P H Y S I C A L R E V I E W V O L U M E 7 2 , N U M B E R 9 N O V E M B E R 1 , 1 9 4 7

Compton Scattering and the Emission of Low Frequency Photons

R E S JOST*

Swiss Federal Institute of Technology, Zurich, Switzerland (Received July 7, 1947)

According to the perturbation theory of the scattering of photons, the cross section for Compton scattering diverges, owing to the multiple scattering effects for low frequency photons. I t is shown that by the application of the method of Block and Nordsieck this dif­ficulty can be evaded, and that, to a first approximation, the cross section for Compton scat­tering is given by the Klein-Nishina formula.

1. INTRODUCTION

RECENTLY Eliezer1 showed that by treating the multiple scattering of a photon col­

liding with a free electron one meets with a diffi­culty similar to that encountered in the theory of bremsstrahlung. This difficulty arises from the unjustified application of the perturbation method to low frequency photons. The cross section for double scattering becomes propor­tional to K~zdzKdQ,y assuming that one quantum with wave number K is scattered into the volume element dsK of momentum space, while the other is scattered into the solid angle dti. Integration over K yields an infinitely large cross section for Compton scattering. We shall show that it is possible to escape this difficulty by performing a canonical transformation similar to that intro­duced by Bloch and Nordsieck2 for the brems­strahlung theory. To a first approximation the cross section for Compton scattering is then given by the Klein-Nishina formula.

2. CROSS SECTION FOR DOUBLE SCATTERING ACCORDING TO THE PERTURBATION THEORY

The process discussed here is the following: a primary photon ko collides with a free electron with momentum p0 and energy TV In the final state, two photons k and K are assumed to be present and the energy and momentum of the electron are T and p, respectively.3 Since we are only interested in the cross section for low fre­quencies of K, we can develop this quantity into a series of ascending powers of K and retain only

* A part of this investigation was elaborated in con­junction with Mr, Bengt Holmberg from Lund, during his stay in Zurich.

1 C. J. Eliezer, Proc. Roy. Soc. A187, 210 (1946). 2 F . Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937). 3 Natural units will always be used; h — c — \.

the lowest order term; i.e., the term in K~~3, which is the only one that gives rise to difficulties. The matrix elements for the absorption of k0 and the emission of k and K are, respectively,4

Hko= —e(2Tr/ko)*(ui*aoU2),

11^ ^-e(2ic/k)*(ui*aU2),

IIK— — e(2w/ K)%(UI*CCU2) ,

(i)

where a0 —(ae0), a= (ae ) , a = (ae), and e<>, e, e are unit vectors in the direction of polarization of k0, k, K, respectively; %\ and u2 are normalized spin functions. The matrix

H=E HAZ\H.Z\ZJEIZ?,F

(EA-Ezi)(EA-Ezt) (2)

is responsible for the transition probability. The sum has to be extended over all permutations of the three processes as well as over both spin orientations and signs of the kinetic energy for the electron in the intermediate states. Develop­ing Eq. (2) in a power series in K, we need retain only the lowest order term proportional to K~*. From (1) it is then evident that the denominator in (2) has to be proportional to K. This is only possible if the quantum K is either emitted at the beginning or at the end of the sequence of the three processes, and then only if the inter­mediate state belongs to a positive energy of the electron. If, for example, K is emitted at the beginning, we have the following states:

Zi' [pi = po —Kjk0, K] , Z2': [p2 ' = po+ko-K;K],

Z 2 " : [p2" = P o - k - K ; k 0 , k , K ] .

4 W. Heitler, Quantum Theory of Radiation (2nd ed.) (Oxford University Press, London, 1944) p. 96,

815

816 RES JO S T

From the above we find

E-EZ1=~-(PO,K)/TO+->* where

(po, K) = 7 V - ( P O K )

Further we have

HAZI— — e(2w/ ic)*(uo*aUi)

where u\ belongs to positive kinetic energy and is an eigenfunction of [a(p0 — K ) + / 3 W ] . By treating («K) as a small perturbation, one gets for u\ a development into a power series in K. We are only interested in the unpertubated eigenfunction, which is UQ. Then (ep0)/To can be substituted for a. The leading term to (2) there­fore becomes

e(2T/K)*tt6f>o)/(po,.K)yS[c (3)

where He is the matrix element of the Compton effect. In the same way, for the case in which K is emitted at the end, we obtain

