Giant resonance in heavy nuclei

Nuclear Physics 66 (1965) 35--48; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

GIANT RESONANCE IN HEAVY NUCLEI A. A. LUSHNIKOV and D. F. ZARETSKY L V. Kurchatov Atomic Energy Institute, Moscow, USSR Received 26 May 1964 giant resonance in the dipole photoabsorption of nuclei is investigated. The main parameters of the photoproduction cross section (the position of the maximum, its width, the sum rule taking into account exchange forces, and the maximum splitting in deformed nuclei) are calculated with the aid of equations of the theory of a Fermi liquid in finite systems.

Abstract: The

1. Introduction

The photoabsorption cross section in heavy nuclei is a smooth curve with a rather narrow peak at a quantum energy ~ 15 MeV (giant resonance). The interpretation of the giant resonance is difficult to a considerable extent because of the need for taking into account the interaction between nucleons, which is not weak. The theory of a Fermi liquid in finite systems allows an evaluation of the interaction between nucleons by introducing phenomenological constants 1). Only one constant connected with the rigidity 13which determines the symmetry energy in the Weizs/icker formula proves essential for dipole photoabsorption. The equations which make it possible to investigate the effect of nuclear photoabsorption are formulated in ref. 2). The solution of these equations is simplified in the case of heavy nuclei. The use of the quasi-classical approximation yields results in the analytical form. The giant resonance parameters thus obtained (the position of the maximum, its width and the sum rule) can be compared with experiment. The analytical solution of the equations presented in ref. 2) allows a deeper insight into the mechanism of dipole photoabsorption. As indicated in ref. 2) the giant resonance width is connected mainly with the fact that the nucleus has not the degree of freedom corresponding to the oscillation of the dipole moment whether there is or not an interaction between quasi-particles. Consideration of the interaction between quasi-particles merely shifts the maximum towards higher energies and broadens the curve.

In sect. 2 the main properties of the equation for the effective field are studied. On the basis of the solution of this equation a very simple procedure is proposed for obtaining the main parameters of the giant resonance for the system of interacting particles if the corresponding parameters of the system of non-interacting quasiparticles are known. 35

36

A. A. LUSHNIKOV AND D. F. ZARETSKY

The effect of the velocity-dependent interaction between quasi-particles on the giant resonance parameters is considered in sect. 3. The cross section curve of the dipole photoabsorption in a rectangular well with infinite walls is calculated in sect. 4. The effect of deformation on the dipole photoproduction curve is investigated in sect. 5. The resonance maximum splitting is obtained. The results obtained are discussed in conclusion, and a comparison with experiment is made.

2. Main Equation and Its Properties; Giant Resonance Cross Section Giant resonance in heavy nuclei is considered in this paper. It is well known that in this case the fine structure which was discussed in detail in ref. 2) is not observed. Therefore, we are seeking the distribution function for the intensities of the absorption lines, without concern for their location. Under the action of a homogeneous periodic electric field with the potential V = Eoxe i'~t in the nucleus there arises, owing to the interaction between nucleons, the effective field 2) Veff which leads to the variation of neutron and proton concentrations in time and hence to a dipole moment. The latter consumes a part of the field energy, which leads to the absorption of quanta. It is shown in ref. 2) that the absorption cross section can be expressed through the imaginary part of nuclear polarizability t a(co) = --2~e2co Im P,

(1)

where /la --/'/a'

P = ~ xa~, aa'

" Xaa,;

g,~ - - e,~, - - co

V *ff is connected with X by the formula l / © f f = ½eEo X. Here 8, are the single-particle levels of quasi-particles in the well U(r), nx the occupation numbers, and xaa, is the matrix element of the operator of the coordinate. For X we can write the equation 2)

xa~, = x~,,+ ~ fa,,aa~ na,- n~ Xaa~, a,aa

(2)

e a t - - e a a - - co

where F is the amplitude describing the interaction of quasi-particles. The amplitude F in tlle coordinate representation has the form

t = ( 2f' ~ a ( r l _ r 2 ) + [ 2f; ] a ( r l _ r 2 ) p l p 2 ' \dn/deo/ ~p~ dn/deo] where the dimensionless c o n s t a n t s f ' andf~ are connected with the Legendre expansion of the quasi-particle zero-angle scattering amplitude (f' is the zero and f~ the * We assume mass).

