- Email: [email protected]
On scattering by nuclei at high energies
ANNALS
OF PHYSICS:
56, 268-294 (1970)
On Scattering
by Nuclei at High Energies*
H. FESHBACH AND J. H~~FNER Lahorafory for Nuclear Science and Department of‘ Physics, Massachusetts Institute of Technology, Cambri&e, Massachusetts 02139
We study how nucleon-nucleon correlations influence high-energy scattering by nuclei. Using the multiple scattering theory of Kerman, McManus, and Thaler, we formulate the scattering problem in terms of an infinite system of coupled equations. For the calculation of elastic scattering, we propose replacing the infinite system by a pair of coupled equations, containing the elastic channel, and another effective one which carries the inelastic strength. The coupling potential is proportional to the nuclear pair correlation function, and a first approximation to the potential is obtained These equations are accurate to the extent that the pair correlations and not higher-order correlations are important. There are no restrictions to forward scattering, nor are any “on the energy shell” approximations made. In order to obtain insight into the structure of the many-channel S-matrix for high energies. the infinite system of coupled equations is solved by semi-classical methods, and explicit formulae for the S-matrix elements as a function of various potentials and the nuclear pair correlation function are obtained. We show how properties of excited target states may complicate a reliable extraction of correlations in high-energy scattering by nuclei. A close relation between our semiclassical solution and Glauber’s multiple scattering is established. A numerical study of high-energy nucleon-nucleus scattering using the methods developed in this paper is under way.
I. INTRODUCTION
The present (I) and anticipated useof high-energy particles to investigate nuclear structure provides the motivation for this study. Because of the relatively high energy incident upon and emerging from the target nucleus, a short-wavelength semiclassical description is indicated. The most frequently used methods are based upon the work of Glauber (2,3), Franc0 and Glauber (4, Bassel and Wilkin (5), Czyi and Lesniac (6), Czyi and Maximon (7), and Formanek and Trefil (8). Briefly, thesepapers assumethat the nucleus can be described as a series of scatterings by the nucleons of the nucleus. More detailed assumptionsare: * This work has been supported in part by the Atomic Energy Commission under Contract No. AT(30-1)2098.
268
SCATTERING
BY
(1) The target nucleons are stationary, the nucleus (“frozen nucleus”).
NUCLEI
while the projectile
269 passes through
(2) The momentum change in the direction of the projectile, the “longitudinal momentum transfer,” is neglected. (3) Each collision between the projectile the collision between free particles.
and a target nucleon is identical to
(4) The total change in phase of the particle wave function equals the sum of the phase changes which occur at each collision with a target nucleon (“additivity hypothesis”). Although this is not the only set of assumptions which leads to Glauber’s results (cf. Czyi and Maximon (7) and Remmler (9)), semiclassical considerations in the spirit of conditions (l)-(4) are always involved. The resulting formulas are relatively simple, easily interpreted physically, and easily evaluated (at least if harmonic oscillator wave functions describe the target state). The results obtained with these theories are thought to be valid in the small-angle high-energy domain. And indeed their agreement with experiment in this range is excellent. However, it is not at all clear that these approximations are sufficiently accurate as to permit the determination of the more subtle features of nuclear structure such as correlations. To obtain these, it is generally necessary to investigate larger-angle scattering where it is at the very least necessary to include the effects of longitudinal momentum transfer as is indicated by recent calculations (10). The additivity hypothesis may also not be sufficiently accurate (II). This hypothesis neglects intermediate off-the-energy shell contributions. For fast particles, these have their greatest significance for short-range correlations which are the interesting ones, since long-range correlations are dominated by the effects of the Pauli exclusion principle. For these as well as other reasons, the conclusions drawn by some of the above papers with respect to nuclear correlations are suspect. A general treatment of the multiple-scattering problem has been given by Foldy (IL’), by Lax (13), by Watson and collaborators (14), and by Kerman, MacManus, and Thaler (15) (the last paper will be referred to as KMT). A detailed review of the multiple-scattering problem may be found in Goldberger and Watson (16). The above formulations of multiple scattering are particularly useful because not only is it possible to introduce the approximations which are appropriate to the high energy domain but they also provide a systematic procedure for determining the subsequent error. In their paper, KMT focus on the optical potential, which, inserted into a Schrodinger equation, yields the exact S-matrix for elastic scattering. The optical potential is a very complicated quantity, since it depends upon both the dynamics of each projectile-nucleon scattering event as well as the dynamic structure of the target nucleus. KMT write the expression for the optical potential as a series. The
270
FESHBACH
AND
HtiFNER
first term depends essentially on the single-particle density of the target nucleus. This term is expected to generate most of the scattering by the nucleus. The next term is proportional to the two-body correlation function. It has a very complicated structure, and we are not aware of any convincing method for evaluating its effect on the scattering. The present study centers around this correlation term. We discuss its structure and propose how to take its effect into account in a numerical calculation. In Section II, we rederive the results of KMT by the method of coupled equations. It is well known that the many-body collision problem can be reduced to an infinite set of coupled equations by expanding the Hamiltonian and the wave function in the eigenstates of the target system. In this paper, we shall follow that procedure with the important difference from the usual practice that an effective potential (closely related to the nucleon-nucleon scattering amplitude) is used in place of the nucleon-nucleon potential (for the case of nucleon-nucleus scattering). We do not propose to solve the infinite system of coupled equations numerically (even if it were possible). But an approximation scheme can be developed. The neglect of all inelastic channels leads to the first term in KMT’s optical potential. Retaining the coupling to the elastic channel but neglecting the coupling between the inelastic channels is equivalent to the optical potential up to second order. We propose to compute elastic scattering for this case by replacing all the inelastic channels by one effective one whose coupling to the elastic channel is proportional to the two-body correlation function. Thus, calculating elastic scattering from the first two terms of KMT’s optical potential is equivalent to solving a pair of coupled equations. In order to provide some insight into the S-matrix for high-energy scattering by a nucleus, we generalize Glauber’s semiclassical methods from potential scattering to the solution of an infinite set of coupled equations (Section III). Then the computation of the S-matrix reduces to an algebraic problem. For certain classes of coupled equations, the algebraic problem can be solved analytically, and explicit formulae for the S-matrix are derived (Section IV), and they then exhibit how the scattering by a nucleus depends on the nuclear two-body correlations. Furthermore, the relation between the semiclassical solution of the coupled equations and Glauber’s expressions for multiple scattering is established.
II.
FORMULATION IN
OF SCATTERING BY TERMS OF COUPLED
NUCLEI AT EQUATIONS
HIGH
ENERGIES
Coupled equations are a familiar tool to describe nucleon-nucleus scattering at low energies, i.e., energies of a few million electron volts. One equation is associated with each state of the target, a prescription which leads to an infinite system of
SCATTERING
BY NUCLEI
271
coupled equations. Actual calculations are successfully performed by truncating the infinite system to a few important channels, usually selected by energy considerations. Highly excited target states and their associated equations are discarded (or included as part of an effective interaction) because they do not influence appreciably the scattering at low energies. The situation is different for highenergy projectiles. Although the system of coupled equations which is used at low energies remains valid at high energies (i.e., energies of or above several hundred million electron volts), a truncation to a few target states would not lead to a good approximation. Compared with the energy of the projectile, the target states appear to be degenerate, and we would expect that a great many target states are equally important. It is, however, possible to derive a set of coupled equations with improved truncation properties. The principal device is to replace the actual nuclear potential by an effective potential which has been derived by KMT. This effective potential takes into account the repeated scattering of the incident particle by the same target nucleon. The omitted terms which now give rise to channel coupling involve collisions with different particles and therefore depend upon nuclear correlations. Truncation of the resulting equations involves neglecting certain nuclear correlations. The following three sections are a reformulation of the multiple scattering theory proposed by KMT. The treatment of nuclear correlations in this formalism is discussed in detail. A. THE SYSTEM OF COUPLED EQUATIONS The Hamiltonian for scattering by a nucleus is H = HN + K, + v, Es H” + v, .
(2-l)
Here HN is the full Hamiltonian for the target nucleus, WN - d @a(rl, r2 ,..., TN) = 0,
(2-4
with properly antisymmetrized eigenfunctions (15= , Q:= 0, 1, 2,... . The kinetic energy of the projectile is denoted by K,, and V, contains the projectile’s interaction with the N target nucleons V, = f u(rO , ri). i=l
(2-3)
The scattering from the target nucleus is described by the Lippmann-Schwinger Equation (17)
~dr, , rl ,..., TN) = h3bs(r,,rl ,..., rN) +
29
E + iS - Ho V,Ul,(r,, r1 ,..., rN), (2-4)
272
FESHBACH
AND
HiiFNER
where I/Q is determined by the boundary condition kp = din(E
-
l a).
P-5)
Here, as everywhere in the following, we neglect the antisymmetrization of the projectile with the target nucleons if the projectile is a nucleon, retaining, however, the antisymmetrization of the target nucleons among themselves (cf. the arguments of Takeda and Watson (18)). The Green’s function in Eq. (2-4) contains the antisymmetrization operator d, allowing only the physical target wave functions as intermediate states: d -zz
E + 8 - H,,
LX=0
E + i8 - K. - E,
Since the interaction V, acts only between antisymmetric replace matrix elements of it everywhere by
’
G-6)
target states @, , we can
<@a I vo I @B> = N(@, I 0 I @l3s>7
(2-7)
with u from Eq. (2-3). The solution of Eq. (2-4) leads to the T-matrix cw
T,o = N<$b I * I %3>.
We shall replace the projectile-nucleon potential v by the effective interaction T which satisfies the equation sd d 7=%L+r (2-9) T=‘VtLL E+i~-Ho E f i6 - Ho ‘v’ The operator 7, which is closely related to the projectile-nucleon scattering amplitude, will be discussed in Section C. We multiply Eq. (2-4) by u and use Eq. (2-9)
d ~1- NT E + i6 - H,, 6’ + 7
d
E + i6 - Ho ‘L’ $0 + (N - lb d = T& + (N - 1)~ tz#g . E + i6 - Ho
=(
E + ,f-
H
0
“yo (2-10)
If a new function ul, is defined by (2-11)
SCATTERING
then the comparison
BY
273
NUCLEI
of Eqs. (2-10) and (2-l 1) leads to
from which equation we conclude
Tm,= NC&t IT I ‘lu,i.
(2-13)
Equation (2-l 1) can be cast in a system of coupled equations for the functions P)~, which are defined by the expansion of the total wave function in terms of target states
y = f do) @& ,..., rd.
(2-14)
‘X=0
The system of coupled equation is
(E-
E,-
Ko)% = f
ffd?& ,
(2-l 5)
B=O
where
v,, = (N- 1X@,, I i- I @k?>.
(2-16)
Although Eqs. (2-15) are formally identical to those used in low-energy nuclear physics, they differ by their interaction. In low-energy nuclear physics V,, is written in terms of the hermitian nucleon-nucleon potential ~1,while in Eq. (2-16) Yap is derived from the complex effective interaction 7. The only approximation in this section has been the neglect of the antisymmetrization-if necessary-between the projectile and the target nucleons. B. THE OPTICAL POTENTIAL, A complete solution of the infinite system of Eqs. (2-15) is impossible, in general. Moreover, even if obtainable, this solution would be “uneconomical,” since it contains everything, elastic scattering and excitation of each target state separately. On the experimental side, elastic scattering of high-energetic protons by nuclei has been studied rather extensively, while data for inelastic scattering are rare. This situation is not likely to change, becauseof the high resolution required to separate excitation of different excited states. Nuclei like 4He, lzC, and 160 show an energy gap between the ground state and the first excited state, thus making it relatively easy to isolate elastic scattering. But few nuclei show such large energy gapsbetween different excited states.
274
FESHBACH
AND
HijFNER
A way to calculate elastic scattering from Eqs. (2-15) without computing inelastic scattering as well is to introduce an optical potential. For the system Eqs. (2-15),
(E- E.- rc,- Vodplo = a#0 f ~o&L (E -
E, -
Ko -
Vm)v,
=
f
B(#d
Vm,vB
;
a: f
(2-17b)
0,
the optical potential is defined by (E - ,zo- K, -
(2-l 8)
Vopt)yo = 0.
By definition, the optical potential yields the same elastic scattering as the infinite system of equations, Eqs. (2-17). We shall calculate Vogt in a series of terms, the expansion parameter being the nondiagonal matrix elements of V,, . To lowest order, the comparison of Eqs. (2-18) and (2-l 7a) yields y(l) opt In next order, the optical potential (2-17b)
v 00.