-e(2xA)*[(ep)/(£, <c)]. (4)

The quantities Tt p are, respectively, the energy and momentum of the electron in the final state F. Combining (3) and (4) we get

H=6{2T/K)* (ep0) (ep)

Hc+-• (Po, K) (p, K)

The corresponding cross section is

d$ = (2ir)~~zdzKw(Ky e)d$c +

(ep0) (ep) with

W(K9 e) = 2wK~1e2

(Po, K) (P, K)

(5)

(6)

(60

d&c denotes the cross section of Klein and Nishina.5 Summing over both directions of polarization e gives

with

W(K) =27r/r"1e2

d$^(2ir)-Wicw(K)d§c

po2 sin2^0 p2 sin V

(7)

(Po, *)2 (P>*)

([po, K][p, K] ) ) 2(LP0,KJLP,KJ)|

*2(p0, K)(P, K) V

<po is the angle (K, p0); <p is the angle (K, p). 5 d&c is the relativistically invariant cross section as

denned for example by C. Miller: Kgl. Danske Vidensk. Selsk. Mat.-fys. Meddel. 23, no. 1 (1945), Section 2.

Equation (7) is, as it must be, invariant under Lorentz transformations since d3Kw(K) may be written in a form which is evidently Lorentz invariant.

2ire2K~"1dzK

with

(Po, P) in* m*

I (PO,K)(P,K) (P,,K)2 (P,K)*

(Po, P) = ToT- (pop).

— (7")

In this section we gave a new derivation of Eliezer's result. At the same time we found that the matrix element for the emission of K may be split into two parts

in such a way that only the first part is re­sponsible for the infra-red difficulties. The second part, HK"9 may be treated along the lines of the usual perturbation theory. This splitting can be done in a more convenient way by starting with the Hamiltonian of radiation theory6

H=(ap)+Pm-e(aA(z)) + Y„N,k, (8)

where A(x) is the transverse part of the vector potential, z, p are the position and momentum vectors of the electron, respectively, Nv is the number of photons with wave number k„ and polarization e„.

Expanding A(x) in a fourier series, we get:

A(x) = L(2fc)"i2rA,(x) +conj. with

A„(x) = e„(27r)* exp[i(kvxv)2,

[<Z»> <Z*'*3 = ^V' and qv*qv = Nv.

Introducing &9 — e(2v/kw)*eVf

the Hamiltonian reads

H = («p)+j8m —£,(aa*)gy exp[i(k„z)]

+conj.+£„ Nvkv

(9)

(10)

(11)

(12)

(13)

Now we split the photons into high and low frequency ones. The high frequency (h.f.) photons are denoted by k and by an index t whereas the low frequency (l.f.) photons are

6 W. Heitler, reference 4, p. 91.

C O M P T O N S C A T T E R I N G 817

denoted by K and a running index s; the number of l.f. photons is ns. To define the l.f. photons, we require that /c<co<<C&o in the center of gravity system (c.g.s.). A more precise definition of the l.f. quanta will be given in section 3. We now perform a canonical transformation involving only the l.f. quanta: F' = S^FS with

5 = « P L » , ( a ) ] (14) which gives

p '=P+L.» .K . , (15)

qj = qs exp [i (KSZ) ] . (16)

Omitting the primes the transformed Hamil-tonian becomes:

11= (op) +pm - E,(«a.) (g.+g.*)

+ Z)s ns(Ks~- (OK,)) — ]C<(«a<)2<

The a in the terms involving the l.f. quanta are split into two parts7

a = ( p / D + «i with r = + ( w 2 + />2)*, (1.8)

£T= («p)+)8w-E.[(pa.)/r](g.+(Z.*)

+ L* »«(£, ^)/r-E«(aaOg«

Xexpp(k|Z)+conj. + E*iVi*i+-Hi, (19)

•ffi= —E»(aia«)(2«*+Q!«) —S ««(aii««). (20)

It is easily seen that already the terms of (19) without Hi give rise to the cross section (7). Hi does not lead to any difficulties if treated along the lines of the usual perturbation method. Therefore we shall omit Hi in the following.