that N =

Z =

½A. T h e s y s t e m o f u n i t s u s e d is h =

c =

M =

1 (M is the nucleon

GIANT

RESONANCE

37

first harmonic of the expansion), p the m o m e n t u m operator, n the density of particles of one kind, s o the Fermi energy and Po the Fermi m o m e n t u m . The quantity f ' is connected with the Weizsticker rigidity fl by the formula a) 1 + 2 f ' = 3fl . So Let us write eq. (2) in the coordinate representation, discarding for the time being the terms proportional to f;:

d__nnX = d n x + 2 f ' E tPa(r)tPx'(r) n x - nx, ds o

ds o

~x,

Xaa,,

(3)

e~ - sx, - o9

where q~x are the single-particle wave functions in the well U(r). With the aid of eq. (3) we transform the expression for the cross section. Multiplying eq. (3) by x on the left and on the right, integrating over the nuclear volume and taking the imaginary part we obtain f r o m eq. (1)

a =

- ne2----~-~I m f dn xXdar. f' Jdso

(4)

Let us transform eq. (3), expanding the difference n~-na, in powers of ea-s~, and keeping only the first term of the expansion (the remaining terms being oz A -~ with respect to the one we keep): d n X = d--n-nx - 2 f ' ~ q~ ¢p~, dn~ s~-e~, X ~ , . dso dso ~' dso s x - s~, - co

(5)

Let us note that eq. (5) can readily be solved at o9 = 0 since

xx'

go;trPx' ~o~ X,w = Xa~ rP2 dn---~= X d_._nn. deo dso

(6)

Using this circumstance we find at co = 0 (see ref. 2)) x X

--

-

-

l+2f' " Eq. (5) can be solved since Xxz, has a sharp m a x i m u m as the function cos - sx-sx, near a frequency COo, which corresponds, as is shown below, to a small width of the giant resonance peak. This makes it possible to take the energy denominators outside the sign o f the sum and perform the summation just as in the case co = 0: tp~ tp~, Aa'

dn A ds 0

s~-sA,

X~,

s ~ - - s A, - - o9

dn~ o9o - co xx, gA>80 gA t < g0

deo

COo

dn~

o90 + o9 xx, gA<£O CA' > gO

-

°92

X dnn.

38

A. A. LUSHNIKOV AND D. F. ZARETSKY

Using eq. (5) we obtain S

O) 2 m ogu 2 2 ~ ogm - - CO

2

o9o (1+2f').

I

where 0.)=

(7)

Now the meaning of the value o90 is clear. Since the coordinate dependence X is the same as when there is no interaction, o9o corresponds to the giant resonance frequency in the system of non-interacting quasi-particles. The half-width t of the giant resonance for heavy nuclei F is small compared with ogre (see ref. 4)). This circumstance can effectively be used for solving eq. (5). The assumption about zero width led to the equation for X being solved and yielding the coordinate dependence x = c(og)x.

This situation is preserved in the case F << o9= as well accurately to ~ F/o9m (the parameter F/o9= has no smallness of the type 2) A-~; its smallness is purely numerical). Let us seek X in the form X = C(og)x. Then from eq. (5) we obtain

c(o9) =

Po(O)

(8)

eo(O)+ 2f'eo(og) '

where Po(

) =

Ix

,l 2

is the polarization operator for independent quasi-particles; Po(0) = - j"(dn/deo)x2d3r. The latter equation is obtained just as eq. (7). Let us assume that the giant resonance cross section calculated by the model of independent quasi-particles can be approximated by the formula

tro(o9) = ½ne2A

Fo

(o9_o9o)2 + F 2 ,

(9)

where F o is the width of the giant resonance calculated by the model of independent quasi-particles. It can be checked straightaway that the polarization operator Vo(og) = Vo(0) 1 Z do9s o9~ ro s ds 0,2- o92_ i~ (o9,- o90)2 + r o~

(10)

leads to a cross section of the form (9) where o9s are the poles Po(o9) and dogJds their density. * I t is m o r e c o n v e n i e n t f o r u s t o u s e a d e f i n i t i o n o f t h e h a l f - w i d t h F g i v i n g h a l f o f t h e v a l u e o f t h a t a d o p t e d in r e f . 6).

