The amplitude of nuclear photon scattering and its Thomson limit

2.1

[ I

Nuclear Physics A I 0 7 (1968) 655--658, (~) North-HollandPubhshmg Co, Amsterdam Not to be reproduced by photoprmt or microfilm without written perm~ssmn from the pubhsher

THE A M P L I T U D E OF N U C L E A R P H O T O N SCATTERING AND ITS THOMSON LIMIT R SILBARt CARL WERNTZtt and H OBERALLt

Departmentof Phystcs, The CathohcUmversttyof Amertca, Washmyton,D C Recmved 5 June 1967 Abstract: It is explicitly shown how, m the long wavelength llmtt of photon scattering by a nucleus, the scattering amphtude separates into a nuclear Thomson term, which remains fimte and represents scattering from the nuclear charge d~stnbutmn as a whole, and a nuclear Rayle~gh term ("nuclear resonance"), which vanishes as coz for w --~ 0 and represents excltatmn of the internal coordinates

In spite of the long standing of this problem, recently of interest again because of the development of high-Intensity p h o t o n beams, the T h o m s o n hmlt of nuclear p h o t o n scattering seems never to have been treated rigorously in the literature Previous w o r k 1-7) m o r e or less has anticipated the expected answer The various amplitudes in the dipole approximation that appear In the literature ~-6) are not all m agreement with each other, while a m o r e general formulation 7) does not treat all the terms consistently and hence arrives at the p r o p e r limit only after an ad&tlonal approxImation In the present note, we wish to show how in the long wavelength hmlt ( L W L ) the expected result, namely a separation o f the amplitude into a T h o m s o n term, which remains finite and represents scattering f r o m the nucleus as a whole, and a Raylelgh term which vantshes as e) 2, and which represents excitation of the internal coordinates, can be obtained by a careful separanon of center-of-mass and internal coordinates The usual gauge-lnvarlant replacement of p by p - eA (r) in the nuclear Hamtltontan leads to an Interaction Hamlltonlan of the form (in C o u l o m b gauge) ~ " n t = -,=~1

[ e p ' ' A ( r ' ) - 2m-e2 AZ(r')l

(1)

Both terms contribute to the second-order scattenng processes 7 + A ~ A ( * ) + 7 ' , the first one in second-order perturbation theory t h r o u g h an intermediate state In) The amphtude, apart from an over-all factor 2zr(kk') -~ (with k and k' the inlnal and t W o r k s u p p o r t e d by a grant o f the N a t m n a l Science F o u n d a t i o n Present address Theoretical D l v , Los A l a m o s Scientific L a b o r a t o r y , Los A l a m o s N M tt W o r k supported by the Office o f Naval Research, U S N a v y

655

g SILBAR et al

656

final photon momenta) is then gwen by Mfo = M1 + M2, e2

1

*

E.o-k-kzF.

e2 ~

(2a)

-ac

(fl Y~ 3 P~ ek,,,e

L

1

"7

* et'z ea' "'ln)(nl ~

E,,o + -U +½tF ( f l Z,

8k';"e

-,w

rJ]O~

(2b)

e2

M2 = - -- ( f ] E 81,,~ 8*,.re'(k-k') "'lO), m

(2c)

where m is the nucleon mass, E.o = E . - E o (nuclear recoil energies having been neglected), F. is the width of In) and ekz the polarization umt vector (polarization index 2) This is written In laboratory coordinates We introduce center-of-mass and Internal coordinates R and p,, respectively, and their conjugate momenta by

r, = R + p , ,

(3a)

p, = A - a P + q .

(3b)

where A

R = A - t Z r,,

(3c)

l=1 A

Z P, = 0

(3d)

l=l

Care must be taken In deahng with the p, coordinates since one of them is redundant Now we assume dipole transitions, exp (tk p,) ~ 1, etc We also go to the LWL, k -~ 0, thus neglecting nuclear recoil terms containing P, since they can be shown to be of order k / m or higher in the L W L compared with terms which we retain Thus, ez

M1 - M'~"' = - - • m 2

e2

+~-~

n

1 E.o-k-½tF

.

E qs 8*,a,ln)(nl ~, q, ~ 1 0 )

(f[

s

1

E.o+k,+kW

(flZq~,

8k.ln)(nlZq,.