3, THE BLOCH-NORDSIECK TRANSFORMATION

It is well known that the breakdown of the perturbation method in the low frequency range is closely connected with the fact that the probability for the emission of a finite number of l.f. photons under the effect considered (e.g. bremsstrahlung or Compton effect) vanishes. The perturbation method should be reliable if restricted to photons with higher energy. In fact, this is not generally true as, for example, Dancoff8 showed in the case of bremsstrahlung.

7 p/T is not exactly the even part of a, since T is always positive.

8 S . M. Dancoff, Phys. Rev. 55, 959 (1939).

But here the failure is connected with the still unsolved difficulties which have their origin in the high frequency region of the spectrum. It is known that neither Dirac's new method of field quantization9 nor the formalism of Heitler-Wilson lead to a solution of this problem.10

In spite of this, we shall restrict the Bloch-Nordsieck transformation to l.f. photons and use the perturbation method for higher frequencies. In fact, it does not help to mix the more formal difficulties connected with l.f. photons with the fundamental problems connected with high energy photons.

The Bloch-Nordsieck transformation consists of the following canonical transformation F' = S~lFS with S = eu

U=ZI(J>*.)/(P,*.)1(<1*-<1**) (21)

<Z.' = 2.-(pa.)/(/>, *,), (22)

(pa.) (pas)2

nj - ns- —(g,+g*) + . (23) (P,K.) ' " (P;K,y

The last term in (23) gives rise to a self-energy:

-E.(pa.)2 /7X/v«.)

e2 c d'^ic v2 sin2?? = I — . (24)

(2TT)2J K2 1-Z/COS#

where we have used the definition of a8 as given in (12) and summed over both directions of polarization; v = p/Ty & is the angle between K and p. In the c.g.s. we have to integrate over /c<a><$C&o and obtain:

e2 / 1 —v2 l + i / \ _ _ / 2 — — In )a>, (25)

2w\ v 1—v/

which is of the order a>e2 and therefore may be neglected. Further, it is consistent to neglect the energy-momentum balance for the l.f. quanta; i.e., to omit the sum 23s n8(p, K)/T. We shall see, that the expectation value of this term is small compared with k0 in the c.g.s.

9 W. Pauli and J. M. Jauch, Phys. Rev. 65, 255 (1944); W. Pauli, Helv. Phys. Acta, 19, 234 (1946).

10 H. A. Bethe and J. R. Oppenheimer, Phys. Rev. 70, 451 (1946).

Xexp[t(k*z)J+conj. + £« Ntkt. (17) leading to

818 RES JO ST

The reduced Hamiltonian reads:

H= («p) +p?n+Zt Ntkt+ V, (26)

where

V= -Ht{<Mt)<lteu exp[ i (k ,z)>^+conj . (27)

The primes have been dropped, and U is the function (21).

Transforming to plane waves for the electron one finds (Pauli and Fierz11 formulas (17)-(2la)):

(•••«,•• - , p | V\ • • •» / - - - ,p / )

= VoIL*(n.,v\K\n;,p') (28)

where Vo is the interaction matrix of the usual radiation theory referring only to h.f. quanta, whereas

(n,p\K\n'9 p;) =«-"/*(»!»'!)*

(i<s/w)n,'~n+2fl

momentum (which we have neglected) but this restriction is not serious because K is very small. Further we see that there is always an infinite number of l.f. photons emitted since ^sws

diverges. Always assuming that the quantum k is scattered into d& we get as the expectation value for the energy of the l.f. quanta

xE /x=o (n — fx)\fx\(n' — n+fx)\ (29)

r(p'a) (pa)-]2

W(K, e ) = • — UP',K) (P,K)\

is the same function as defined in (6'). In Eq. (29) 1/cr! is zero if <r<0.

4. THE CROSS SECTION FOR THE COMPTON SCATTERING

We start with the reduced Hamiltonian (26) and interpret p as the momentum of the electron and ns as number of the free l.f. photons, whereas the photons eliminated in (23) are bound photons giving rise to the self-energy (24). Along the lines of the usual perturbation theory we get the cross section for the effect, that a quantum k is scattered into the solid angle dQ> and that beside this ns l.f. photons are emitted. Using

£(»,p|2q»',p')(»\p'l*l»",p")

= («,p|2s:|»",p") d$(- ••«„•• ')=TJ.(w.nt/n.!)eru"-d$c.