GIANTRESONANCE

39

Since the distance between adjacent poles (in the case of spherical nuclei) oc e0 A -+ and F oc e o A -+ the sum in eq. (10) can be replaced by an integral. Then we have

~o-Wo

V

Po(09) = ½Po(O) /

+

I.O9o-- 09 -- iFo

09o-Wo

7

I.

090 + 09 -- iroA

Substituting this equation into eq. (8) we find C(09) = 09~-092 -2i1-'o 090 2 09m - 092 _ 2iF09m

2 = 09o2(1+2f,). where F 2 = F~(1 + 2 f ' ) and 09m Using eq. (4) we can now find the cross section (accurately to ,~ F/09m) a(09) = ½rce2A

(11)

F

(09- 09m)2 + r ~" The coefficient in eq. (11) is obtained taking into account the sum rule

fo®409)009 -- i °~o(09)d09 -- ½~2¢~A. 3. Effect of the Velocity-Dependent Interaction of Quasi-Particles on the Position and Width of the Resonance Let us write the equation for X including the terms discarded earlier proportional to

f;:

d n X = d n x + 2 f ' Z
deo

deo

xx'

n 2 - - n 2,

Xx2,+

e 2 - ~2' - 09

2f~

Z pxx,(r)

po2 22'

n 2 ~ 114,

X22,P,

(12)

e 2 - ex, -

where pxx,(r) ~ q~x(r)ptpx,(r). Eq. (12) can be solved in exactly the same way as eq. (5) in sect. 2. Let us put F = 0 and seek the solution in the f o r m X = Ax+

iBpx

(13)

09

Substituting eq. (13) into eq. (1) we express the cross section through A and B: t' 02209 = -2~¢21 x~ d__~_nda r Im (A-B).

J

d%

COo 2 _ cos

(14)

4o

A.A. Lvmmtov AND D. r. ZARETS~:V

The sums necessary for obtaining this expression and further calculations are computed in appendix C. For A and B we obtain from eq. (12) £O2

A = 1 - 2 f ' °92°9°2_092(A-B),

B(l+]f;)

= ~f; 0 ~ _ 0 2 (A-B).

Let us solve this set of equations with respect to A - B : ca02-- ca 2

A-B

= (l+]f/)

2

2"

cam - - Ca

Hence from eq. (14) we obtain the cross section

~(Ca) = (l +~f~)~(Ca-Cam)~2e2Ca2f ~-eosoX2dar. Let us determine the factors in the formula for a ( o ) . This can readily be done, a s s u m i n g f ' andf~ to be zero. Then we have f cro(ca)dca =

, e Ca f oX2d r.

On the other hand

fo

rgO(ca)dca = --27z2e2 E

[xxx'12(nx- nx,)(sa--ax,)

,aA.' 8 A > 83, t

_

2~2e 2 1_ y. M*

nx(p~x-xp~)x -- 7r2e2A

2i a

2M*

Comparing these two equations we find a(Ca) .= (1 + ]f~')

n2e2A 6(ca- Cam),

(15)

where 2 cara

= ca2(l+2f')(1

]f~)

and

o,2=

A

2M*f~ooX2@r

(16)

In the case of a flat-bottom well we have ca2 -- 15hA%

4RSp~ ' where e o =

p212M*.

Besides, Po determines the number of particles of the same

41

GIANT RESONANCE

kind by the formula ½A = ~z~R3p3/3rc 2. Hence we readily obtain (.02

=

(17)

1 0 80

Using eq. (17) we have for COrn COrn= I0 R-~ (1+~f~'),

(18)

which coincides with the result of ref. 2). The formulae obtained can easily be extended to the case when the width is not zero. We do not solve the equations anew for this purpose, but in the formula for A - B add - i F o to coo. Then after simple operations we obtain

ne2A

F

o = (l+~-f~) 2M-~ (com_co)2+F2,

(19)

where = COrn

coco

(1 +

}fl).

80

Thus, if F o and coo determined by the model of non-interacting quasi-particles are known, eqs. (15) and (19) yield giant resonance parameters taking into account the interaction between quasi-particles. Using eq. (15) we can write the sum rule in the presence o f exchange forces

~2e2A

fo (co)dco 2M*

(20)

This result was obtained in ref. 2) from different considerations. 4. Giant Resonance Cross Section in a Rectangular Well With Infinite Walls To solve eq. (5) we can make use of eq. (8). In the rectangular well we have

(21) d%

(r > R).