(2-19)
is obtained by eliminating
ya = (E + i8 - E, - Ko -
Voct-’
qar from the Eqs. (2-20)
f Vm’olay~, B(#a)
and inserting it into (2-17a) keeping only the /I = 0 term
vd’dt + vgt = v,, + afO f vOcrE +
1 i8 - E, - K, -
V,,
V ao *
(2-21)
Similarly, all the higher orders of the optical potential can be obtained. It is hoped that the series for Vopt converges rapidly. The second-order potential V,-$ is already a complicated nonlocal operator. There have been estimates of this term, but we are not aware of a fully satisfactory treatment of V,$i in the literature. Therefore, in the remaining part of this section, we will introduce a series of approximations with the aim of making V$ + V$,“l amenable to numerical computation of nucleon-nucleus scattering. Our first approximation is generally employed, namely, to replace the excitation energies E, and the diagonal potential V,, in the denominator of Eq. (2-21) by average values d and v. Then closure is applied in
1 VA%= c vo, E + is - c _ K. - p “a ’ a#0
SCATTERING
BY
275
NUCLEI
and leads to the correlation function in the numerator of Eq. (2-22). We use a result of KMT in order to simplify the discussion: In the momentum representation, each matrix element V,, may be approximated by a product Pm’,,= (N -
1) ?<4Jp,(rl ,..., rN)I eeiq”l / QB(r, ,..., I-~)),
(2-23)
where i is the Fourier transform of the effective interaction Eq. (2-9) and the second factor carries all the dependence on the particular states 01,/?. The tilde indicates the momentum representation, q denotes the momentum transfer. The main assumption leading to the factorization Eq. (2-23) is neglecting the momenta of the target nucleons (in the laboratory system). If one also assumes that i is a function of momentum transfer only, then V&q) is a local operator. Using Eq. (2-23), we have for the closure expression in the numerator of Eq. (2-22) E;. ~o&l) ~‘O(S’) = (N - 1)” T(n) ?(a with
e(q, q’) as the nuclear correlation
z;(s, s’>,
(2-24)
function
C(q, 9’) ==X$ [ C (CD,/ $J eeiq”j 1@,)i@, / F ewiq”‘” 1@,)I a+0
__ “,y + ;
1 [‘p,
I=1
k=l
, e-i(a.r,+e’.r,)
I(@,, / e-i(s+s’).rl
/ Qo) - (dj, 1e-iq’rl
1 @o)(@o 1 e-iq”rl
~ Qo)]
/ @& - (@‘, 1e-iq’rl 1@,>(dj, / epiq’rlj Qo)]. (2-26)
The first term in Eq. (2-26) proportional to (N - 1)/N, contains the true two-body correlations, which originate (a) from the fact that the wave function O. is antisymmetric (“Pauli correlations”) and (b) from the nature of the nuclear forces, which generally lead to short-range correlations. The term proportional to l/N in Eq. (2-26) describesthe “self-correlations,” and would be present even if @, is a product of single-particle wave functions, in which case the true correlations vanish. As shown in Section C, the self-correlations in Vd,“i cancel against a term of vd,‘;, such that the optical potential depends only on the true correlations. By inspection of Eqs. (2-25) and (2-26), one proves WI, 9’) = wl’, 4,
(2-27)
and C(Sl 0) = 0,
(2-28)
relations which are true for each term of Eq. (2-26) separately. Even after closure has been performed in Eq. (2-22), V&$ is still a complicated operator, mainly because of the appearance of c and v in the denominator. Nevertheless, as the
276
FESHBACH
AND
HijFNER
discussion in Section IV shows, the average potential Vcannot be dropped but is necessary to compute V$\ properly. If in the sum of Eq. (2-24) only one state, say 01,,, contributes significantly (the choice of the intermediate states @, in Eq. (2-25) being rather arbitrary), the correlation function ii factorizes and the Schrodinger equation containing I’;$ + I$$ is equivalent to a pair of coupled equations. In actual nucleon-nucleus scattering, this situation may not be realized exactly, but it is conceivable that a group of states with very similar properties carries most of the inelastic strength. We think here especially of the quasi-elastic scattering, i.e., the (p, 2~) reaction. Guided by this picture, we propose to factorize the correlation function and write C(q, s’> = WI,
WI,
Q') = mI)
s’> + d 2((4> q’)
(2-29)
* WI').
(2-30)
The part of k’i$ which is proportional to z‘, can be treated by solving a pair of coupled equations which now replace Eq. (2-17):
The remainder, dc, may be amenable to perturbation theory. A numerical investigation of whether this is true is under way. A support for the ansatz Eq. (2-29) with d c’ = 0 is given in Section IV. There we solve the infinite system of coupled equations (which generates I’$\ + I’,$) by semi-classical methods and find that the elastic S-matrix can always by computed from a pair of coupled equations like the one of Eqs. (2-31) and an expression for A and P is derived. The discussion in this section assumed that the summation over the nuclear states Nin Eq. (2-24) is the only source of correlations. As shown in the next section, the effective interaction T already depends on the nucleon-nucleon correlations; so does @it = %-I) p(q).
(2-32)
The correlations can be extracted frown T, and a new effective interaction t, Eq. (2-45) (without correlations), can be iatroduced. All expressionsand conclusions of this section remain unchanged if one replaces Q-by t everywhere and drops the self-correlation term in Eq. (2-26). The main approximations in this section were (i) computing the optical potential only to second order in the nondiagonal matrix elements of the interaction; (ii) factorizing pea4 into the effective interaction and the nuclear form factor;
277
SCATTERING BY NUCLEI
Parallel to the present study, Kujawski (29) investigated the possibility to replace V$ by an equivalent local operator. C. THE EFFECTIVE INTERACTION
We shall now compare the effective interaction 7, Eq. (2-9), with the free projectile-nucleon scattering amplitude t’. We define it by (2-33) The indices “0” and “1” refer to the projectile and the struck nucleon, respectively. Schematically, the first terms of the Born seriesof Eq. (2-33) can be represented by the graphs of Fig. I. Here the vertical lines stand for the projectile and the nucleons
FIG. 1. The graphs represent the first terms of the Born series for the free scattering amplitude for the projectile “0” and a nucleon “1” (Eq. (2-33)).
of the target, and the dashed lines indicate the action of the potential ~1.Next we define a scattering amplitude t by the equation
to,
=
1'01
+ 2'01
1 E + i8 - K. - HN to1’
(2-34)
where the Green’s function resemblesthe one for T, Eq. (2-9), except that the antisymmetrization operator for the intermediate states is missing. The graphs for the first terms of the Born seriesare shown in Fig. 2. The double barred Iine represents the (N - 1) nucleons, which are not struck by the projectile. As the Hamiltonian HN in Eq. (2-34) contains the kinetic energy operator Kl of particle 1, the Born seriesof Eq. (2-34) contains all graphs of Fig. 1 (with the (N - 1) nucleons being unlinked) plus those where the struck nucleon interacts with all the other nucleons. This is called the binding energy correction (20), the first graph of this type being of third order in 1~(Fig. 2). Becauseof the antisymmetrization operator :d, the effective interaction EfiS-
K,-HN
r
(2-9)
278
FESHBACH
0
1 (N-l)
0
AND
1
HiiFNER
0
(N-l)
1 (N-l)
0
1 (N-l)
FIG. 2. The graphs represent the first terms of the Born series for scattering of a projectile “0” and a nucleon “l”, the nucleon being able to interact with the remaining (N - 1) target nucleons (Eq. (2-34)).
is symmetric in the labels of all target nucleons. This fact is expressed by the averaging procedure (l/N) Ci in Fig. 3. Furthermore, the antisymmetrization operator & in the intermediate states of Eq. (2-9) leads to the exchange diagram in secondorder of Fig. 3. As the projectile interacts with particles i and k, this diagram contains some information about nucleon-nucleon correlations in the target. The binding energy corrections which were shown in Fig. 2 will also appear in the thirdorder terms of Eq. (2-9).
0
i (N-l)
n#k
0
i
k (N-21
0
i
k
FIG. 3. The graphs represent the first terms of the Born series for the scattering of a projectile “0” from a nucleus, where the antisymmetrization of the target nucleons is properly taken into account (Eq. (2-9)).
Both terms of the optical potential (2-35) contain information about the nucleon-nucleon correlations. By expressing the operator T in terms of t defined in Eq. (2-34) and combining terms in Eq. (2-35), we extract the correlations from the first-order potential. We write 71 = f, + t,
d t E + i6 - K. - HN -
1 E + i6 - K, - HN 1 r’ *
(2-36)
Then
v:;t= (N- l)(@o I-I-I@o) = (iv - l)(Qo j t, j Qo) + (N - 1) 1 a#0 (@0I~11@0x@0
-t’N-ll)l -
(~olh
I Tll@O>
E+i6-Ko-~o 1 E+i&--Ko-HN
(2-37)
SCATTERING
BY
279
NUCLEI
The first term in Eq. (2-37) gives rise to the scattering from the single-particle density of the nucleus, and no approximation has to be made. The remaining terms are of the order of (N - 1)-l compared with the second-order optical potential V$,[ . Some approximations have to be introduced in order to combine these terms with V$$ and obtain a simple result. The main difference between the correction terms in Eq. (2-37) and Eq. (2-35) is the average potential Fappearing in the denominator of the expression for V,,,(2) , Eq .(2-35). We will simply introduce Binto the denominators of the terms in Eq. (2-37). The error associated with this procedure is of the order of (V,,J2 . (r/(N - 1)) in the total expression for the optical potential. If we count F/(N - 1) to be of the same order as an inelastic matrix element V,, , then the correction is of third-order in the inelastic matrix elements. Terms of similiar order have been neglected in our derivation of the optical potential from the beginning. Furthermore, we replace T by t in all second-order terms of the optical potential. Then the second term in Eq. (2-37) readily combines with the expression for V$ , and the third term (in curly brackets) cancels the self-correlation term in the correlation function Eq. (2-26). This result has already been stated in the appendix of KMT. In actual calculations one will usually also neglect the binding-energy corrections in the scattering amplitude Eq. (2-34).
III.
SEMICLASSICAL
SOLUTION
FOR
A GENERAL
SYSTEM
OF
EQUATIONS
Thresholds and compound resonances characterize and complicate the lowenergy solutions of a system of coupled equations like the one of Eqs. (2-15). Therefore, in most cases their solution can be obtained only by numerical integration of the equations. At high energies, nuclear thresholds are far from the energy of scattering, and the time which the projectile spends inside the nucleus is too small to develop pronounced compound resonances. Therefore, the solutions of the coupled equations at high energies are expected to vary smoothly with energy, and semiclassical methods may be appropriate. In this section, we closely follow the reasoning of Glauber (2) when generalizing his treatment of potential scattering to the solution of the set of coupled equations, Eqs. (2-15). A pair of coupled equations, as well as a case with an infinite number of channels, has recently been studied in Glauber’s approximation by Bassichis, Feshbach, and Reading (10). The following derivation generalizes their result and discusses in detail the problems arising from the noncommutativity of the interaction matrix. We start from the set of Eqs. (2-15),
(3-l)
280
FESHBACH
where we have explicitly which is
exhibited
AND
HtiFNER
the dependence on the boundary
for
Z---CO.
condition,
(3-2)
The source of the projectiles is located at z = - co. By h we denote the state of the target before the scattering event. The wave number k, is given by d2m(E - en) in the nonrelativistic limit, the additional factor (m/k,)1/2 ensures flux normalization. Because of its high energy, the projectile is not appreciably deflected during its way through the target. Therefore, following Glauber (2), we set (3-3) and approximations will be made for the functions #y’(r). the coupled equations leads to the system of equations
Inserting
Eq. (3-3) into
We assume that $(r) varies slowly over distances k-l; therefore, to first approximation the second derivatives in Eqs. (3-4) are neglected. There exist various proposals, e.g., Feshbach (II) or Hahn (22), as to how to improve the approximations, i.e., how to partially include the effect of the second derivatives. We will not follow these lines but will simply neglect A/2m. Then $ b(r)
= --i 1 Kdr> 6
(3-5)
949
The system of Eqs. (3-5) and (3-6) is the obvious generalization of Glauber’s expression for one-channel potential scattering. The correct inclusion of the nuclear excitation energies multiplies the potential matrix V,, with a hermitian factor exp[--i(k,
- k&z] ~exp
[+id2n2E*
( Ea2iEa )z],
(3-7)
the value of which might differ appreciably from one for large nuclei. The factor m/(k,ks)li2 generalizes the coefficient m/k which appears in Glauber’s expressions. The accuracy of the approximations which lead to Eqs. (3-5) are difficult to assess, because matrices appear instead of the scalar potentials in Glauber’s treatment.
