nucleus by the innermst electrns effectively reduces the psitive electric charge f the nucleus by sme amunt (prprtinal t the number f electrns shielding the nucleus?) hich e rite at ^ W, here W is the effective shielding number. It is fund that fr transitin t the lest energy level, the next highest level, W 7.4. W ". Fr transitins t

15 Evidence fr Phtns 15 Cmptn Scattering f X-rays and Gamma Rays As the study f x-rays cntinued, it as discvered that hen x-rays scatter frm slids, the avelength f the scattered x-ray is different frm the riginal avelength. Experimentally, ne uld use an x-ray surce ith a strng O α line f knn avelenth. The x-rays frm this surce uld be directed at a target and the avelength f the scattered x-rays uld be analyzed using a crystal as a grating. It as fund that the shift in the avelength as prprtinal t the scattering angle, accrding t the equatin - - º " cs) In 1925 Cmptn explained this avelength shift and derived the Cmptn equatin - - œ 2- " cs) 7- using the assumptin that the energy f the electrnmagentic ave as carried in a quantized packet hich e call the phtn. T derive this equatin Cmptn made use f the fact that a phtn must have mmentum as described by Einstein's thery f special relativity % I œ:- 7- œ:- even thugh the rest mass f a phtn is zer. Using the Einstein equatin fr the energy f a phtn, e btain frm hich e btain the relatin 2- Iœ2 œ œ:- - :œ 2 - Using this relatinship and the assumptin that phtns act smething like billiard balls hen they strike ther bjects, Cmptn applied the principles f cnservatin f mmentum and energy in a simple cllisin prblem t derive his n famus equatin. T begin the analysis f Cmptn scattering, cnsider the diagram bel. An incident phtn (avelength - Ñ cllides ith an electrn hich is initially at rest ithin the target material. The scattered phtn (avelength -') mves ff at an angle ) relative t the directin f the incident phtn, hile the electrn is given a kinetic energy X and mves ff at an angle 9 ith respect t the directin f the incident phtn.

16 Evidence fr Phtns 16 y λ' λ ο θ φ x We n apply the cnservatin f relativistic energy and mmentum t this prblem in bth the Band the C directin. Befre the cllisin, the energy and mmentum f the incident phtn is given by als, the target electrn has rest mass 7- given by : œ 2-2- I œ 2 œ œ : I œ 7 - œ: T, s the ttal energy befre the cllisin is - After the cllisin, the energy f the system can be expressed by the equatin The cnservatin f energy requires that 2- I œ? 7 - œ : - I - 0 I œ I 3 0 r : œ:- I frm hich e can derive the energy f the electrn I œ : : The cnservatin f mmentum requires that : œ : cs 9 : cs )

17 Evidence fr Phtns 17 and : sin 9 œ : sin ) This leaves us ith three equatins and fur unkns ()9,, I, I ß here : and : are nt independent unknns). We cannt slve this uniquely, but e can eliminate the dependence n the electrn energy and the angle 9. T eliminate the angle 9, e rite the t mmentum cnservatin equatins : cs 9 œ : : cs ) : sin 9 œ : sin ) N squaring bth equatins and adding gives : œ : : : : cs ) T eliminate the ttal energy and the mmentum f the electrn frm ur equatins, e make use f the relativistic equatin hich relates the ttal energy f the electrn t the mmentum f the electrn % I œ :- 7 - Simply plugging in the expressins fr the ttal energy f the electrn and the mmentum f the electrn, e btain : : œ : : : : cs ) % Expanding the left-hand side f this equatin, cancelling dividing by c gives Expanding : : % 7- : : : : 7 -œ: : : : cs ) and regruping gives :: " cs ) : : 7-œ! n bth sides, and N e substitute the relatin :œ2î-fr the mmentum f the phtns and btain " cs ) œ Œ - - hich can be simplified t the frm f Cmptn's equatin - - œ 2 " cs ) 7- N the cmbinatin f cnstants 2Î7 - must have units f length. By multiplying the tp and the bttm f this grup by - e btain 2- "%! Z 87 œ œ Þ%$ "! &"" "! Z $ $

18 Evidence fr Phtns 18 This is the s-called G97:>98 90 >2 6-><98. Ntice that the maximum change in avelength ccurs fr backscattered electrns, and the maximum change in avelength is nly tice the Cmptn avelength f the electrn - hich is very small. The Cmptn scattering equatin can als be ritten in terms f the incident and scattered phtn energies. By ding this, e can determine the energy given t the target electrn in the cllisin prcess. The maximum energy imparted t the electrn ccurs hen the phtn is scattered directly backard (at ) œ ")! Ñ. This means that the maximum energy that can be imparted t an electrn in the target material has a simple dependence upn the energy f the incident phtn. The maximum energy hich can be imparted t an electrn ithin a target material shs up in gamma-ray spectra as the Cmptn knee fr a given phtn. Exercise: Write ut an expressin fr Cmptn scattering in terms f the energies f the incident and scattered phtns. Using this expressin determine the maximum energy that can be imparted t a target electrn as a functin f the incident phtn.