(30)

The emission of the l.f. quanta obeys a Poisson distribution, that is, they are emitted inde­pendently. As a matter of fact, the numbers ns

are restricted by conservation of energy and

and for the momentum

(31)

(32)

Using the expression for ws and then summing over both directions of polarization, (31) and (32) becomes

4TT2 /

a3* r (£o, p)

e2 /» ddK r « = I K

4TT2 J K L

'(Po, K)(P, K)

m2

(Po, K) 2

(Po, P) ) '{Po, K)(P, K)

mi

\p,«A (33)

{pa,Ky (P,K) (34)

(e, «) is a four-vector since we integrate over a relativistically invariant domain: K<CO in the c.g.s. Equation (33) reads in the c.g.s.:

e(0) = (2e2/7r)w l + 2c

a ( l+« 2 )*

X l n [ a + ( l + a 2 ) i ] - l ) , (35)

11 W. Pauli and M. Fierz, Nuovo Cimento 15, 1 (1938).

with a = p/msin^Oi 9 angle between k0 and k. For a<<Cl

e = 8e2'0)-a2/37r-

In particular, for the non-relativistic case

€=(8e/37r)2-co-z;2sin2p, v = p/m (36)

while for a£M

e-(2e/7r)2o;{2ln2a-li . (37)

The expression (2/V) {• • • } in (37) is of the order of unity up to energies of 104 m in the laboratory system. In this energy range we may

C O M P T O N S C A T T E R I N G 819

therefore conclude

€«a>(«&o). (38)

From (34) it may be seen that m lies in the plane (po, k) and is symmetrical with respect to these two vectors, One easily finds:

T V - (p0«) - r - €(TT) sin2f # (39)

and from this

T \e(d)-e(ir)sin2^d\ | *(0) | = - —«*> . (40)

p cos|0

Now we have to show that, after performing the Bloch-Nordsieck transformation, it is possible to calculate the double Compton effect. At the same time, we get a lower limit for «. As already mentioned, a consistent calculation of the e6-correction to the Klein-Nishina formula is impossible since this correction diverges. From the point of view of the perturbation theory (as well as from experimental data), it is reasonable to assume (as long as damping effects are ex­cluded) that corrections to the Klein-Nishina formula (including corrections from multiple scattering) are of the relative order e2. It will be proved that under this assumption a closer deter­mination of co does not affect the results of physical interest, admitting errors of the relative order e2. To calculate the double Compton scat­tering, we start with the Hamiltonian (26). We write the cross section for the emission of a second quantum k' (k'>co in the c.g.s.) into the element dzkf in the form

d&(k') = Q(k')d*k,d$> (41)

with d$ from (30). Assuming k'<ZJd0 in the c.g.s. we may expand Q(kf) in a power series in kf and restrict ourselves to the lowest order term. As in Section 2 we get:

d*kf

QQs!)dW = w(k', e') + • • • (42) (2TT)3

with wQsf, e') as in (60, e' denoting the direction of polarization of k'. The correction to the .Klein-Nishina formula due to the double scat­tering is:

JT.Q(P)d>k',

with the summation sign X) denoting summation over both directions of polarization. The term (42) gives a contribution of the order of mag­nitude

r d*k' E w(k', e)S(€/«) InOVco)-^2 ln(*0/«),

J (2TT)S

with e determined from Eq. (35).

It is reasonable to assume that the other terms in (44) give contributions of the order of mag­nitude e2. We have, therefore, the condition

^ln(^0 /cu)«l . (43)

The value of co can be chosen so small that the use of development (42) may be justified. If (43) is fulfilled we get for the probability that under a Compton effect, in which a quantum k is scattered into dtt and a second quantum k' (k'<Kko in the c.g.s.) with polarization e' is emitted into the element dzkf:

w(k , ,e ,)^3^7(2 ?r)3 .