SCATTERING
For his case, Glauber finds two classical solution. k-l,
BY
sufficient
281
NUCLEI
criteria
for the validity
of the semi-
(i) The potential varies slowly in space, slowly compared with the wavelength or k.R> 1 (3-8)
(R characterizes
the size of the nucleus).
(ii) The strength
V of the potential
satisfies
V/E<
1.
(3-9)
Tn the many-channel case, the condition (i) can be taken over without any change. As for (ii), we remark that the potential matrix I’,, is nearly diagonal in the representation in which the projectile-nucleon scattering amplitude t is the basic interaction. Therefore, Glauber’s conditions have to hold separately for the diagonal part of each channel. As Reading (22) has commented, the structure of the differential equation Eq. (3-5) is reminiscent of the equation for time evolution in the interaction picture. Here the space derivative replaces the time derivative. Therefore, in complete analogy, a space evolution operator L’(r) can be introduced by Q’(x,
y, z) = 1 U&(X, 4’, z) yp(x, I3
y, z = -co).
(3-10)
It takes the wave function from its value at z = - CCto its actual value. Following this analogy, we anticipate that the S-matrix is given by (3- 11)
&3(x, v) = &3(.-G y, 2 = + co),
a result which will be proven at the end of this chapter. Here x and y are treated as parameters, b = (x, y) being the impact parameter. Using Eq. (3-10) together with Eq. (3-5) we derive an integral equation for U,, : U,,(r)
= &, - i c 1’ dz’ ,YL,,(b, z’) U,,,(b, z’). Y -m
(3-12)
In general, the solution of Eq. (3-12) cannot be found explicitly but has to be evaluated by the series U,,(r) = 6,, - i jZ
--rli
$
C-i)"
f
dz’ 9ia(b, c
--m Y
Fi,(b,
z’) z’) dz’ jZ’ V$(b,
--0
z")
dz” + ... .
(3-13)
282
FESHBACH AND HijFNER
However, if the two matrices Y(r) and Jtm Y’(r) dz’ commute, i.e.,
[v(x,y,z),r;,dz’9“(x,Y,z’,]=0,
(3-14)
the seriesEq. (3-14) can be summedup to give (3-l 5) a form which is familiar from Glauber’s one-channel case, where Eq. (3-14) holds trivially. A sufficient condition for Eq. (3-14) to be valid is that Y can be written as a sum of matrices (3-16) with [A(“), fp’]
zr 0
and where the matrices Atn) do not depend on the spatial coordinates. Another condition for Eq. (3-14) to hold usesa “random phase” assumption for the matrix elements1YoB(r) if they are evaluated at different points of space, namely,
c8 T&x,y,z’>%3?J(x, Y,z>K L%z- z’>.
(3-l 7)
One of the requirements entering the models which we are going to study in Section IV will be that the solution of our system of coupled equations can be written in the form of Eq. (3-15). Before turning to these models we prove relation (3-11) for the S-matrix in the small angle approximation. We start from the scattering amplitude .&(k,
k’) = - g
j d3r eickek’)” c 9:.,(r) #F)(r),
Y
and insert (3-10): f&k,
k’) = -
$f
i d3r eickek’).‘~ V&(r) U,,,(r).
Y
(3-19)
1 Similar arguments are invoked to justify the locality of the imaginary potential in low-energy reaction theory (23).
SCATTERING
BY
283
NUCLEI
The small-angle approximation, together with the assumption of small longitudinal momentum transfer (for inelastic processes it is proportional to k(ca - c,)/2E) leads to the expression .f&(k, k’) = - $
s d2b ei(k-k’)‘b
s 1 T,(b,
Y
z) U,,(b, z) dz.
(3-20)
The integral in (3-20) is performed by using the integral Eq. (3-12) and yields fJk,
k’) = - $
J^d2b ei(k-k’)“[8aB
-
U&b, z = + a)].
(3-21)
Therefore the S-matrix is given by Eq. (3-11) and in the casewhen Eq. (3-14) holds, ‘v S,,(b) = /exp [--i 1:: dz’ Y’(b, zj)] ino.
(3-22)
Although Eq. (3-22) is an explicit expression for the S-matrix, each element of the matrix i,sa very complicated function of the elements J
dz’ F&(b, z’).
After the neglect of secondderivatives in Eq. (3-4) the infinite system of coupled equations could be solved by quadratures only. The further assumption of commutativity ‘of the potential matrix at different points in space, Eq. (3-14) led to the S-matrix as the exponential of the interaction matrix.
IV.
SOLVABLE
MODELS
FOR
MANY-CHANNEL
SCATTERING
One aim of our study has been to find models of a genuine many-channel situation simple enough so that explicit formulae for each S-matrix element are obtained. Thereby, we hope to provide someinsight into the structure of the manychannel S-matrix for high-energy scattering by nuclei. For this purpose, even the general expressionfor the S-matrix Eq. (3-22) is not simple enough, sinceit involves the exponentiation of an infinite dimensional matrix. Therefore, we shall discuss several caseswhere the exponentiation can be performed in closed form. A. THE BORDERED INTERACTION MATRIX The optical potential for elastic scattering up to secondorder in the nondiagonal matrix elements is given by Eq. (2-21). We observe that Eq. (2-22) is the exact
284 expression interaction
FESHBACH
for the optical potential matrix
AND
HiiFNER
of a scattering
problem
with a “bordered”
(4-l)
i.e., where only matrix elements between the ground state and the excited states are important and where the interaction between different excited states is neglected. In the eikonal approximation, the S-matrix for the interaction (4-l) can be solved exactly. We define M,,(b)
= j.+m 9$(b,
-m
z’) dz’,
(4-2)
where the quantities 9’& and V,, are related by Eq. (3-6). The functional dependence of the S-matrix elements on the MaiBis computed by determining eigenvectors and eigenvalues of the matrix M,,(b):
in terms of which the S-matrix is given by
In general, M,, is a non-hermitian but symmetric matrix and is diagonalized by a complex orthogonal transformation. The eigenvectors n” are adjoint to n but not their complex conjugates. The specific form of M,, , Eq. (4-2), leads to the eigenvalue problem C Moon? = (A - Moo)@ O#O Moons)
= (A -
R)n~‘,
(4-5a) R = Ml1
= M,,
= .*- .
(4-5b)
If h # &?i,the component nr’, 6 > 0 can be eliminated from Eq. (4-5b):
MBO IP’B =X--Rno*
(A)
(4-b)
When inserted into Eq. (4-5a), it leads to the secular equation (A - Moo)0 - ml = C MO&f,, , B#O
(4-7)
SCAT’I‘ERING
BY
NUCLEI
285
with the solutions
Y’ = c MORMRO< P#O
(4-9)
Because of relation Eq. (4-6) the eigenvectors related to h,,, are
ini-)
= N;; . (A,,, - R), Ml0 ) M2, ,.,. 3,
(4-10)
and the normalization is calculated to be
N& = (A,*, - ET)”+ Y2.
(4-11)
All other solutions of the system Eq. (4-5) have degenerate eigenvalues h, = ;i;l for k > 2 and, therefore, according to Eq. (4-5b), $J
= 0
k > 2.
(4-12)
The eigenvectors for k > 2 need not be constructed as long as one wants only the S-matrix elementsS,, , Q:= 0, 1, 2,..., which are the only elements of our model with physical consequences: (A, - AT)” e-ih’ so0= 2 ,,=I 0, - m2 + Y2
(4-13a)
2
Gb - 2) so, = MO=/bxl (A, _ a>2 + y2 e-ihkYa f 0
(4-13b)
Using Eq. (4-7) one arrives at s
00
=
e-i(M,,+m/2
so, = -i
M,,
jcos x _ i Moo - 33 I sin X ) t 2 x t . e-i(Mo,+W)12
SifX
(4-14) (4-l 5) (4-16)
All quantities Moo , &i, and X are complex numbers. Trigonometric functions of complex arguments rise exponentially. Since the effective interaction T satisfies a Lippmann-Schwinger equation and therefore is a scattering amplitude, its imaginary part is negative definite. This property propogates into the matrix M,, (as it is a
286
FESHBACH
AND
HijFNER
matrix representation of T). Therefore, the eigenvalues of M,, have negative definite imaginary parts and lead to exponentials with values smaller than 1 in Eqs. (4-l 3). E.g., the case MO, = I%?leads to j So, 1 = + 1 epiA1+ eviA2 1 < 4 [I e-“’ / + 1e-+ I] < 1, which shows that unitarity complex. B.
is not violated even if the interaction
(4-17)
matrix M,, is
DISCUSSION
The eikonal approximation leads naturally to the prescription to use a pair of coupled equations to compute elastic scattering for the interaction matrix Eq. (4-l). Employing the methods described in Section A, one proves that ,I$,, , Eq. (4-14), is the semiclassical solution to a pair of coupled equations with the twodimensional interaction matrix (4-l 8) The potential Y which couples the two channels is however different from the one proposed in Eq. (2-36). We define a coupling potential V, by the integral equation
&
,I,” VAb,2’) dz’ = Y(b) = dB& Mm(b)Wm(b>,
(4-19)
where Y is proportional to the correlation function, cf. Eq. (2-24). The wave number k = 1/2m(E - E) relates to the average excitation energy <. The integral equation Eq. (4-19) for V, can be solved under the assumption that V, is spherically symmetric (Glauber (2)): v(r) c =-
da -__
m
1 m
d m Y@)bdb -. dr I r d/b2 - r2
(4-20)
For a comparison with the coupling potential (2-36), we take the Fourier transforms of Eq. (4-19). Each integration over z’ is replaced by the condition qz = 0 in the Fourier transformed quantities: d2q d2q’ eib.(Q+q’)A&(q, 0) &f&q,
0). (4-27)
Therefore, s
d2(q- q’)[~c(q,0) t%f, 0) - W - II2 %q,0) dq’, 0) ec
(4-22)
SCATTERING
BY
287
NUCLEI
The factorization condition Eq. (4-22) should be compared with the one proposed in Eq. (2-35). As long as (q - q’) remains small, both conditions are equivalent. We recall that Eq. (2-35) holds not only for q - q’ = 0 but also for the first derivative in (q - q’) at q = q’. However, the two terms in the bracket of Eq. (4-22) will decay differently as a function of q - q’. To that extent conditions (2-35) and (4-22) are different. The total momentum transfer during the double scattering event is not given by (q - q’) but by (q + q’), and no condition has to be imposed on this quantity. The difference (q - q’) measures only how the total momentum transfer is distributed between the first and second scattering. The elastic scattering amplitude S,,,(b) depends on the nuclear two-body correlation function via the term Y2(b). We ask: Given the cross section for elastic scattering of a nucleon from a nucleus, can one unambiguously extract information about the correlations in the nucleus? According to Eqs. (4-14) and (4-16) the correlations in Y2 always appear together with the difference in the diagonal potentials (M,, - 1i!i)~/4. Only if one knows this quantity to an accuracy much better than the expected value of Y2, can one learn about the correlations by studying elastic scattering. M,, - R may be different from zero for two reasons: (i) The average excitation energy E - Edis different from zero. Then (MO, - R)2/4 N M&((E - E,)/4E)?
(4-23)
The factor (C - c,)/E seems to go to zero with increasing energy E of the projectile. This conclusion is misleading if the average excitation energy E is a function of E. In the case of “quasi-free scattering,” the average excitationrenergy is given by E - Eg _ 1 q2 ----= 4E 2M 4E
4k2 sin2W> 2ME.4
= sinz(e/z>
(4-24)
and is therefore a function of the angle 8, independent of E. (ii) The diagonal potentials M,,, and R differ because the excited states will have a matter distribution which differs from the one for the ground state. We use Gaussian matter distributions p(/3, r) cc ecBr2
(4-25)
and approximate MO, - R Moo
II i
SP(B,r)dz-sP(B+6B,r)dz I
=-
s P@, r) dz W2) (r2)
[
1 _ 2 2 (r2)
1 ’
(4-26)
288
FESHBACH
AND
HtiFNER
While for one particle-one hole excitations the quantity 6(r2)/(r2) has a value of 2.5 y0 for 160, we find 15 y0 for 4He. In general, the expression Eq. (4-26) will decrease with increasing number of nucleons in the target. In order to know how much the difference M,,, - R may disturb the correlations, we have to estimate the magnitude of the correlation term
Ih$)12=
I
.fdzdz’,4b,z>0, z’)f(l z - z’ I> ,~5 .f dz p(b, z) j dz’ p(b, z’)
R’
(4-27)
Here, the nuclear correlation function has been approximated in the form of Eq. (2-41), where the function f(z) vanishes for 1z 1> re . The term re is called the correlation length and may be assumedto be of the order of 0.5-l.Of. (We denoted the nuclear radius by R.) According to Eq. (4-27), the correlation Yis only a small fraction of the diagonal term M,, . Therefore, M,, and M have to be rather well known in order to determine Y2 reliably. Some further insight is obtained if we make the reasonable assumption to start with that M,, = &i. Then so0= e-~wO(l _
+y2
+ ...)
for
1 Y2j<<1.
(4-28)
Equation (4-28) hasto be contrasted with the expressionfor the S-matrix computed for R = 0:
+ je-i(Moo’2) sin(M,,,/2) A&,/2
Y2 MO, + .** *
(4-29)
The influence of the correlations on the S-matrix is completely different for the two choices R = Moo and I% = 0 (except for Moo< 1). Although the possibility R = 0 does not appear to be a reasonable prescription in the framework of our models, it doesnot seemto have thesedrastic consequencesin the multiple scattering formalism reviewed in Section II. Indeed, Chalmers and Saperstein (24) use the prescription &? = 0 in their numerical calculations of nucleon-nucleus scattering. Although the possibility % = 0 may also lead to a fit to the experimental data, the interpretation in terms of nuclear correlations seemsquestionable. Concerning the inelastic matrix elements, Eq. (4-13, we find it remarkable that So, for j3 # 0 is proportional to the “Born matrix element” MO, , although elastic and inelastic scattering are treated to all orders. The factors which multiply MO0originate from initial and final state interactions and also ensure the unitarity of the S-matrix. The fact that Soaisos= Mom/Moo seemsto be a specific feature of the bordered matrix Eq. (4-l).
