Corrections to Glauber theory in the optical limit

Volume 58B, number 1

PHYSICS LE'ffrERS

18 August 1975

C O R R E C T I O N S T O G L A U B E R T H E O R Y IN T H E O P T I C A L L I M I T D.R. HARRINGTON Department of Physics, Rutgers UniversiO,,New Brunswick, New Jersey 08903, USA Received 13 May 1975 In the optical limit the leading f'mite energy correction to Glauber theory for scattering from a composite system in the fLxedscatterer approximation takes a form similar, but not identical, to the leading correction to scattering from an elementary system via the optical potential.

The eikonal approximation in non-relativistic potential theory is now known to be the leading term in an expan sion in powers of the inverse momentum [ 1 - 4 ] . If we keep only the lowest order correction the T-matrix element can be written as (k + q[ Tlk) =

ikm-lf d2b exp ( - i q "b) {exp [ix0(b) + ix1 (b)] -

1},

(1)

where oa

X o ( b ) = - m k -1 f --

dz V(b,z)

(2)

oo

is the uncorrected eikonal and XI(b) =m2(2k3) -1 f

d~Jl d/J2 {@([J2 -~Jl)(/J2 -~Jl) VbV(b,~2)'Vb V(b, [ J l ) - ~([J2 - ~ l ) V ( b , ~ 2 ) V ( b , ~ l ) } (3)

is the lowest order correction to it. If V is spherically symmetric (3) reduces to [2] oo

Xl(b) = - m 2 k -3 f

dz (d/dr2)[r 2 V2(r)],

(4)

--oo

so that the corrected amplitude can be calculated using the uncorrected eikonal formula with the modified potential Vmod = V+ mk -2 (d/dr2)[r 2 V21.

(5)

For scattering from a nucleus in the fixed scatterer approximation we simply set

V = V(r, {r/}) = ~ V(r - r/) /

(6)

and then use the nuclear wave function to average over the nucleon coordinates el. Then [4]

XO = XO(b, {bj}) = ~ . Xo(b - bj), ]

(7)

while :

l,]

where

×l (b;

(8)

Volume 58B, number 1

PHYSICS LETTERS

XI (b; ri, t)) = m2(2k3) - 1 f d ~ 2 d~Jl {0(/j2 _ ~Jl)(~2 -~11 - 5(g2 - ~I)V( b - bi, ~2 -

zi)V(b -

x(b - b/) =x0(b -

Vb V(b -bi, ~2 -zi)'~Tb V(b -bj, ~1 - zj)

b i, ~I - z/)}.

With terms in (8) with i = j can be added to

18 August 1975

(9)

Xo(b - hi) to give

b/) + Xl(b - bi),

(10)

which is simpl~, the corrected eikonal for scattering from the jth nucleon. Combining these results and expanding the exponential containing the i :/:j terms the elastic scattering amplitude is

T(q)=ikm-l fd2bexp(-iq'bl(Glexp[i~x(b-b/I][l+i~Xl(b;r],rill-llG) j j~i

(Ill

where IG)is the nuclear ground state. To take the optical limit of(111 we first assume the A nucleons in the nucleus are completely uncorrelated so that

ikm -1

s

d2b exp ( - i q ' b )

{Is :_:

d3r P(rl) exp(ix(b

+iA(A - l) I f d3rlP(rl)exp(ix(b-b11) ]

f d3rl d3r2 P(r2)P(rl)exp[ix(b-bl)+ix(b-b2)] Xl(b;r2,rl)- l},

where p is the single-nucleon distribution function. When A is large and the range of the elementary interaction is short compared with the distance over which p changes appreciably we have Ifd3rl P(rl) exp (ix(b - b I 1)1 A ~ exp[iXopt,0(b)]

(131

where [5] iXopt,0(b) = -

~Ao(1 -

ia)fdz p(r/.

(14)

Here o is the total cross section and a the ratio of real to imaginary part of the forward amplitude for scattering from a single nucleon. If we define the optical potential Vopt(r) = - ik(2 m ) - 1A o(1 - its) p(r)

(15)

then the leading terms in (12) have exactly the form of the uncorrected eikonal approximation for scattering from an elementary object via this optical potential [5]. If this description of the scattering were taken seriously then the leading correction to Xopt,0 would be, according to (4),

x~elem) pt,0 = _m2k_3

fdz (did r 2) [r 2 V2opt].

