Compton Scattering. There are three related processes. Thomson scattering (classical) Rayleigh scattering (coherent)

Transcription

1 Comton Scattring Tr ar tr rlatd rocsss Tomson scattring (classical) Poton-lctron Comton scattring (QED) Poton-lctron Raylig scattring (cornt) Poton-atom Tomson and Raylig scattring ar lasticonly t dirction of t oton cangs, not its nrgy Plus Tomson and Raylig scattring ar only imortant at low nrgis wr t otolctric ffct dominats 1

2 Tomson Scattring In Tomson scattring an lctromagntic (EM) wav of frquncy f is incidnt on an lctron Wat ans to t lctron? Tus t lctron will mit EM wavs of t sam frquncy and in as wit t incidnt wav T lctron absorbs nrgy from t EM wav and scattrs it in a diffrnt dirction In articular, t wavlngt of t scattrd wav is t sam as tat of t incidnt wav

3 3 Tomson Scattring ( ) ( ) 4 T t owr subtractd from t incoming bam tis is 3 8 owr mittd sin sin owr/ara 8 8 cm r mc S mc P m E c a c P m t E a t E E F c B c E S π π σ π ω ω π π r r r

4 Raylig Scattring Raylig scattring is scattring of ligt from a armonically bound lctron Assuming SHO wit frquncy ω0 for an lctron in an atom σ Raylig σ 4 ω T om son ( ω ω ) 0 You may rcall t robability for Raylig scattring gos as 1/λ 4 Wy is t sky blu? 4

5 Comton Scattring Comton scattring is t scattring of ligt (otons) from fr lctrons 5

6 6 Comton Scattring Calculations T cang in wavlngt can b found by alying Enrgy consrvation Momntum consrvation ( ) 1/ 4 c m c E m c ν ν ν cosθ r r r r r

7 7 Comton Scattring From nrgy consrvation From momntum consrvation Eliminating ( ) ( ) ν ν ν ν ν ν ν ν ν ν m c c c c c m m c c m ) ( 4 4 θ ν ν ν ν θ cos cos c c c c r r ( ) ( ) θ ν ν ν ν cos 1 m c

8 Continuing on ν ν νν Comton Scattring (1 cosθ ) mc And using vc/λ w arriv at t Comton ffct λ λ mc ( 1 cosθ ) And /mc is calld t Comton wavlngt λ C mc m 8

9 Comton Scattring Summarizing and adding a fw otr usful rsults ar λ λ ν v 1 mc T ν ν m c cotϕ 1 ( 1 cosθ ) v m c v ( 1 cosθ ) θ tan 9

10 10 Comton Scattring T diffrntial and total cross sctions ar calculatd in a straigtforward mannr using QED Calld t Klin-Nisina formula ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) Ω ln 1 ln cos 1 1 cos 1 cos 1 cos γ γ γ γ γ γ γ γ γ γ π σ θ γ θ γ θ θ γ σ Comton r r d d

11 Comton Scattring On t rvious slid At low nrgis σ σ Comton At ig nrgis σ Comton T om son γ 8π r 3 v m c 8πr 3 ln γ 3 8γ 1 11

12 Comton Scattring Tus at ig nrgis, t Comton scattring cross sction σ C gos as σ Comton ~ Z v 1

13 Comton Scattring Graically, dσ/dω 13

14 Comton Scattring In olar form, assum a oton incidnt from t lft 14

15 Comton Scattring At ig nrgis, say > 10 MV, most of t otons ar scattrd in t forward dirction Bcaus of t ig forward momntum of t incidnt otons, most of t lctrons will also b scattrd in t forward dirction 15

16 Comton Scattring Concrning krma and absorbd dos, w ar articularly intrstd in t scattrd lctron bcaus it is ionizing W can slit t Comton cross sction into two arts: on giving t fraction of nrgy transfrrd to t lctron and t otr t fraction of nrgy containd in t scattrd oton 16

17 Comton Scattring σ C tr C tr T v v σ C σ C σ C ν ν sc v σ C σ C ν similarly for t mass nrgy transfr tr μc ρ σ T ν σ μc ρ sc C attnuation cofficint T N ν σ A Av C 17

18 Comton Scattring Hr σ n σ tr 18

19 Comton Scattring Anotr usful form of t diffrntial cross sction is dσ/dt, wic givs t nrgy distribution of t lctron 19

20 Comton Scattring T maximum lctron kintic nrgy is givn by T max v T and for v T γ v and γ 1 γ max max γ v 1 1 γ v larg mc v m c 0.555MV vmc m c v 0