(4-30)
SCATTERING C. STATISTICAL
BY NUCLEI
289
ASSUMPTIONS
The bordered interaction matrix Eq. (4-l) described a situation where the elastic channel played a dominant and singular role: only this channel interacted with all the other channels. A complimentary situation, where the interaction matrix elements between all channels are equally important, is treated in this section. However, an explicit formula for each S-matrix element can be obtained only, if one makes certain statistical assumptions concerning the interaction matrix Mma = i__ t/;k6
!
‘+I dz’ Y(b, 2’) dz’.
We split A4,, into an average diagonal matrix ,?!i and into the remainder. We define R such that statistical arguments apply for the difference x,, = Iv,, - Ii? a,, .
(4-32)
(i) Gaussian Distribution: If a Gaussian probability distribution (4-33) holds for each matrix element X, then the ensembleaverage over S,, is diagonal and (S,,)
= a,, exp(-i?i? - J-(X2>),
(4-34)
where (X2) denotes the ensembleaverage of X2. (ii) Random Phase Assumptions: Tf the matrix elements X,, have random signs and one assumesthat (4-35) i.e., the diagonal elements of X2 are independent of 01,then one obtains for the higher powers of X,, c &343,J-,d = xX2? x,,, 2 BY
(4-36)
1 xaJ~yxvsx~~~ = 2(X2)2 a,,, )... . B,Y,6 The S-matrix is evaluated to --
S,, = e--iM I?&(&+ + cos d2(X2))
1
595/56/1-19
sin 2/2(X2) - X,, ---. 2/2(X2> 1
(4-37)
290
FESHBACH
In both cases, (i) and (ii), the quantity function, since
AND
HtiFNER
(X2) contains the two-body correlation
averaged over the states 01. When comparing the model S-matrices for elastic scattering as derived from the bordered interaction matrix and the two statistical models, one observes a close similarity. They all express the element S,,,, in terms of three parameters, M, the deviation from the average potential, e.g., M, - M, and the correlation term. Furthermore, to lowest order in the correlation term, all models show the same dependence (we assume M,,, - M = 0), S,,, = cim (1 - 4 c
M,,M,,
+ *s-j,
(4-39)
u&a,
since we treated the second order of the inelastic matrix elements exactly. Deviations between the models occur for higher orders of the correlation term. Although none of these models depends explicitly on nuclear correlations of higher than second order, each of these models implicitly contains an assumption about how to express higher-order correlations in terms of densities and two-body correlations. For instance, terms of fourth order in X,, contain some information about fourthorder correlations, and in all the previously discussed models the fourth power of X,, has been expressed in terms of the two-body correlation function only. If we introduce
z=a
Y2
z = $ z=D
Y2
for the bordered matrix
(4-41a)
for the Gaussian distribution
(4-41 b)
for the random phase model.
(4-41c)
The comparison of the three results, Eqs. (4-41), should lead to some caution. If a numerical investigation of the optical potential, Eq. (2-21), shows that the secondorder potential cannot be treated in perturbation theory, higher-order correlations among nucleons may be important for the scattering. At least the rather formal
SCATTERING BY NUCLEI
291
truncation leading to the bordered interaction matrix may be suspect. Statistical models start from a certain picture for the nucleus, and therefore their results may have greater significance. Nevertheless, we cannot be sure, whether or not the statistical models will describe correctly so delicate an object as four-body correlations, especially since they also differ in their predictions. We conclude from the agreement of all different models up to first order in the correlations that the results of the optical potential V,$ + V$$ may at least be trusted up to first order in V$ ,. D. COIVIPARISON WITH GLAUBER'S MULTIPLE
SCATTERING
Given the free nucleon-nucleon scattering amplitude and the wave function for the ground state of the target, Glauber’s multiple scattering formalism provides a method for computing the S-matrix by integrations only. No equation with an optical potential has to be solved. Each S-matrix element for elastic scattering can be written as &,,(b) = ,+‘db),
(4-42)
where b either denotes the partial wave Z, (I N kb), or is the continuous impact parameter. From his expression for SO, , Glauber (2) derives a series for x&b) xopt(b)
:= N s p(r)(--ir(b
- s)) d37
+ ; NW- 1)j [pt2)(r, )r2)- y&- P@I) P(G)] [--ir@ - s,)] X [--ir(b
- sz)] + 3rd order in r + me*.
(4-43)
Here, the “profile function” ris proportional to the two-body scattering amplitude t (in configuration space), s is the projection of r on the impact parameter plane, and pc2)(r1, rJ =
i
/ c&Jr1 , rz , r3 ,..., rN)12d3r3d3r., a*-d3r, (4-44)
p(r) = i d3r’ p(‘)(r, r’).
The convolution integrals involving p and I’ in Eq. (4-43) are the reflection of a product representation ,6(q) F(q) in momentum space-equivalent to the assumption introduced in Eq. (2-23). Except for a factor N/(N - I), which we will comment upon below, the first term in Eq. (4-43) is identical to M,,, as defined in Eq. (4-2). Similarly, the next term in Eq. (4-43) corresponds to the correlation term Y2. We
292
FESHBACH
AND
HtiFNER
have compared Glauber’s result, Eq. (4-43) with the S-matrix obtained by solving the infinite system of coupled equations. We find a one-to-one correspondence (up to second order in Y2) only if we assume M,,, = a. That means that Glauber’s formalism treats the nuclear levels to be all degenerate (the average excitation energy E = E,,). Furthermore, the formalism does not allow for a change in nuclear shape during the virtual excitation of intermediate states; i.e., the average potential P in the excited states is identical to that for the nuclear ground state. Both these restrictions are usually referred to as the “frozen nucleus” assumption. The origin of the factor (N - 1)/N is traced as follows: We recall that the transformations in Eqs. (2-8)-(2-l 3) replaced the projectile-nucleon potential Nu by the effective interaction (N - l)~, from the scattering function !PO , Eq. (2-11), is computed. In order to calculate the T-matrix, Eq. (2-13) the interaction which appears in the matrix element is NT, not (N - 1). Therefore, the S- or T-matrix which had been derived and discussed in the semi-classical approximation did not represent Eq. (2-13) but should have been replaced by
Tmo = -&
((c+"),,
- S,,)&.
We neglected this point, which led to the slight difference in the comparison with Glauber’s results. We do not propose the semi-classical solutions of Section IV as a practical means to compute cross sections but rather as a model to give insight into the structure of the S-matrix at high energies. The computations which are under way use the formalism outlined in Section TT.
V. SUMMARY
AND CONCLUSIONS
We have presented a theoretical study of high-energy scattering by nuclei with particular emphasis on the effect of nucleon-nucleon correlations. Most of the calculations in this field use Glauber’s semi-classicalmethods. The validity of these results become doubtful for larger angles,where one would expect the correlations to show up most. For this reason, we used the more fundamental approach to multiple scattering as worked out by Kerman, McManus, and Thaler. An infinite system of coupled equations determines the scattering wave function. As employed by KMT, the actual interaction potential with the target nucleons is replaced exactly by a many-body operator which describes the interaction between the projectile and a target nucleon as modified by the presenceof the remaining target nucleons. In lowest approximation the effective interaction reduces to the free target nucleon transition operator. At this level of accuracy and neglecting the excitation of target states, the optical potential describing the projectile-nucleus
SCATTERING BY NUCLEI
293
interaction is proportional to the target-nucleon density and to the forward scattering amplitude. This “single-scattering” optical potential is a well-known result. If one retains the coupling of the elastic channel to the inelastic ones but neglects the interaction between all inelastic states,the second-order, or double-scattering, term for the optical potential is obtained. It depends bilinearly on the free nucleon scattering amplitude. As expected, this term is directly related to the two-particle correlation function. The result does include off-the-energy-shell effects, and no approximations need be made as to longitudinal momentum transfer. An effective potential describing the motion of the projectile between collisions is introduced as well. Evaluation of this second-order optical potential is by no means trivial. We propose to describe the single- and double-scattering process by a pair of coupled equations, one for the incident channel and the other for an effective channel. Since the probability for leaving the incident channel and exciting another state is directly proportional to the two-particle correlation function, it is no surprise that the coupling potential between the incident and the effective channel is proportional to this correlation function. This approximation of a factorization of the correlation functions is currently being used to analyze recent high-energy experiments. Insight into the dependenceof the S-matrix at high energy is obtained by solving the infinite systemof coupled equations by semi-classicalmethods, which are closely related to those which Glauber has developed for potential scattering. This procedure neglects the longitudinal momentum transfer. It assumescommutativity between the interaction matrix at two different points along the classicaltrajectory. However, it includes the effect of correlations. For the casewhere one neglects all interactions between inelastic channels(a caseequivalent to the single- and doublescattering optical potentialj, the S-matrix is evaluated explicitly. It turns out that this S-matrix for elastic scattering can always (i.e., without factorization assumptions on the correlation function) be obtained from a pair of coupled equations. The discussion also shows that the energies and average potentials of virtually excited target states may interfere with an accurate extraction of nucleon-nucleon correlations in high-energy scattering. Another model for scattering by nuclei, where one does not neglect the interaction between inelastic channels but imposesstatistical requirements on them, leads to similar conclusions. The correspondence to Glauber’s formalism is established.
ACKNOWLEDGMENTS We are very manuscript.
grateful
to Dr.
RECEIVED: June 17, 1969
A. Gal
for a number
of discussions
and for critically
reading
the
294
FESHBACH AND HiiFNER REFERENCES
in “High Energy Physics and Nuclear Structure” (G. Alexander, Ed.), p. 151. North-Holland, Amsterdam, 1967. H. PALEVSKY et al., Phys. Rev. Letters 18, 1200 (1967). G. W. BENNETT et al., Phys. Rev. Letters 19,387 (1967). G. J. IGO et al., Nucl. Phys. B3, 181 (1967). 2. R. J. GLAUBER, in “Lectures in Theoretical Physics” (W. E. Brittin and L. G. Dunham, Ed%), Vol. I. p. 315. Wiley (Interscience), New York, 1959. R. J. GLAUBER, Phys. Rev. 100, I. H. PALEVSKY,
242 (1955). 3. R. J. GLAUBER,
in “High Energy Physics and Nuclear Structure” (G. Alexander, Ed.), p. 311. Amsterdam, 1967). V. FRANCO AND R. J. GLAUBER, Phys. Rev. 142, 1195 (1966). R. J. GLAUBER AND V. FRANCO, Phys. Rev. 156, 1685 (1967). R. H. BASSEL AND C. WILKIN, Phys. Rev. Letters 18, 871 (1967); Phys. Rev. 174, 1179 (1968). W. Czvi AND L. LESNIAK, Phys. Letters 24B, 227 (1967); Phys. Letters 25B, 319 (1967). W. Czyi AND L. C. MAXIMON, Phys. Letters 27B, 354 (1968); Ann. Phys. (N.Y.) 52,59 (1969). J. FORMANEK AND J. S. TREFIL, Nucl. Phys. B3, 155 (1967). J. S. TRERL, MIT preprint CEP 15 (1968), submitted to Phys. Rev. E. A. REMMLER, preprint (1968). W. BASSICHIS, H. FESHBACH, AND J. F. READING (to be published). Also H. FESHBACH, “International School of Physics Enrico Fermi 38th Course.” Academic Press, New York, 1967. H. FESHBACH, MIT preprint CTP 44 (1968). L. L. FOLDY, Phys. Rev. 67, 107 (1945). M. LAX, Rev. Mod. Phys. 23,287 (1951). K. M. WATSON, Phys. Rev. 89, 575 (1953). N. FRANCE AND K. M. WATSON, Phys. Rev. 92, 291 (1953). W. B. RIESENFELD AND K. M. WATSON, Phys. Rev. 102, 1157 (1956). K. M. WATSON, Rev. Mod. Phys. 30, 565 (1958). A. K. KERMAN, H. MCMANUS, AND R. M. THALER, Ann. Phys. (N.Y.) 8, 551 (1959) (referred to as KMT). M. L. GOLDBERGER AND K. M. WATSON, “Collision Theory” Wiley, New York, 1964. B. A. LI~PMANN AND J. SCHWINGER, Phys. Rev. 79,469 (1950). G. TAKEDA AND K. M. WATSON, Phys. Rev. 97, 1336 (1965). E. KUJAWSKI, Thesis, MIT (1969). G. F. CHEW AND M. L. GOLDBERGER, Phys. Rev. 87,778 (1952). G. F. CHEW AND G. C. WICK, Phys. Rev. 83, 636 (1952). Y. HAHN, Phys. Rev., 174, 1135 (1968). J. F. READMG, private communication. C. BLOCH, J. Phys. Radium 17, 510 (1956); Nucl. Phys. 3, 137 (1957). J. S. CHALMERS AND A. M. SAPERSTEIN, Phys. Rev. 168, 1145 (1968). (North-Holland,
4. 5. 6.