Then we obtain

f d~o8 ° xZd3 r _ 4re Io R 4 15 where l o - p o R

and R is the nuclear radius.

n2 '

42

A. A. L U S H N I K O V

AND

D . F, Z A R E T S K Y

For C(o~) we obtain from eq. (8) I C(co) =

dna

10f'7~ 2

loR4 ~](rYlo)ax,l 2 d%

1+

ea-ea, l - 1 8a-ex,-o~_l "

(22)

(The imaginary part in the denominator l) - i a sign o~ is omitted for brevity.) The levels ex in the spherical well do not depend on the magnetic quantum number m. This makes it possible to make the summation over m and m' in eq. (22) using the well-known expressions ,[ Yl,,,Yrm"Y1o dO through the 3j-symbols. As a result we have in the quasi-classical approximation

~ If Y,mY,.-.Ylod~2 --~l

(a,,,~+l+a,, ,-1).

Let us introduce the dimensionless variables v = -o~lo -, 280

¢±

=

12_~o(en, t + l - s n , , z

)

l

y = '

t = -r- . R

To '

(23)

The matrix element of r with radial wave functions as well as ~± are calculated in the appendix (see eqs. (A.6) and (A.8)). Using the variables (23) and eqs. (A.6) and (A.8) we find that the sum on the right-hand side of eq. (22) is 1

(24)

The coefficient in this expression is obtained taking into account two directions of the spin. The summation over I has been replaced by integration since at a fixed 6n and change of l by unity ¢ changes by ~ 1/lo whereas at a fixed l and change of 6n by unity ~ changes by ,~ 1. This follows from the explicit form for ~ (A.8). The sum and integral in eq. (24) are calculated in appendix B. Using (B.6) and (B.10) we obtain C(co)-where

1-~f'

3 I(~

1 + 2v6

+~ 5~f' F(~ v

I(v) and F(v) are the functions given by eqs. (B.7) and

(B.10). Hence we have

57~f'F(v) Im C = -

3

v

2

[ 1 - - ~ f ' ( ~ - t - 2v6 I(v))?.-b Eqs. (4) and (25) yield a(v) ----7t~/3n2e2r2 AtG(v),

(25)

~22 f'2F2(v)

43

GIANT RESONANCE

where G(v) =

F(v)

3 I(v))12+ ~22f'2F2(v ) "

I I - ~ 9 - f ' ( ~ + 2v--g

The function G(v) is represented in the diagram of fig. 1. For comparison we plot (fig. 2) giving the giant resonance cross section in the model of non-interacting

F(v)

°21

i

I

.t6

Fig. 1. Plot of the function G(v) giving the photoabsorption cross section for the system of interacting quasi-particles.

F(Y)

a6

Q~

Fig. 2. Plot of the function F(v) giving the photoabsorption cross section for the system of noninteracting quasi-particles. quasi-particles:

Oo(V) = x~/3~2e2r 2 atF(v). We further obtain frO)

O)~---

V

V.

The numerical calculation gives S~° G(v)dv = ~o F(v)dv ~, 0.3. Thus 5~° a(O))dO) does not practically depend o n f ' . In our calculations we take 2f' = 1.5 (see ref. a)). Then (1 + 2 f ' ) ~ = 1.5. Calculation o f the ratio of the width of the function G(v) to that of F(v) gives F/Fo = 1.5. The location of the maximum of the function F(v) is Vo = 1.15 and the maximum of G(v) is at vm = 1.7. This corresponds to Vm/VO = 1.5. All these results correspond to the conclusion of sect. 2. I f we determine O)m by eq. (7) we can using v o = 1.15 obtain

tom ~ ~/9fl/R 2, which approximately corresponds to eq. (18).

(26)

44

A. A. LUSHNIKOV AND D. F. ZARETSKY

5. D e f o r m e d

Nuclei

The scheme presented above makes it also possible to consider deformed nuclei. In this case the polarizability of the nucleus is a tensor. The cross section for a giant resonance on a non-polarized nucleus has the form

tr = -}neZo9 I m Tr P

=

-27ze20)

I m (Pxx+2P.),

(27)

where n~-

Px,, = ~ x~z' 2A'

n2,

Xaa',

Pry = ~ Y~a'

~j. ~ / ~ j , - - O9

22'

n a - na,

Y~'

/~A - - 8Z, ~ O9

and X and Y satisfy the equations

dn X = x - dn - + 2 f ' ~ ~oa~02, n a - n~, Xaa,, de o _

de o

aa'

ea - 8a, - o9

dn y = Y _dn_ + 2f' ~ ¢Paq~a' na-nx,

deo

deo

aa'

(28) Yaa,.