,w,,lO), *

(4)

where now If), In) and 10) designate only internal states The integration over daR leads to the over-all momentum conservation, and the corresponding 6-function is not written explicitly If the dipole limit were carried out straight-forwardly in M2 also, we would obtain the coherent Thomson limit of Z free protons, Ze 2 M 2 ~

8kZ

gt,,z, 6 f 0 ,

(5)

m

which is actually a high 4) (but not too high s)) energy limit In LWL, however, we

NUCLEAR PHOTON SCATTERING

657

expect a Thomson cross section of the nucleus as a whole. This result, as we shall see, comes about by M2 breaking up into a center-of-mass part ~denUcal to the expected Thomson ltmlt, and an internal part which combines with M I "t to form the Raylelgh amplitude proportional to kk' (1 e 0 ) 2 for elastic scattering) To do this, we rewnte M2 [eq (2c)] using the identity 9) t =

Z

*

• r,]

(6a)

J

_-- ~Z 8kl 8*'a' + ' [ Zj ~

q,,/~*a'

P,]

(68)

Again, the daR integration at this point leads to over-all m o m e n t u m conservation, and we get from eq (6b). M2 = MTh + M 2mt,

(7a)

with the expected Thomson term Z2e 2

MTh = -- _ _ Am

8k2 "

* 8k'2'~fO ,

(7b)

and an internal term l~fint

•.-2

e2

= - t -- ~ ( f l ~ S k 2 m

n

~

•

q,ln)(n]~ ~,,~, *

• p~[0)

j

O2

+ ' -- 2 ( f l 2 .k'a, m . s

pjln)(n[ 2 e,z

p,]O)

(7c)

Use of Helsenberg's equaUon of motion (noting that the elgenvalues of the Hamlltonian are complex because of the widths of the intermediate states) and of energy conservation 9) allow us to combine M [ "t, eq (4), and M 2,.t , eq (7c), into M1lnt + M 2lnt ---~ M r a y ,

Mray -- e2kk,(~"

1 ( f [ ~ , *.'2' p~ln)(nl~2n.a E.o - k - ½IF. ~

1 +EE.o+k'+½W. (fi ~."~,a,

p, ln)(n[ ~a *

p30)

p,105

}

(8)

and the total amplitude is Mr0 = MTU + Mray,

(9)

going to MTh In the L W L The form of eq (8) is justwhat one would obtain by using an effective dipole interaction Hamlltoman, - e E r We wash to thank Drs E Hayward, N Meshkov and H Relss for discussions t T h e d e f i n m o n s m e q (3) i m p l y the c o m m u t a t i o n r e l a t i o n s [P~, R/~] ~ --t6~t ~, [P~, p ~ ] = [q~, Rp] = 0, so t h a t eq (6b) f o l l o w s i m m e d i a t e l y f r o m e q (6a) T h e r e m a i n i n g c o m m u t a U o n rel a U o n s c a n be s h o w n t o be [q~, pj~] = - - z 6 ~ B ( d z j - ( l / A ) )

658

R SILBARet al

Note added m p r o o f 1) D r M D a n o s has p o i n t e d out to us that a separation of c m a n d relative coordinates m nuclear radiative transitions has been carried out previously (Sachs a n d A u s t e r n lo)) This m e t h o d , however, is based on a n as-

sumed form of the c m i n t e r a c t i o n term m o t i v a t e d by gauge mvarlance, whereas our t r e a t m e n t leads to the right separation of T h o m s o n a n d Rayleigh cross section b y direct calculation 2) It was stated by Brenlg 11) that the T h o m s o n h m l t should arise from the A 2 term m eq (1), a n d the Raylelgh limit from the p A term I n the light of our present study, Brenlg's statement must be consxdered erroneous

References 1) 2) 3) 4) 5) 6) 7)

8) 9) 10) 11)

G Placzek, Marx Handbuch der Ra&ologle VI, 2 (1934) p 205 J S Levmger, Phys Rev 84 (1951) 523 U Fano, NBS Technical Note 83 (1960) E Hayward, Lectures on photonuclear reactions, Scottish Umversltles' Summer School, ed by N MacDonald (Ohver and Boyd, Edinburgh, 1965) J Le Tourneux, Phys Lett 13 (1964) 325 H Arenhovel and W Gremer, Phys Lett 18 (1965) 136, H Arenhovel and H J Weber, Nuclear Physics Agl (1967) 145 H Arenhovel, thesis, Umverslty of Frankfurt/Mare (October 1965) unpubhshed, H Arenhovel and W Grelner, Nuclear Physics 86 (1966) 193, H Arenhovel, M Danos and W Grelner, Phys Rev 157 (1967) 1109 J D Jackson, Classical electrodynamlcs (John Wiley & Sons, New York, 1962) p 491 P A M Dlrac, The principles of quantum mechanics, 4th ed (Oxford University Press, London, 1958) p 247-248 R G Sachs and N Austern, Phys Rev 81 (1961) 705 W Brenlg, Advances m theoretical physics (Academic Press, New York, 1965) vol 1, p 59