7. 8. 9. 10.
Il. 12.
13. 14.
15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
OF PHYSICS:
56, 268-294 (1970)
On Scattering
by Nuclei at High Energies*
H. FESHBACH AND J. H~~FNER Lahorafory for Nuclear Science and Department of‘ Physics, Massachusetts Institute of Technology, Cambri&e, Massachusetts 02139
We study how nucleon-nucleon correlations influence high-energy scattering by nuclei. Using the multiple scattering theory of Kerman, McManus, and Thaler, we formulate the scattering problem in terms of an infinite system of coupled equations. For the calculation of elastic scattering, we propose replacing the infinite system by a pair of coupled equations, containing the elastic channel, and another effective one which carries the inelastic strength. The coupling potential is proportional to the nuclear pair correlation function, and a first approximation to the potential is obtained These equations are accurate to the extent that the pair correlations and not higher-order correlations are important. There are no restrictions to forward scattering, nor are any “on the energy shell” approximations made. In order to obtain insight into the structure of the many-channel S-matrix for high energies. the infinite system of coupled equations is solved by semi-classical methods, and explicit formulae for the S-matrix elements as a function of various potentials and the nuclear pair correlation function are obtained. We show how properties of excited target states may complicate a reliable extraction of correlations in high-energy scattering by nuclei. A close relation between our semiclassical solution and Glauber’s multiple scattering is established. A numerical study of high-energy nucleon-nucleus scattering using the methods developed in this paper is under way.
I. INTRODUCTION
The present (I) and anticipated useof high-energy particles to investigate nuclear structure provides the motivation for this study. Because of the relatively high energy incident upon and emerging from the target nucleus, a short-wavelength semiclassical description is indicated. The most frequently used methods are based upon the work of Glauber (2,3), Franc0 and Glauber (4, Bassel and Wilkin (5), Czyi and Lesniac (6), Czyi and Maximon (7), and Formanek and Trefil (8). Briefly, thesepapers assumethat the nucleus can be described as a series of scatterings by the nucleons of the nucleus. More detailed assumptionsare: * This work has been supported in part by the Atomic Energy Commission under Contract No. AT(30-1)2098.
268
SCATTERING
BY
(1) The target nucleons are stationary, the nucleus (“frozen nucleus”).
NUCLEI
while the projectile
269 passes through
(2) The momentum change in the direction of the projectile, the “longitudinal momentum transfer,” is neglected. (3) Each collision between the projectile the collision between free particles.
and a target nucleon is identical to
(4) The total change in phase of the particle wave function equals the sum of the phase changes which occur at each collision with a target nucleon (“additivity hypothesis”). Although this is not the only set of assumptions which leads to Glauber’s results (cf. Czyi and Maximon (7) and Remmler (9)), semiclassical considerations in the spirit of conditions (l)-(4) are always involved. The resulting formulas are relatively simple, easily interpreted physically, and easily evaluated (at least if harmonic oscillator wave functions describe the target state). The results obtained with these theories are thought to be valid in the small-angle high-energy domain. And indeed their agreement with experiment in this range is excellent. However, it is not at all clear that these approximations are sufficiently accurate as to permit the determination of the more subtle features of nuclear structure such as correlations. To obtain these, it is generally necessary to investigate larger-angle scattering where it is at the very least necessary to include the effects of longitudinal momentum transfer as is indicated by recent calculations (10). The additivity hypothesis may also not be sufficiently accurate (II). This hypothesis neglects intermediate off-the-energy shell contributions. For fast particles, these have their greatest significance for short-range correlations which are the interesting ones, since long-range correlations are dominated by the effects of the Pauli exclusion principle. For these as well as other reasons, the conclusions drawn by some of the above papers with respect to nuclear correlations are suspect. A general treatment of the multiple-scattering problem has been given by Foldy (IL’), by Lax (13), by Watson and collaborators (14), and by Kerman, MacManus, and Thaler (15) (the last paper will be referred to as KMT). A detailed review of the multiple-scattering problem may be found in Goldberger and Watson (16). The above formulations of multiple scattering are particularly useful because not only is it possible to introduce the approximations which are appropriate to the high energy domain but they also provide a systematic procedure for determining the subsequent error. In their paper, KMT focus on the optical potential, which, inserted into a Schrodinger equation, yields the exact S-matrix for elastic scattering. The optical potential is a very complicated quantity, since it depends upon both the dynamics of each projectile-nucleon scattering event as well as the dynamic structure of the target nucleus. KMT write the expression for the optical potential as a series. The
270
FESHBACH
AND
HtiFNER
first term depends essentially on the single-particle density of the target nucleus. This term is expected to generate most of the scattering by the nucleus. The next term is proportional to the two-body correlation function. It has a very complicated structure, and we are not aware of any convincing method for evaluating its effect on the scattering. The present study centers around this correlation term. We discuss its structure and propose how to take its effect into account in a numerical calculation. In Section II, we rederive the results of KMT by the method of coupled equations. It is well known that the many-body collision problem can be reduced to an infinite set of coupled equations by expanding the Hamiltonian and the wave function in the eigenstates of the target system. In this paper, we shall follow that procedure with the important difference from the usual practice that an effective potential (closely related to the nucleon-nucleon scattering amplitude) is used in place of the nucleon-nucleon potential (for the case of nucleon-nucleus scattering). We do not propose to solve the infinite system of coupled equations numerically (even if it were possible). But an approximation scheme can be developed. The neglect of all inelastic channels leads to the first term in KMT’s optical potential. Retaining the coupling to the elastic channel but neglecting the coupling between the inelastic channels is equivalent to the optical potential up to second order. We propose to compute elastic scattering for this case by replacing all the inelastic channels by one effective one whose coupling to the elastic channel is proportional to the two-body correlation function. Thus, calculating elastic scattering from the first two terms of KMT’s optical potential is equivalent to solving a pair of coupled equations. In order to provide some insight into the S-matrix for high-energy scattering by a nucleus, we generalize Glauber’s semiclassical methods from potential scattering to the solution of an infinite set of coupled equations (Section III). Then the computation of the S-matrix reduces to an algebraic problem. For certain classes of coupled equations, the algebraic problem can be solved analytically, and explicit formulae for the S-matrix are derived (Section IV), and they then exhibit how the scattering by a nucleus depends on the nuclear two-body correlations. Furthermore, the relation between the semiclassical solution of the coupled equations and Glauber’s expressions for multiple scattering is established.
II.
FORMULATION IN
OF SCATTERING BY TERMS OF COUPLED
NUCLEI AT EQUATIONS
HIGH
ENERGIES
Coupled equations are a familiar tool to describe nucleon-nucleus scattering at low energies, i.e., energies of a few million electron volts. One equation is associated with each state of the target, a prescription which leads to an infinite system of
SCATTERING
BY NUCLEI
271
coupled equations. Actual calculations are successfully performed by truncating the infinite system to a few important channels, usually selected by energy considerations. Highly excited target states and their associated equations are discarded (or included as part of an effective interaction) because they do not influence appreciably the scattering at low energies. The situation is different for highenergy projectiles. Although the system of coupled equations which is used at low energies remains valid at high energies (i.e., energies of or above several hundred million electron volts), a truncation to a few target states would not lead to a good approximation. Compared with the energy of the projectile, the target states appear to be degenerate, and we would expect that a great many target states are equally important. It is, however, possible to derive a set of coupled equations with improved truncation properties. The principal device is to replace the actual nuclear potential by an effective potential which has been derived by KMT. This effective potential takes into account the repeated scattering of the incident particle by the same target nucleon. The omitted terms which now give rise to channel coupling involve collisions with different particles and therefore depend upon nuclear correlations. Truncation of the resulting equations involves neglecting certain nuclear correlations. The following three sections are a reformulation of the multiple scattering theory proposed by KMT. The treatment of nuclear correlations in this formalism is discussed in detail. A. THE SYSTEM OF COUPLED EQUATIONS The Hamiltonian for scattering by a nucleus is H = HN + K, + v, Es H” + v, .
(2-l)
Here HN is the full Hamiltonian for the target nucleus, WN - d @a(rl, r2 ,..., TN) = 0,
(2-4
with properly antisymmetrized eigenfunctions (15= , Q:= 0, 1, 2,... . The kinetic energy of the projectile is denoted by K,, and V, contains the projectile’s interaction with the N target nucleons V, = f u(rO , ri). i=l
(2-3)
The scattering from the target nucleus is described by the Lippmann-Schwinger Equation (17)
~dr, , rl ,..., TN) = h3bs(r,,rl ,..., rN) +
29
E + iS - Ho V,Ul,(r,, r1 ,..., rN), (2-4)
272
FESHBACH
AND
HiiFNER
where I/Q is determined by the boundary condition kp = din(E
-
l a).
P-5)
Here, as everywhere in the following, we neglect the antisymmetrization of the projectile with the target nucleons if the projectile is a nucleon, retaining, however, the antisymmetrization of the target nucleons among themselves (cf. the arguments of Takeda and Watson (18)). The Green’s function in Eq. (2-4) contains the antisymmetrization operator d, allowing only the physical target wave functions as intermediate states: d -zz
E + 8 - H,,
LX=0
E + i8 - K. - E,
Since the interaction V, acts only between antisymmetric replace matrix elements of it everywhere by
’
G-6)
target states @, , we can
<@a I vo I @B> = N(@, I 0 I @l3s>7
(2-7)
with u from Eq. (2-3). The solution of Eq. (2-4) leads to the T-matrix cw
T,o = N<$b I * I %3>.
We shall replace the projectile-nucleon potential v by the effective interaction T which satisfies the equation sd d 7=%L+r (2-9) T=‘VtLL E+i~-Ho E f i6 - Ho ‘v’ The operator 7, which is closely related to the projectile-nucleon scattering amplitude, will be discussed in Section C. We multiply Eq. (2-4) by u and use Eq. (2-9)
d ~1- NT E + i6 - H,, 6’ + 7
d
E + i6 - Ho ‘L’ $0 + (N - lb d = T& + (N - 1)~ tz#g . E + i6 - Ho
=(
E + ,f-
H
0
“yo (2-10)
If a new function ul, is defined by (2-11)
SCATTERING
then the comparison
BY
273
NUCLEI
of Eqs. (2-10) and (2-l 1) leads to
from which equation we conclude
Tm,= NC&t IT I ‘lu,i.
(2-13)
Equation (2-l 1) can be cast in a system of coupled equations for the functions P)~, which are defined by the expansion of the total wave function in terms of target states
y = f do) @& ,..., rd.
(2-14)
‘X=0
The system of coupled equation is
(E-
E,-
Ko)% = f
ffd?& ,
(2-l 5)
B=O
where
v,, = (N- 1X@,, I i- I @k?>.