The expectation value of energy momentum emitted into dzkf reads

*'w/(k', e/)d8*//(2ir)«l

k /w(k ,,e ,)d8*7(2T)8, (44)

which is identical with Eqs. (31) and (32). Con­sequently the choice of co is arbitrary provided that condition (43) is satisfied. This discussion can also be developed by using the function 5(e,«) introduced by Pauli and Fierz.12 We are interested in the cross section for the process that a quantum k is scattered into the solid angle d£l, while the sum of energy momentum of the l.f. quanta lies between e and e+de, « and iz+die. This cross section can be written in the form13

d$ = S(€,*)dei*«»c (45)

where d$c is the cross section of Klein and Nishina, while

S(e, Tz)ded*T = £ t t i , n ^ l / » « ! - (46)

12 W. Pauli and M. Fierz, reference 11, p. 13. 13 It has to be noted that Pauli and Fierz take into

account the correction of the cross section due to the energy loss €, whereas we neglect the influence of energy-momentum loss (€ ,«) . Compare (26) where we have neglected the term Zn9T

y~1(pSf K,).

820 RES JO ST

The sum has to be extended over all quanta As C<3C1 it is easily seen that S(e) decreases very which fulfill the inequalities: rapidly as e increases,

^s^ \s . \ i ^-v- ^ x J To discuss the influence of the choice of «, we go back to Eqs. (47) and (48). Again we have to

It is easily 6een, that14 take into account the double scattering. Writing the cross section in the form:

5(€, TT) = (2TT)~4J dxd*x d i ^ 5 ( € > w ) d r f i 1 r t » c (55)

Xexp[i(xo€~x«)]exP[/(xo, x)]» (4?) where ^ is the cross section for which apart w j t ^ from the quantum k which is emitted into dQ,

there is an energy-momentum loss (e, w; e+de, j(Xo? £= ^Y.(2TT)~"WKW{K, e) tt+dto) arising from emission of If. quanta K or

J ' from additional emission of a quantum kf. Under X {exp[-i(xoK-~(2CK))]-l}. (48) condition (43) restricting k' to an interval co<k'

r r u ^ . - . u - <coi<<C o in the c.g.s., we get: 1 he sum 2* again denoting summation over both directions of polarization. If (%o, 2c) transforms __ n like a four vector, ./(xo, 2c) *s invariant, as is 5(c, «) =5(e, « ) + I ]C5(€—T?, >x— <) 5(e, «). If we are only interested in the energy loss in the c.g.s. we may define Xw(C» e ' ) ^ —f)dM3f. (56)

/

/• Putting d37r5(€, *) - (1/2*0 I dxe^efM (49)

with S(e,<x) = (2ic)-*jdx«l*x

/ (x )= f E W V ( 2 x ) » > ( K , e ) ( ^ x i c - i ) . (50) J XexpL^(xo€-(3c«))J6r(xo, x) (57)

one finds The integration over the angles yields (compare Eqs. (33) and (35)): G(Xo, X) = exp[/(xo, x)] 1 + j £ ( 2 i r ) - W

/(x) = c f dic(e-**-l)A (51)

with Xw(k', e') exp[~-i(xo^~(xkO] (58)

2e2( l+2a2 1 which, to a good approximation, is equal to £ - . - ^ _ j _ - - i n ^ - ^ ^ + ^ i j — i i (52)

T U ( 1 + a 2 ) 5 J . -pC/(xo,x)+/ ' (xo,2c)] The integral (51) can be evaluated by using with tabulated functions, but it is easier to introduce „ another method of cutting off at the frequency f(xo, x) = I 7L(2ir)~*d*k'w(k', e') (o>): introducing the function erK,<* we get

«dK X{exp[~-i(xo^-(xkO)]}, (59) /(x) = C | ~-*r<i»(eri*<--l)=ln(l+ixu)-c (53)

J0 K where co <fe'< o)i in the c.g.s. Therefore we get the same result as if we had

which yields for S(e): extended the domain of the l.f. quanta to K < « I , 5(€)=co-1(e/o))<?~1«-^/r(C). (54)15 in the c.g.s.

The author wishes to express his sincere thanks * 4 C o m p a r e w - P a u l i a n d M- F i e r z ' r e f e r e n c e n> E^ to Professor W. Pauli, who suggested this 16 W. Pauli and M. Fierz, reference 11, p. 14, Eq. (36). problem, for many valuable discussions.