_

~a-- ca' - 02

The matrix dements Xaa, and Yaa. have a sharp m a x i m u m as functions of o9, = c a - ca, near the frequencies o9o= and ogoy respectively, and o9o= < ogoy if the nucleus is elongated along the axis x. Eq. (27) is solved if the parameter F/co= is assumed to be zero. It is evident that

x = o9 ._o92 2

2 x

ogo y_o, and

g

-

-

ogmx - - o9

2

2 Y'

(/)my -- o9

where ~ max

~

o9o2x(1+ 2f'),

2 o9my

= o9o2y(1+ 2f').

For the resonance cross section we obtain

~r(o9) = -~r~2e2A [6(og-ogm,,) + 2 ~ 6(og-o9my)l ,

(29)

where a and b are the major and minor semi-axes of the ellipsoid. We obtain the resonance m a x i m u m splitting t~O9m =

ogmy - - ogmx =

( ( D 0 x - - Og0y)( 1 "]- 2f') +-

Calculating 090. and o9oy by eq. (16) we obtain

6o9m = x/16-fl(1/b- l/a) 2.

(30)

This result was obtained in ref. 2) from different considerations. The generalization for the non-zero width case is very simple. I f it is assumed that the width depends weakly on deformation the cross section curves are obtained f r o m superposition of two curves with the same width F = Fo(1 + 2 f ' ) + but different m a x i m u m positions (Wmx and ogmy). The quantity Fo can be calculated without taking into account the deformation.

G I A N T RESONANCE

45

6. C o n c l u s i o n

The photoabsorption cross section versus the frequency of incident quanta has bccn measured for a large number of nuclei. However, these measurements do not have a high accuracy (~ 20~). Therefore, the quasi-classicalaccuracy (~ A -~) is quite sufficientfor a comparison of the theory with experiment. The experimental dependence of the m a x i m u m position on A is 82A -~ M c V (for A > 50) 4). The resultswc have obtained (cqs. (18) and (26)) give approximately the same figure. Comparison of our width with experiment is more interesting.In the calculation by the model of independent particless) the value 2F o = 3 M c V is obtained for lead. Recalculation by cq. (19) (iff~ = 0) gives 2F = 4.2 M e V (fl = 23 M c V and eo = 40 M c V ) which is in agreement with the result of rcf. 4). In conclusion wc wish to thank Professor A. B. Migdal for interestand stimulating advice and A. I. Larkin for discussions. Appendix

A

Let us calculate the matrix element r.~+l ' ..t with radial wave functions R,, t of an infinitely deep rectangular well which have the form, in the quasi-classical approximation,

R., =

~2

08.~ 1 -~ On ~/28.,-12/r 2 cos ~.,(r)

(A.1)

(the energy e.t is calculated from the bottom of the well),

f" rRnl+iRn,ldr,

(A.2)

rnt+ l,n,i ----

Q/ Pn|

where r., = l/,/2T.,. Let us represent the product R. t+ 1R.,~ as a sum of two terms, one of which varies over distances o f the order R and the other oscillates rapidly. The integral from the second term is quasi-classically small. The first term can be written as 1 1 08n~ cos an ~/2e.l_12/r 2

q~,z+1- ~,'l _ 0~"l ~8,1+1. ,'t+

adp. z _ 08.1

a~.z

,

(A.3)

a_ f " x/2e.l_12/r,2dr, - ~/28.1r2--12

2e.l

ae.t J ,.~

ac~.z Ol

-

arccos

I rx/ 2~.z

(A.4)

46

A.A.

LUSHNIKOV AND D. F. ZARETSKY

Since dn.l/dso = - 6 ( 8 . 1 - 5 0 ) we can in eqs. (A.3)and (A.4)and subsequent operations put 8. z = 50 in the corresponding places. Introducing the dimensionless variables (23) we obtain

fl "~ ~ ~/tt2dt .~rRnt+lgn'tdr = Rrc2 t38.,1of 8n 2-_y2 cos (~x/i2"y2+arccos t-ly).