(2-16)
Although Eqs. (2-15) are formally identical to those used in low-energy nuclear physics, they differ by their interaction. In low-energy nuclear physics V,, is written in terms of the hermitian nucleon-nucleon potential ~1,while in Eq. (2-16) Yap is derived from the complex effective interaction 7. The only approximation in this section has been the neglect of the antisymmetrization-if necessary-between the projectile and the target nucleons. B. THE OPTICAL POTENTIAL, A complete solution of the infinite system of Eqs. (2-15) is impossible, in general. Moreover, even if obtainable, this solution would be “uneconomical,” since it contains everything, elastic scattering and excitation of each target state separately. On the experimental side, elastic scattering of high-energetic protons by nuclei has been studied rather extensively, while data for inelastic scattering are rare. This situation is not likely to change, becauseof the high resolution required to separate excitation of different excited states. Nuclei like 4He, lzC, and 160 show an energy gap between the ground state and the first excited state, thus making it relatively easy to isolate elastic scattering. But few nuclei show such large energy gapsbetween different excited states.
274
FESHBACH
AND
HijFNER
A way to calculate elastic scattering from Eqs. (2-15) without computing inelastic scattering as well is to introduce an optical potential. For the system Eqs. (2-15),
(E- E.- rc,- Vodplo = a#0 f ~o&L (E -
E, -
Ko -
Vm)v,
=
f
B(#d
Vm,vB
;
a: f
(2-17b)
0,
the optical potential is defined by (E - ,zo- K, -
(2-l 8)
Vopt)yo = 0.
By definition, the optical potential yields the same elastic scattering as the infinite system of equations, Eqs. (2-17). We shall calculate Vogt in a series of terms, the expansion parameter being the nondiagonal matrix elements of V,, . To lowest order, the comparison of Eqs. (2-18) and (2-l 7a) yields y(l) opt In next order, the optical potential (2-17b)
v 00.
(2-19)
is obtained by eliminating
ya = (E + i8 - E, - Ko -
Voct-’
qar from the Eqs. (2-20)
f Vm’olay~, B(#a)
and inserting it into (2-17a) keeping only the /I = 0 term
vd’dt + vgt = v,, + afO f vOcrE +
1 i8 - E, - K, -
V,,
V ao *
(2-21)
Similarly, all the higher orders of the optical potential can be obtained. It is hoped that the series for Vopt converges rapidly. The second-order potential V,-$ is already a complicated nonlocal operator. There have been estimates of this term, but we are not aware of a fully satisfactory treatment of V,$i in the literature. Therefore, in the remaining part of this section, we will introduce a series of approximations with the aim of making V$ + V$,“l amenable to numerical computation of nucleon-nucleus scattering. Our first approximation is generally employed, namely, to replace the excitation energies E, and the diagonal potential V,, in the denominator of Eq. (2-21) by average values d and v. Then closure is applied in
1 VA%= c vo, E + is - c _ K. - p “a ’ a#0
SCATTERING
BY
275
NUCLEI
and leads to the correlation function in the numerator of Eq. (2-22). We use a result of KMT in order to simplify the discussion: In the momentum representation, each matrix element V,, may be approximated by a product Pm’,,= (N -
1) ?<4Jp,(rl ,..., rN)I eeiq”l / QB(r, ,..., I-~)),
(2-23)
where i is the Fourier transform of the effective interaction Eq. (2-9) and the second factor carries all the dependence on the particular states 01,/?. The tilde indicates the momentum representation, q denotes the momentum transfer. The main assumption leading to the factorization Eq. (2-23) is neglecting the momenta of the target nucleons (in the laboratory system). If one also assumes that i is a function of momentum transfer only, then V&q) is a local operator. Using Eq. (2-23), we have for the closure expression in the numerator of Eq. (2-22) E;. ~o&l) ~‘O(S’) = (N - 1)” T(n) ?(a with
e(q, q’) as the nuclear correlation
z;(s, s’>,
(2-24)
function
C(q, 9’) ==X$ [ C (CD,/ $J eeiq”j 1@,)i@, / F ewiq”‘” 1@,)I a+0
__ “,y + ;
1 [‘p,
I=1
k=l
, e-i(a.r,+e’.r,)
I(@,, / e-i(s+s’).rl
/ Qo) - (dj, 1e-iq’rl
1 @o)(@o 1 e-iq”rl
~ Qo)]
/ @& - (@‘, 1e-iq’rl 1@,>(dj, / epiq’rlj Qo)]. (2-26)
The first term in Eq. (2-26) proportional to (N - 1)/N, contains the true two-body correlations, which originate (a) from the fact that the wave function O. is antisymmetric (“Pauli correlations”) and (b) from the nature of the nuclear forces, which generally lead to short-range correlations. The term proportional to l/N in Eq. (2-26) describesthe “self-correlations,” and would be present even if @, is a product of single-particle wave functions, in which case the true correlations vanish. As shown in Section C, the self-correlations in Vd,“i cancel against a term of vd,‘;, such that the optical potential depends only on the true correlations. By inspection of Eqs. (2-25) and (2-26), one proves WI, 9’) = wl’, 4,
(2-27)
and C(Sl 0) = 0,
(2-28)
relations which are true for each term of Eq. (2-26) separately. Even after closure has been performed in Eq. (2-22), V&$ is still a complicated operator, mainly because of the appearance of c and v in the denominator. Nevertheless, as the
276
FESHBACH
AND
HijFNER
discussion in Section IV shows, the average potential Vcannot be dropped but is necessary to compute V$\ properly. If in the sum of Eq. (2-24) only one state, say 01,,, contributes significantly (the choice of the intermediate states @, in Eq. (2-25) being rather arbitrary), the correlation function ii factorizes and the Schrodinger equation containing I’;$ + I$$ is equivalent to a pair of coupled equations. In actual nucleon-nucleus scattering, this situation may not be realized exactly, but it is conceivable that a group of states with very similar properties carries most of the inelastic strength. We think here especially of the quasi-elastic scattering, i.e., the (p, 2~) reaction. Guided by this picture, we propose to factorize the correlation function and write C(q, s’> = WI,
WI,
Q') = mI)
s’> + d 2((4> q’)
(2-29)
* WI').
(2-30)
The part of k’i$ which is proportional to z‘, can be treated by solving a pair of coupled equations which now replace Eq. (2-17):
The remainder, dc, may be amenable to perturbation theory. A numerical investigation of whether this is true is under way. A support for the ansatz Eq. (2-29) with d c’ = 0 is given in Section IV. There we solve the infinite system of coupled equations (which generates I’$\ + I’,$) by semi-classical methods and find that the elastic S-matrix can always by computed from a pair of coupled equations like the one of Eqs. (2-31) and an expression for A and P is derived. The discussion in this section assumed that the summation over the nuclear states Nin Eq. (2-24) is the only source of correlations. As shown in the next section, the effective interaction T already depends on the nucleon-nucleon correlations; so does @it = %-I) p(q).
(2-32)
The correlations can be extracted frown T, and a new effective interaction t, Eq. (2-45) (without correlations), can be iatroduced. All expressionsand conclusions of this section remain unchanged if one replaces Q-by t everywhere and drops the self-correlation term in Eq. (2-26). The main approximations in this section were (i) computing the optical potential only to second order in the nondiagonal matrix elements of the interaction; (ii) factorizing pea4 into the effective interaction and the nuclear form factor;
277
SCATTERING BY NUCLEI
Parallel to the present study, Kujawski (29) investigated the possibility to replace V$ by an equivalent local operator. C. THE EFFECTIVE INTERACTION
We shall now compare the effective interaction 7, Eq. (2-9), with the free projectile-nucleon scattering amplitude t’. We define it by (2-33) The indices “0” and “1” refer to the projectile and the struck nucleon, respectively. Schematically, the first terms of the Born seriesof Eq. (2-33) can be represented by the graphs of Fig. I. Here the vertical lines stand for the projectile and the nucleons
FIG. 1. The graphs represent the first terms of the Born series for the free scattering amplitude for the projectile “0” and a nucleon “1” (Eq. (2-33)).
of the target, and the dashed lines indicate the action of the potential ~1.Next we define a scattering amplitude t by the equation
to,
=
1'01
+ 2'01
1 E + i8 - K. - HN to1’
(2-34)
where the Green’s function resemblesthe one for T, Eq. (2-9), except that the antisymmetrization operator for the intermediate states is missing. The graphs for the first terms of the Born seriesare shown in Fig. 2. The double barred Iine represents the (N - 1) nucleons, which are not struck by the projectile. As the Hamiltonian HN in Eq. (2-34) contains the kinetic energy operator Kl of particle 1, the Born seriesof Eq. (2-34) contains all graphs of Fig. 1 (with the (N - 1) nucleons being unlinked) plus those where the struck nucleon interacts with all the other nucleons. This is called the binding energy correction (20), the first graph of this type being of third order in 1~(Fig. 2). Becauseof the antisymmetrization operator :d, the effective interaction EfiS-
K,-HN
r
(2-9)
278
FESHBACH
0
1 (N-l)
0
AND
1
HiiFNER
0
(N-l)
1 (N-l)
0
1 (N-l)
FIG. 2. The graphs represent the first terms of the Born series for scattering of a projectile “0” and a nucleon “l”, the nucleon being able to interact with the remaining (N - 1) target nucleons (Eq. (2-34)).
is symmetric in the labels of all target nucleons. This fact is expressed by the averaging procedure (l/N) Ci in Fig. 3. Furthermore, the antisymmetrization operator & in the intermediate states of Eq. (2-9) leads to the exchange diagram in secondorder of Fig. 3. As the projectile interacts with particles i and k, this diagram contains some information about nucleon-nucleon correlations in the target. The binding energy corrections which were shown in Fig. 2 will also appear in the thirdorder terms of Eq. (2-9).
0
i (N-l)
n#k
0
i
k (N-21
0
i
k
FIG. 3. The graphs represent the first terms of the Born series for the scattering of a projectile “0” from a nucleus, where the antisymmetrization of the target nucleons is properly taken into account (Eq. (2-9)).
Both terms of the optical potential (2-35) contain information about the nucleon-nucleon correlations. By expressing the operator T in terms of t defined in Eq. (2-34) and combining terms in Eq. (2-35), we extract the correlations from the first-order potential. We write 71 = f, + t,
d t E + i6 - K. - HN -
1 E + i6 - K, - HN 1 r’ *
(2-36)
Then
v:;t= (N- l)(@o I-I-I@o) = (iv - l)(Qo j t, j Qo) + (N - 1) 1 a#0 (@0I~11@0x@0
-t’N-ll)l -
(~olh
I Tll@O>
E+i6-Ko-~o 1 E+i&--Ko-HN
(2-37)
SCATTERING
BY
279
NUCLEI
The first term in Eq. (2-37) gives rise to the scattering from the single-particle density of the nucleus, and no approximation has to be made. The remaining terms are of the order of (N - 1)-l compared with the second-order optical potential V$,[ . Some approximations have to be introduced in order to combine these terms with V$$ and obtain a simple result. The main difference between the correction terms in Eq. (2-37) and Eq. (2-35) is the average potential Fappearing in the denominator of the expression for V,,,(2) , Eq .(2-35). We will simply introduce Binto the denominators of the terms in Eq. (2-37). The error associated with this procedure is of the order of (V,,J2 . (r/(N - 1)) in the total expression for the optical potential. If we count F/(N - 1) to be of the same order as an inelastic matrix element V,, , then the correction is of third-order in the inelastic matrix elements. Terms of similiar order have been neglected in our derivation of the optical potential from the beginning. Furthermore, we replace T by t in all second-order terms of the optical potential. Then the second term in Eq. (2-37) readily combines with the expression for V$ , and the third term (in curly brackets) cancels the self-correlation term in the correlation function Eq. (2-26). This result has already been stated in the appendix of KMT. In actual calculations one will usually also neglect the binding-energy corrections in the scattering amplitude Eq. (2-34).
III.