21 Comton Scattring In cass wr t scattrd oton lavs a dtctor witout intraction on would obsrv 1

22 Comton Scattring

23 Comton Scattring v v θ π θ π 1 v v / 55kV m c mc 3

24 Pair Production Pair roduction is t dominant oton intraction at ig nrgis (> 10 MV) In ordr to crat a air, t oton must av > m 1.0 MV In ordr to consrv nrgy and momntum, air roduction must tak lac in t Coulomb fild of a nuclus or lctron For nuclar fild, E trsold > x m For atomic lctron fild, E trsold > 4 x m 4

25 Pair Production 5

26 Pair Production Enrgy and momntum consrvation giv Enrgy f E Momntum (x) Momntum (y) E Enrgy consrvation can b r-writtn But momntum consrvation (x) sows Tus nrgy and momntum ar not simultanously consrvd f cosθ c 0 sinθ cosθ sinθ 4 f c m c c f max c c m c 4 6

27 Pair Production T rocsss of air roduction and brmsstralung ar rlatd (crossd rocsss) Tus w d xct t cross sction to dnd on t scrning of atomic lctrons surrounding t nuclus Dos t oton s nuclar carg Z or 0 or somting in btwn? T rlvant scrning aramtr is ξ 100m c E E Z v 1/3 7

28 Pair Production In t Born aroximation (wic is not vry accurat for low nrgy or ig Z) on finds No scrning σ 4Z ( ξ >> 1) 7 9 Comlt scrning σ air air 4Z αr αr 7 9 ln and ν m c m c ( ξ 0) ( Z ) ( 1/3 ln183z ) f ( Z ) f << ν << 137m c and ν >> 137m c 1 54 Z Z 1/3 1/3 8

29 Nots σ air ~ Z Pair Production Abov som oton nrgy (say > 1 GV), σ air bcoms a constant In ordr to account for air roduction from t Coulomb fild of atomic lctrons, Z is rlacd by Z(Z1) aroximatly sinc t cross sction is smallr by a factor of Z Usually w don t distinguis btwn t sourc of t fild 9

30 Nots Pair Production In t cas of t nuclar fild and for larg oton nrgis, t man scattring angl of t lctron and ositron is θ T For m c T ν 1.0 ν 5MV T MV andθ o 15 30

31 Pair Production T robability for air roduction 31

32 Pair Production m (1.0 MV) of t oton s nrgy gos into crating t lctron and ositron T lctron will tyically b absorbd in a dtctor T ositron will tyically anniilat wit an lctron roducing two anniilation otons of nrgy m (0.511 MV) ac If ts otons ar not absorbd in t dtctor tan t air roduction nrgy sctrum will look lik 3

33 Pair Production 33

34 Pair Production Similar to t otolctric ffct and Comton scattring w dfin t mass attnuation and mass nrgy transfr cofficints as μ air ρ N A Av σ air μ tr air ρ v mc v μ air ρ 34

35 Potonuclar Intractions Hr a nuclus is xcitd by t absortion of a oton, subsquntly mitting a nutron or roton Most imortant wn t nrgy of t oton is aroximatly t binding nrgy of nuclons (5-15 MV) Calld giant nuclar diol rsonanc Still a small fraction comard to air roduction owvr 35

36 Potonuclar Intractions Giant diol rsonanc 36

37 Potonuclar Intractions Ts intractions would b obsrvd wit igr nrgy x-ray macins A 5 MV x-ray bam will contain nutron contamination from otonuclar intractions Small ffct comard to t oton bam itslf Also imortant in dsigning silding sinc ~MV nutrons ar difficult to contain 37

38 Tyical oton cross sctions Poton Intractions 38

39 Tyical oton cross sctions Poton Intractions 39

40 Poton Intractions Nots Of cours diffrnt intractions can occur at a givn oton nrgy σ σ Zσ σ μ ρ μ ρ Zμ Comton Comton A olynrgtic bam suc as an x-ray bam is not attnuatd xonntially Lowr nrgy x-rays av igr attnuation cofficints tan igr nrgy x-rays Tus t attnuation cofficint cangs as t bam rocds troug matrial An ffctiv attnuation lngt μ ff can b stimatd as μ ff HVL ρ air μ air ρ 40

41 Bam Hardning 41

42 Poton Intractions Lt s rturn to our first slid I I 0 μx As w v sn in t diffrnt oton intractions Scondary cargd articls ar roducd Potons can los nrgy troug Comton W dfin Narrow bam gomtry and attnuation Only rimaris strik t dtctor or ar rcordd Broad bam gomtry and attnuation All or som of t scondary or scattrd otons strik t dtctor or ar rcordd Effctiv attnuation cofficint μ < μ 4

43 Poton Intractions 43

44 Poton Intractions In idal broad bam gomtry all surviving rimary, scondary, and scattrd otons (from rimaris aimd at t dtctor) is rcordd In tis cas μ μ n 44