(A.5)

The calculation of the integral in eq. (A.5) leads to the expression 1 (C)"

Ir. ~- 1, .,zl 2 = R2

For the calculation of ¢ and

08.|~an we can

(A.6)

use the quantization condition

f R 4 2 ~ . , - 12/r=dr

= =n.

( A.7 )

rnl

Differentiating eq. (A.7) over n and l we obtain ~± _

~ 6 n ± arccosy ~/1-y 2 ~/1-y 2

(A.8)

From eqs. (A.8) and (23) it is obvious that 08._j = 2~r8o On /0~/1 - y2 "

(A.9)

Eqs. (A.8) and (A.9) were used in the calculation of the integral in eq. (A.5). Appendix B

The sum in eq. (24) transforms easily to 1~

1

1

_

1

-~~.foY(1-y2)[~6n_arccosy_vx/-~_y2 ~6n+arccosy+vx/1---y2]dy _ 2Z

(1

y(1_y2)~dy

v26. Jo (~6n + arccos y)2 ~.fo

Y(1-y2)'dY

(~5n+arccosy)Z

=

. '

y2)~dy gJo siny(12-arccos -f: y

03.1) yx/-~--y2dy

:~"

(B.2)

Using the expansion ctg ~ in common fractions we obtain for eq. (B.1) S -

la /~ y(1 -- y2)[ctg (v41 _ y2 _ arccos y) + ctg (vx/1 -

2 y2 + arccos y)]dy - 3v--~ •

(B.3)

GIANT RESONANCE

47

After simple trigonometrical transformations and the substitution o f the variable ~/1 _ y 2 = z we obtain S = --

z a cos vz

+

in v z - z

dz-

sin

(B.4)

3v 2 "

T o find correctly the imaginary part S we must prescribe n o w the path o f integration must avoid the poles in the integrand. F o r this purpose let us add - / 5 (v > 0) to v. Then we have

flz3

S = _ _I v 3 ,Jo

V

1

cos vz [_sin vz - z - i5 cos vz

1

+

sin vz + z - i 6

] dzcos v z

2_

(B.5)

3V2"

After the substitution o f variables t = vz we obtain for I m S: (B.6)

I m S = ~- F(v), where

F(v) =

0,

0~v
1 to V - - t o v 6 1 -- x / v - ~ _to2 '

l__
1 t3o~/V2--t~

--

,

v6 i + x/v-Y~toz

2 ~ > - -

(B.7)

n

= 2

and to is the r o o t o f the equation

to(V )

= v sin

Re S = I(v)-P

to(V),

(B.8)

vl_gI ( v ) - - 3v2, 2 t3 c o s t

-

+ sin t - t v sin t +

(B.9)

dt.

(B.10)

The integral in eq. (B.10) has been calculated numerically.

Appendix C

I n the calculation o f the sums in sects. 2 and 3 we used the fact that the operators x and p give transitions only t h r o u g h one shell. This makes it possible to take the frequency dependent factors outside the sum. Thus we have nx--nx, xx,

8x-ea,-o9

= xx,

dn~ ( e x - e ; ¢ ) z

dn

deo e x - e x , - o 9

deo (.02-(`02 °

o9o22

48

A. A. LUSHNIKOV AND D. F. ZARETSKY

The other sums are calculated on a similar pattern. We merely give the final results: nx-n~,

dn

qT~ (/7,v

~x'

0 2

X ~ ~ , "= - - X

~-

"~ ( p x ( r ) ) ~ ,

8~, - o

de0 COo ~ - 0 2, 3

" ~ - "~'

~'

8x--sx,--o

;tX'

SX-- g;t' -- O

p~ X22'

o

= ~ 3~ 2 0 2 - - 0 2

3~ 2

0 2 -- 0 2 "

References 1) 2) 3) 4.) 5)

A. A. A. E. D.

B. Migdal and A. L Larkin, JETP 45 (1963) 1036 B. Migdal, A. A. Lushnikov and D. F. Zaretsky, Nuclear Physics 66 (1965) 193 B. Migdal, JETP 43 (1962) 1940 G. Fuller and E. Hayward, Nuclear Physics 33 (1962) 43 H. Wilkinson, Physica 22 (1956) 1039