SEMICLASSICAL
SOLUTION
FOR
A GENERAL
SYSTEM
OF
EQUATIONS
Thresholds and compound resonances characterize and complicate the lowenergy solutions of a system of coupled equations like the one of Eqs. (2-15). Therefore, in most cases their solution can be obtained only by numerical integration of the equations. At high energies, nuclear thresholds are far from the energy of scattering, and the time which the projectile spends inside the nucleus is too small to develop pronounced compound resonances. Therefore, the solutions of the coupled equations at high energies are expected to vary smoothly with energy, and semiclassical methods may be appropriate. In this section, we closely follow the reasoning of Glauber (2) when generalizing his treatment of potential scattering to the solution of the set of coupled equations, Eqs. (2-15). A pair of coupled equations, as well as a case with an infinite number of channels, has recently been studied in Glauber’s approximation by Bassichis, Feshbach, and Reading (10). The following derivation generalizes their result and discusses in detail the problems arising from the noncommutativity of the interaction matrix. We start from the set of Eqs. (2-15),
(3-l)
280
FESHBACH
where we have explicitly which is
exhibited
AND
HtiFNER
the dependence on the boundary
for
Z---CO.
condition,
(3-2)
The source of the projectiles is located at z = - co. By h we denote the state of the target before the scattering event. The wave number k, is given by d2m(E - en) in the nonrelativistic limit, the additional factor (m/k,)1/2 ensures flux normalization. Because of its high energy, the projectile is not appreciably deflected during its way through the target. Therefore, following Glauber (2), we set (3-3) and approximations will be made for the functions #y’(r). the coupled equations leads to the system of equations
Inserting
Eq. (3-3) into
We assume that $(r) varies slowly over distances k-l; therefore, to first approximation the second derivatives in Eqs. (3-4) are neglected. There exist various proposals, e.g., Feshbach (II) or Hahn (22), as to how to improve the approximations, i.e., how to partially include the effect of the second derivatives. We will not follow these lines but will simply neglect A/2m. Then $ b(r)
= --i 1 Kdr> 6
(3-5)
949
The system of Eqs. (3-5) and (3-6) is the obvious generalization of Glauber’s expression for one-channel potential scattering. The correct inclusion of the nuclear excitation energies multiplies the potential matrix V,, with a hermitian factor exp[--i(k,
- k&z] ~exp
[+id2n2E*
( Ea2iEa )z],
(3-7)
the value of which might differ appreciably from one for large nuclei. The factor m/(k,ks)li2 generalizes the coefficient m/k which appears in Glauber’s expressions. The accuracy of the approximations which lead to Eqs. (3-5) are difficult to assess, because matrices appear instead of the scalar potentials in Glauber’s treatment.
SCATTERING
For his case, Glauber finds two classical solution. k-l,
BY
sufficient
281
NUCLEI
criteria
for the validity
of the semi-
(i) The potential varies slowly in space, slowly compared with the wavelength or k.R> 1 (3-8)
(R characterizes
the size of the nucleus).
(ii) The strength
V of the potential
satisfies
V/E<
1.
(3-9)
Tn the many-channel case, the condition (i) can be taken over without any change. As for (ii), we remark that the potential matrix I’,, is nearly diagonal in the representation in which the projectile-nucleon scattering amplitude t is the basic interaction. Therefore, Glauber’s conditions have to hold separately for the diagonal part of each channel. As Reading (22) has commented, the structure of the differential equation Eq. (3-5) is reminiscent of the equation for time evolution in the interaction picture. Here the space derivative replaces the time derivative. Therefore, in complete analogy, a space evolution operator L’(r) can be introduced by Q’(x,
y, z) = 1 U&(X, 4’, z) yp(x, I3
y, z = -co).
(3-10)
It takes the wave function from its value at z = - CCto its actual value. Following this analogy, we anticipate that the S-matrix is given by (3- 11)
&3(x, v) = &3(.-G y, 2 = + co),
a result which will be proven at the end of this chapter. Here x and y are treated as parameters, b = (x, y) being the impact parameter. Using Eq. (3-10) together with Eq. (3-5) we derive an integral equation for U,, : U,,(r)
= &, - i c 1’ dz’ ,YL,,(b, z’) U,,,(b, z’). Y -m
(3-12)
In general, the solution of Eq. (3-12) cannot be found explicitly but has to be evaluated by the series U,,(r) = 6,, - i jZ
--rli
$
C-i)"
f
dz’ 9ia(b, c
--m Y
Fi,(b,
z’) z’) dz’ jZ’ V$(b,
--0
z")
dz” + ... .
(3-13)
282
FESHBACH AND HijFNER
However, if the two matrices Y(r) and Jtm Y’(r) dz’ commute, i.e.,
[v(x,y,z),r;,dz’9“(x,Y,z’,]=0,
(3-14)
the seriesEq. (3-14) can be summedup to give (3-l 5) a form which is familiar from Glauber’s one-channel case, where Eq. (3-14) holds trivially. A sufficient condition for Eq. (3-14) to be valid is that Y can be written as a sum of matrices (3-16) with [A(“), fp’]
zr 0
and where the matrices Atn) do not depend on the spatial coordinates. Another condition for Eq. (3-14) to hold usesa “random phase” assumption for the matrix elements1YoB(r) if they are evaluated at different points of space, namely,
c8 T&x,y,z’>%3?J(x, Y,z>K L%z- z’>.
(3-l 7)
One of the requirements entering the models which we are going to study in Section IV will be that the solution of our system of coupled equations can be written in the form of Eq. (3-15). Before turning to these models we prove relation (3-11) for the S-matrix in the small angle approximation. We start from the scattering amplitude .&(k,
k’) = - g
j d3r eickek’)” c 9:.,(r) #F)(r),
Y
and insert (3-10): f&k,
k’) = -
$f
i d3r eickek’).‘~ V&(r) U,,,(r).
Y
(3-19)
1 Similar arguments are invoked to justify the locality of the imaginary potential in low-energy reaction theory (23).
SCATTERING
BY
283
NUCLEI
The small-angle approximation, together with the assumption of small longitudinal momentum transfer (for inelastic processes it is proportional to k(ca - c,)/2E) leads to the expression .f&(k, k’) = - $
s d2b ei(k-k’)‘b
s 1 T,(b,
Y
z) U,,(b, z) dz.
(3-20)
The integral in (3-20) is performed by using the integral Eq. (3-12) and yields fJk,
k’) = - $
J^d2b ei(k-k’)“[8aB
-
U&b, z = + a)].
(3-21)
Therefore the S-matrix is given by Eq. (3-11) and in the casewhen Eq. (3-14) holds, ‘v S,,(b) = /exp [--i 1:: dz’ Y’(b, zj)] ino.
(3-22)
Although Eq. (3-22) is an explicit expression for the S-matrix, each element of the matrix i,sa very complicated function of the elements J
dz’ F&(b, z’).
After the neglect of secondderivatives in Eq. (3-4) the infinite system of coupled equations could be solved by quadratures only. The further assumption of commutativity ‘of the potential matrix at different points in space, Eq. (3-14) led to the S-matrix as the exponential of the interaction matrix.
IV.
SOLVABLE
MODELS
FOR
MANY-CHANNEL
SCATTERING
One aim of our study has been to find models of a genuine many-channel situation simple enough so that explicit formulae for each S-matrix element are obtained. Thereby, we hope to provide someinsight into the structure of the manychannel S-matrix for high-energy scattering by nuclei. For this purpose, even the general expressionfor the S-matrix Eq. (3-22) is not simple enough, sinceit involves the exponentiation of an infinite dimensional matrix. Therefore, we shall discuss several caseswhere the exponentiation can be performed in closed form. A. THE BORDERED INTERACTION MATRIX The optical potential for elastic scattering up to secondorder in the nondiagonal matrix elements is given by Eq. (2-21). We observe that Eq. (2-22) is the exact
284 expression interaction
FESHBACH
for the optical potential matrix
AND
HiiFNER
of a scattering
problem
with a “bordered”
(4-l)
i.e., where only matrix elements between the ground state and the excited states are important and where the interaction between different excited states is neglected. In the eikonal approximation, the S-matrix for the interaction (4-l) can be solved exactly. We define M,,(b)
= j.+m 9$(b,
-m
z’) dz’,
(4-2)
where the quantities 9’& and V,, are related by Eq. (3-6). The functional dependence of the S-matrix elements on the MaiBis computed by determining eigenvectors and eigenvalues of the matrix M,,(b):
in terms of which the S-matrix is given by
In general, M,, is a non-hermitian but symmetric matrix and is diagonalized by a complex orthogonal transformation. The eigenvectors n” are adjoint to n but not their complex conjugates. The specific form of M,, , Eq. (4-2), leads to the eigenvalue problem C Moon? = (A - Moo)@ O#O Moons)
= (A -
R)n~‘,
(4-5a) R = Ml1
= M,,
= .*- .
(4-5b)
If h # &?i,the component nr’, 6 > 0 can be eliminated from Eq. (4-5b):
MBO IP’B =X--Rno*
(A)
(4-b)
When inserted into Eq. (4-5a), it leads to the secular equation (A - Moo)0 - ml = C MO&f,, , B#O
(4-7)
SCAT’I‘ERING
BY
NUCLEI
285
with the solutions
Y’ = c MORMRO< P#O
(4-9)
Because of relation Eq. (4-6) the eigenvectors related to h,,, are
ini-)
= N;; . (A,,, - R), Ml0 ) M2, ,.,. 3,
(4-10)
and the normalization is calculated to be
N& = (A,*, - ET)”+ Y2.
(4-11)
All other solutions of the system Eq. (4-5) have degenerate eigenvalues h, = ;i;l for k > 2 and, therefore, according to Eq. (4-5b), $J
= 0
k > 2.
(4-12)
The eigenvectors for k > 2 need not be constructed as long as one wants only the S-matrix elementsS,, , Q:= 0, 1, 2,..., which are the only elements of our model with physical consequences: (A, - AT)” e-ih’ so0= 2 ,,=I 0, - m2 + Y2
(4-13a)
2
Gb - 2) so, = MO=/bxl (A, _ a>2 + y2 e-ihkYa f 0
(4-13b)
Using Eq. (4-7) one arrives at s
00
=
e-i(M,,+m/2
so, = -i
M,,
jcos x _ i Moo - 33 I sin X ) t 2 x t . e-i(Mo,+W)12
SifX
(4-14) (4-l 5) (4-16)
All quantities Moo , &i, and X are complex numbers. Trigonometric functions of complex arguments rise exponentially. Since the effective interaction T satisfies a Lippmann-Schwinger equation and therefore is a scattering amplitude, its imaginary part is negative definite. This property propogates into the matrix M,, (as it is a
286
FESHBACH
AND
HijFNER
matrix representation of T). Therefore, the eigenvalues of M,, have negative definite imaginary parts and lead to exponentials with values smaller than 1 in Eqs. (4-l 3). E.g., the case MO, = I%?leads to j So, 1 = + 1 epiA1+ eviA2 1 < 4 [I e-“’ / + 1e-+ I] < 1, which shows that unitarity complex. B.
is not violated even if the interaction
(4-17)
matrix M,, is
DISCUSSION
The eikonal approximation leads naturally to the prescription to use a pair of coupled equations to compute elastic scattering for the interaction matrix Eq. (4-l). Employing the methods described in Section A, one proves that ,I$,, , Eq. (4-14), is the semiclassical solution to a pair of coupled equations with the twodimensional interaction matrix (4-l 8) The potential Y which couples the two channels is however different from the one proposed in Eq. (2-36). We define a coupling potential V, by the integral equation
&
,I,” VAb,2’) dz’ = Y(b) = dB& Mm(b)Wm(b>,
(4-19)
where Y is proportional to the correlation function, cf. Eq. (2-24). The wave number k = 1/2m(E - E) relates to the average excitation energy <. The integral equation Eq. (4-19) for V, can be solved under the assumption that V, is spherically symmetric (Glauber (2)): v(r) c =-
da -__
m
1 m
d m Y@)bdb -. dr I r d/b2 - r2
(4-20)
For a comparison with the coupling potential (2-36), we take the Fourier transforms of Eq. (4-19). Each integration over z’ is replaced by the condition qz = 0 in the Fourier transformed quantities: d2q d2q’ eib.(Q+q’)A&(q, 0) &f&q,
0). (4-27)
Therefore, s
d2(q- q’)[~c(q,0) t%f, 0) - W - II2 %q,0) dq’, 0) ec
(4-22)
SCATTERING
BY
287
NUCLEI
The factorization condition Eq. (4-22) should be compared with the one proposed in Eq. (2-35). As long as (q - q’) remains small, both conditions are equivalent. We recall that Eq. (2-35) holds not only for q - q’ = 0 but also for the first derivative in (q - q’) at q = q’. However, the two terms in the bracket of Eq. (4-22) will decay differently as a function of q - q’. To that extent conditions (2-35) and (4-22) are different. The total momentum transfer during the double scattering event is not given by (q - q’) but by (q + q’), and no condition has to be imposed on this quantity. The difference (q - q’) measures only how the total momentum transfer is distributed between the first and second scattering. The elastic scattering amplitude S,,,(b) depends on the nuclear two-body correlation function via the term Y2(b). We ask: Given the cross section for elastic scattering of a nucleon from a nucleus, can one unambiguously extract information about the correlations in the nucleus? According to Eqs. (4-14) and (4-16) the correlations in Y2 always appear together with the difference in the diagonal potentials (M,, - 1i!i)~/4. Only if one knows this quantity to an accuracy much better than the expected value of Y2, can one learn about the correlations by studying elastic scattering. M,, - R may be different from zero for two reasons: (i) The average excitation energy E - Edis different from zero. Then (MO, - R)2/4 N M&((E - E,)/4E)?