(16)

To see to what extent this result is valid we look at the optical limit of the remaining terms in the curly brackets in (12). The first factor is identical to (13) except that the exponent is A - 2 instead of A: In the large A limit we can ignore this difference. Using (9) the remaining integrals over r 2 and r I lead to integrals of the form - i m k -1 f d 3 r l P(rl) Vb V(b - b l , ~1 - Zl) exp [ix(b - b l ) ]

~fd2bl P(bl,~l ) V b {explix(b-bl) 6

(17)

] - 1} ~ -7bP(b, ~l)lO(1 -iti)

Volume 58B, number 1

PHYSICS LETTERS

18 August 1975

V R=I

I

R=.G4 5 R=O

?-

-I

'

~

'

~,

, r

&

'

~

'

IFMI

Fig. 1. The solid curves show tile shape of Vopt, 1, the leading correction to the optical potential, for three values of R. The dashed curve shows the Woods-Saxon shape assumed for the nuclear density. The curves have been normalized so that the nuclear density and Vopt, 1 for R = 1 take the value unity at the center of the nucleus.

and - i m k -1

fd3rl P ( r l ) V ( b - b l , ~ l

- Z l ) e x p O x ( b - b l ) } ~- - p(b,~l)½O(1 - i a ) R ,

(18)

where

R = f d 2 b ix(b) exp (ix(b)}

2b [exp (ix(b)) - 1]

(19)

and we have used the short-range approximation for the z 1 as well as the b 1 integrations. Using these results we find that, in the optical limit, to first order,

T(q) = ikm -1

fd2b exp (-iq'b)

{exp[iXopt,0(b) + iXopt,l(b)] - 1},

(20)

where Xopt,O is given by (14) and • (elem) ( b ) - i m 2 ( 2 k 3 ) - l ( R 2 iXopt, l(b) = 1Xopt,1

1)f V2opt(r)dz.

(21)

Another way to look at this result is to note that in this limit the corrections to Glauber theory are included by using the uncorrected eikonal approximation with the modified optical potential Vopt,mod = Vopt + Vopt,l '

(22)

where Vopt, 1 =ink -2 ((d/dr2)[r2V2pt] + ~ ( R 2 - 1) Vopt),2

(23)

which is just the result (5) for scattering from an elementary system except for the additional term proportional to R2-1. To determine the importance o f this term we must first see what values are reasonable for the ratio R. From its definition (19) we see that in general R is a dimensionless complex number which depends on the strength and

Volume 58B, number 1

PHYSICS LETTERS

18 August 1975

shape of the elementary interaction, and which equals one for weak interactions. Most elementary interactions at intermediate and high energies can be approximately described by the Gaussian form exp {ix(b)} = 1 + 3'0 exp

(-b2[2a2),

(24)

for w.hich oo

R = 1 -

(-3"o)"/n(n

n=l

,

+

1)2.

(25)

When the central absorption is complete 3'0 = - 1 and R = (rr2/6) - 1 ~ 0.645. We have plotted a function proportional to Vopt, 1 for a Woods-Saxon potential, with parameters [6] chosen to fit 208pb, forR = 1, 0.645, and 0 in fig. 1, along with a function which has the same shape as the original WoodsSaxon potential. The curves for Vopt, 1 differ considerably in the interior of the nucleus but are nearly identical in the surface because the derivative of V times the radius is larger than V itself in the nuclear surface. Since the uncorrected optical potential alone gives large absorption in the center of the nucleus the magnitude of Vopt, 1 in the nuclear interior will have little effect on the cross sections. In the optical limit, therefore, the leading corrections to Glauber theory can apparently be estimated rather well from the optical potential using (16), in spite of the fact that the composite nature of the nucleus is ignored, except perhaps at large angles where small differences in the surface shape can produce large differences in the differential cross section. The above conclusions of course depend upon the fixed target assumption* and the use of uncorrelated wave functions. Correlations are probably important in determining the effect of the delta-function term in (9) which contributes only when the two potentials due to nucleons i and/overlap. However, the correlations will probably tend to reduce the importance of these terms, which are already small in the nuclear surface, and thus, in fact, reduce the importance of the composite nature of the nucleus in calculating corrections to Glauber theory. * The fixed target approximation is discussed in ref. [7]. In a recent preptint (ref. [8] ) Wallace suggests that corrections to this approximation may be comparable to the corrections considered here, at least for light nuclei.

References [1] [2] [3] [4] [5] [6] [7] [8]

S.J. Wallace, Ann. Phys. (N.Y.) 78 (1973) 190. A.R. Swift, Phys. Rev. D9 (1974) 1740. U. Weiss, DESY preprint DESY 74/9. S.J. Wallace, Phys. Rev. C8 (1973) 2043. R.J. Glauber, Lectures in Theoretical Physics, ed. W.E. Brittin and L.G. Dunhara (Interscience, New York, 1959) p. 3i5. J . h Friar and J.W. Negele, NucL Phys. A212 (1973) 93. L L Foldy and J.D. Walecka, Ann. Phys. (N.Y.) 54 (1969) 447. S.J. Wallace, Univ. of Maryland preprint ORO-4856-2.