(4-23)
The factor (C - c,)/E seems to go to zero with increasing energy E of the projectile. This conclusion is misleading if the average excitation energy E is a function of E. In the case of “quasi-free scattering,” the average excitationrenergy is given by E - Eg _ 1 q2 ----= 4E 2M 4E
4k2 sin2W> 2ME.4
= sinz(e/z>
(4-24)
and is therefore a function of the angle 8, independent of E. (ii) The diagonal potentials M,,, and R differ because the excited states will have a matter distribution which differs from the one for the ground state. We use Gaussian matter distributions p(/3, r) cc ecBr2
(4-25)
and approximate MO, - R Moo
II i
SP(B,r)dz-sP(B+6B,r)dz I
=-
s P@, r) dz W2) (r2)
[
1 _ 2
1 ’
(4-26)
288
FESHBACH
AND
HtiFNER
While for one particle-one hole excitations the quantity 6(r2)/(r2) has a value of 2.5 y0 for 160, we find 15 y0 for 4He. In general, the expression Eq. (4-26) will decrease with increasing number of nucleons in the target. In order to know how much the difference M,,, - R may disturb the correlations, we have to estimate the magnitude of the correlation term
Ih$)12=
I
.fdzdz’,4b,z>0, z’)f(l z - z’ I> ,~5 .f dz p(b, z) j dz’ p(b, z’)
R’
(4-27)
Here, the nuclear correlation function has been approximated in the form of Eq. (2-41), where the function f(z) vanishes for 1z 1> re . The term re is called the correlation length and may be assumedto be of the order of 0.5-l.Of. (We denoted the nuclear radius by R.) According to Eq. (4-27), the correlation Yis only a small fraction of the diagonal term M,, . Therefore, M,, and M have to be rather well known in order to determine Y2 reliably. Some further insight is obtained if we make the reasonable assumption to start with that M,, = &i. Then so0= e-~wO(l _
+y2
+ ...)
for
1 Y2j<<1.
(4-28)
Equation (4-28) hasto be contrasted with the expressionfor the S-matrix computed for R = 0:
+ je-i(Moo’2) sin(M,,,/2) A&,/2
Y2 MO, + .** *
(4-29)
The influence of the correlations on the S-matrix is completely different for the two choices R = Moo and I% = 0 (except for Moo< 1). Although the possibility R = 0 does not appear to be a reasonable prescription in the framework of our models, it doesnot seemto have thesedrastic consequencesin the multiple scattering formalism reviewed in Section II. Indeed, Chalmers and Saperstein (24) use the prescription &? = 0 in their numerical calculations of nucleon-nucleus scattering. Although the possibility % = 0 may also lead to a fit to the experimental data, the interpretation in terms of nuclear correlations seemsquestionable. Concerning the inelastic matrix elements, Eq. (4-13, we find it remarkable that So, for j3 # 0 is proportional to the “Born matrix element” MO, , although elastic and inelastic scattering are treated to all orders. The factors which multiply MO0originate from initial and final state interactions and also ensure the unitarity of the S-matrix. The fact that Soaisos= Mom/Moo seemsto be a specific feature of the bordered matrix Eq. (4-l).
(4-30)
SCATTERING C. STATISTICAL
BY NUCLEI
289
ASSUMPTIONS
The bordered interaction matrix Eq. (4-l) described a situation where the elastic channel played a dominant and singular role: only this channel interacted with all the other channels. A complimentary situation, where the interaction matrix elements between all channels are equally important, is treated in this section. However, an explicit formula for each S-matrix element can be obtained only, if one makes certain statistical assumptions concerning the interaction matrix Mma = i__ t/;k6
!
‘+I dz’ Y(b, 2’) dz’.
We split A4,, into an average diagonal matrix ,?!i and into the remainder. We define R such that statistical arguments apply for the difference x,, = Iv,, - Ii? a,, .
(4-32)
(i) Gaussian Distribution: If a Gaussian probability distribution (4-33) holds for each matrix element X, then the ensembleaverage over S,, is diagonal and (S,,)
= a,, exp(-i?i? - J-(X2>),
(4-34)
where (X2) denotes the ensembleaverage of X2. (ii) Random Phase Assumptions: Tf the matrix elements X,, have random signs and one assumesthat (4-35) i.e., the diagonal elements of X2 are independent of 01,then one obtains for the higher powers of X,, c &343,J-,d = xX2? x,,, 2 BY
(4-36)
1 xaJ~yxvsx~~~ = 2(X2)2 a,,, )... . B,Y,6 The S-matrix is evaluated to --
S,, = e--iM I?&(&+ + cos d2(X2))
1
595/56/1-19
sin 2/2(X2) - X,, ---. 2/2(X2> 1
(4-37)
290
FESHBACH
In both cases, (i) and (ii), the quantity function, since
AND
HtiFNER
(X2) contains the two-body correlation
averaged over the states 01. When comparing the model S-matrices for elastic scattering as derived from the bordered interaction matrix and the two statistical models, one observes a close similarity. They all express the element S,,,, in terms of three parameters, M, the deviation from the average potential, e.g., M, - M, and the correlation term. Furthermore, to lowest order in the correlation term, all models show the same dependence (we assume M,,, - M = 0), S,,, = cim (1 - 4 c
M,,M,,
+ *s-j,
(4-39)
u&a,
since we treated the second order of the inelastic matrix elements exactly. Deviations between the models occur for higher orders of the correlation term. Although none of these models depends explicitly on nuclear correlations of higher than second order, each of these models implicitly contains an assumption about how to express higher-order correlations in terms of densities and two-body correlations. For instance, terms of fourth order in X,, contain some information about fourthorder correlations, and in all the previously discussed models the fourth power of X,, has been expressed in terms of the two-body correlation function only. If we introduce
z=a
Y2
z = $ z=D
Y2
for the bordered matrix
(4-41a)
for the Gaussian distribution
(4-41 b)
for the random phase model.
(4-41c)
The comparison of the three results, Eqs. (4-41), should lead to some caution. If a numerical investigation of the optical potential, Eq. (2-21), shows that the secondorder potential cannot be treated in perturbation theory, higher-order correlations among nucleons may be important for the scattering. At least the rather formal
SCATTERING BY NUCLEI
291
truncation leading to the bordered interaction matrix may be suspect. Statistical models start from a certain picture for the nucleus, and therefore their results may have greater significance. Nevertheless, we cannot be sure, whether or not the statistical models will describe correctly so delicate an object as four-body correlations, especially since they also differ in their predictions. We conclude from the agreement of all different models up to first order in the correlations that the results of the optical potential V,$ + V$$ may at least be trusted up to first order in V$ ,. D. COIVIPARISON WITH GLAUBER'S MULTIPLE
SCATTERING
Given the free nucleon-nucleon scattering amplitude and the wave function for the ground state of the target, Glauber’s multiple scattering formalism provides a method for computing the S-matrix by integrations only. No equation with an optical potential has to be solved. Each S-matrix element for elastic scattering can be written as &,,(b) = ,+‘db),
(4-42)
where b either denotes the partial wave Z, (I N kb), or is the continuous impact parameter. From his expression for SO, , Glauber (2) derives a series for x&b) xopt(b)
:= N s p(r)(--ir(b
- s)) d37
+ ; NW- 1)j [pt2)(r, )r2)- y&- P@I) P(G)] [--ir@ - s,)] X [--ir(b
- sz)] + 3rd order in r + me*.
(4-43)
Here, the “profile function” ris proportional to the two-body scattering amplitude t (in configuration space), s is the projection of r on the impact parameter plane, and pc2)(r1, rJ =
i
/ c&Jr1 , rz , r3 ,..., rN)12d3r3d3r., a*-d3r, (4-44)
p(r) = i d3r’ p(‘)(r, r’).
The convolution integrals involving p and I’ in Eq. (4-43) are the reflection of a product representation ,6(q) F(q) in momentum space-equivalent to the assumption introduced in Eq. (2-23). Except for a factor N/(N - I), which we will comment upon below, the first term in Eq. (4-43) is identical to M,,, as defined in Eq. (4-2). Similarly, the next term in Eq. (4-43) corresponds to the correlation term Y2. We
292
FESHBACH
AND
HtiFNER
have compared Glauber’s result, Eq. (4-43) with the S-matrix obtained by solving the infinite system of coupled equations. We find a one-to-one correspondence (up to second order in Y2) only if we assume M,,, = a. That means that Glauber’s formalism treats the nuclear levels to be all degenerate (the average excitation energy E = E,,). Furthermore, the formalism does not allow for a change in nuclear shape during the virtual excitation of intermediate states; i.e., the average potential P in the excited states is identical to that for the nuclear ground state. Both these restrictions are usually referred to as the “frozen nucleus” assumption. The origin of the factor (N - 1)/N is traced as follows: We recall that the transformations in Eqs. (2-8)-(2-l 3) replaced the projectile-nucleon potential Nu by the effective interaction (N - l)~, from the scattering function !PO , Eq. (2-11), is computed. In order to calculate the T-matrix, Eq. (2-13) the interaction which appears in the matrix element is NT, not (N - 1). Therefore, the S- or T-matrix which had been derived and discussed in the semi-classical approximation did not represent Eq. (2-13) but should have been replaced by
Tmo = -&
((c+"),,
- S,,)&.
We neglected this point, which led to the slight difference in the comparison with Glauber’s results. We do not propose the semi-classical solutions of Section IV as a practical means to compute cross sections but rather as a model to give insight into the structure of the S-matrix at high energies. The computations which are under way use the formalism outlined in Section TT.
V. SUMMARY
AND CONCLUSIONS
We have presented a theoretical study of high-energy scattering by nuclei with particular emphasis on the effect of nucleon-nucleon correlations. Most of the calculations in this field use Glauber’s semi-classicalmethods. The validity of these results become doubtful for larger angles,where one would expect the correlations to show up most. For this reason, we used the more fundamental approach to multiple scattering as worked out by Kerman, McManus, and Thaler. An infinite system of coupled equations determines the scattering wave function. As employed by KMT, the actual interaction potential with the target nucleons is replaced exactly by a many-body operator which describes the interaction between the projectile and a target nucleon as modified by the presenceof the remaining target nucleons. In lowest approximation the effective interaction reduces to the free target nucleon transition operator. At this level of accuracy and neglecting the excitation of target states, the optical potential describing the projectile-nucleus
SCATTERING BY NUCLEI
293
interaction is proportional to the target-nucleon density and to the forward scattering amplitude. This “single-scattering” optical potential is a well-known result. If one retains the coupling of the elastic channel to the inelastic ones but neglects the interaction between all inelastic states,the second-order, or double-scattering, term for the optical potential is obtained. It depends bilinearly on the free nucleon scattering amplitude. As expected, this term is directly related to the two-particle correlation function. The result does include off-the-energy-shell effects, and no approximations need be made as to longitudinal momentum transfer. An effective potential describing the motion of the projectile between collisions is introduced as well. Evaluation of this second-order optical potential is by no means trivial. We propose to describe the single- and double-scattering process by a pair of coupled equations, one for the incident channel and the other for an effective channel. Since the probability for leaving the incident channel and exciting another state is directly proportional to the two-particle correlation function, it is no surprise that the coupling potential between the incident and the effective channel is proportional to this correlation function. This approximation of a factorization of the correlation functions is currently being used to analyze recent high-energy experiments. Insight into the dependenceof the S-matrix at high energy is obtained by solving the infinite systemof coupled equations by semi-classicalmethods, which are closely related to those which Glauber has developed for potential scattering. This procedure neglects the longitudinal momentum transfer. It assumescommutativity between the interaction matrix at two different points along the classicaltrajectory. However, it includes the effect of correlations. For the casewhere one neglects all interactions between inelastic channels(a caseequivalent to the single- and doublescattering optical potentialj, the S-matrix is evaluated explicitly. It turns out that this S-matrix for elastic scattering can always (i.e., without factorization assumptions on the correlation function) be obtained from a pair of coupled equations. The discussion also shows that the energies and average potentials of virtually excited target states may interfere with an accurate extraction of nucleon-nucleon correlations in high-energy scattering. Another model for scattering by nuclei, where one does not neglect the interaction between inelastic channels but imposesstatistical requirements on them, leads to similar conclusions. The correspondence to Glauber’s formalism is established.
ACKNOWLEDGMENTS We are very manuscript.
grateful
to Dr.
RECEIVED: June 17, 1969
A. Gal
for a number
of discussions
and for critically
reading
the
294
FESHBACH AND HiiFNER REFERENCES
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