on high-energy elastic scattering of protons by nuclei

ANNALS OF PHYSICS 76, 80-l 15 (1973)

On High-Energy Elastic Scattering of Protons by Nuclei*

ERIC LAMBERT+ AND HERMAN FESHBAC~

Laboratory for Nuclear Scienre and Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts

Received June 14, 1972

The two coupled channel formalism for high energy elastic scattering [l] is extended to include spin and isospin effects. For a spin and isospin zero nucleus these manifest themselves by additional spin-orbit terms in the potentials. Explicit formulas for these potentials are obtained in terms of the fully spin and isospin dependent nucleon-nucleon scattering amplitude, the ground state nuclear form factor and the state dependent correlation functions. The coupling potential except for a small term arising from double spin and isospin flip process involving nuclear excitation depends only upon the pair correlations.

Numerical calculations are performed for the elastic scattering of 1 GeV protons incident on 4He. Various phenomenological dynamical two-body correlations as well as correlations generated from the Reid soft-core and Tabakin potentials in an approximate Brueckner-Hartree-Fock calculation are considered. The angular distribution beyond its first diffraction minimum as well as the polarization in the same angular range are shown to be sensitive to these correlations. However, the present accuracy of the experimental data and the lack of knowledge of the nucleon-nucleon scattering amplitude prevent any definitive conclusion about their nature.

I. INTRODUCTION

In this paper the study of the high energy elastic nucleon-nucleus collision begun in [I] and [2] is continued. These employ the optical model potential developed by Kerman, McManus and Thaler [3]. This potential is essentially an expansion in terms of target nucleus correlation functions of increasing order. The first term describes the process in which after the scattering of the incident particle by a nucleon of the target nucleus, the target nucleus remains in the ground state. This term depends directly upon the target nuclear density and linearly on the nucleon-

* This work supported in part through funds provided by the Atomic Energy Commission under Contract AT(ll-1)3069.

+ On leave from the University of Neuchltel, Switzerland on a grant from the Swiss Institute for Nuclear Research. Present address: Department of Theoretical Physics, Institut de Physique de l’Universit6, Neuchbtel, Switzerland.

80 Copyright 0 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.

ON HIGH-ENERGY ELASTIC SCATTERING OF PROTONS BY NUCLEI 81

nucleon lscattering amplitude. Repetitions of this process are automatically included when the Schroedinger equation is solved. Unitarity is guaranteed. The second term involves a process which after the first scattering of the incident particle, the target nucleus does not remain in the ground state, returning only after a second scattering. This term depends bilinearly upon the nucleon-nucleon amplitude and upon the: target nuclear pair correlation function. Because of the propagation which occurs between the two scattering events this term in the potential is nonlocal and energy dependent. In this paper as in [l] and [2] we shall stop at this second term using the results in the form given by Kerman, McManus, and Thaler [3]. Ullo [4] has extended their analysis as well as that of [l] and [2] to triple scattering before the nucleus returns to the ground state. This term in the optical potential depends upon the target nucleus triple correlation function. This work will be discussed in a paper now in preparation.

We restrict the following discussion to the first two terms assuming that the effects of the triple correlation functions are small. This is certainly the case for the heavier nuclei. The aforementioned energy dependence and nonlocality of the optical potential make the direct solution of the Schroedinger equation for the elastic channel impractical. In the coupled channel approximation developed in [l] and [2] the single Schroedinger equation is replaced by a number of coupled Schroedinger equations with local diagonal and coupling potentials. When terms up to pair correlations are included in the optical potential, there are two coupled equation?. The first of these describes the elastic channel the second an “average” inelastic channel. In [I] and [2] where a spin and isospin independent nucleon- nucleon amplitude was used, it was shown that the potential coupling the two channels is directly dependent upon the pair correlation function, vanishing if it vanished.

Using this method, experiments involving the elastic scattering of high energy protons by a number of nuclei were discussed in [2] and the sensitivity to dynamic pair correlations was explored. It was found that these correlations would be most easily observed in the lighter nuclei. Moreover, and this is an effect due to “overlapping” potentials [5] correlation lengths of the order and greater than the range of nuclear forces would have the greatest effects. As noted in [2] the restriction to a spin and isospin independent elementary amplitude which is commonly employed (see [6], for example) is not qualitatively correct and should be removed in order to achieve a more reliable interpretation of the experiments. This is one of the principal goals of this paper. The inclusion of spin-dependent terms permits the prediction of polarization phenomena. The inclusion of isospin dependent terms permits the prediction of charge exchange scattering. This is not treated in detail in this paper.

1 When triple correlations are included, four equations are required.

82 LAMBERT AND FESHBACH

In this paper the theory is applied to the proton-4He elastic scattering and com- pared with the experiment performed by Palevsky et al [7] using protons with 1.69 GeV/c incident laboratory momenta. These calculations are made more realistic by employing pair correlation functions which are generated by the Brueckener-Hartree-Fock method [8] for finite nuclei. This method employs phenomenological nuclear forces. We shall obtain the correlations when these are the Reid [9] and the Tabakin potentials [lo]. This is in contrast to earlier papers where ad-hoc correlations were employed.

At the present time the experimental information needed to exploit these improvements is not available. The nucleon-nucleon amplitude which is obviously of particular importance is not well known. Hopefully when the new proton accelerators such as LAMPF become operational, these nucleon-nucleon (as well as nucleon-nucleus), experiments which are of course important in their own right will be performed.

The motivations behind the coupled-channel approach have been discussed earlier in Refs. [l] and [2]. We need then only briefly allude to them here. It has become clear that new information on nuclear structure in terms of correlations and densities can be extracted from these experiments only with a sufficiently accurate scattering and reaction theory. Validity at large angles in the presence of the “overlapping potentials,” which are present when the incident particle is a nucleon, and when the elementary amplitude is spin and isospin dependent is necessary. The extent to which these goals are met is discussed to some extent in [2] but further study is needed. However, it is clear that the coupled channel formulation is an improvement upon the diffraction model [6, 1 l] which it approaches in the limit of very high energies.

In Section II we briefly recall some steps leading to the expression for the optical potential. Next we give the first and second order terms explicitly written in terms of the fully spin and isospin dependent elementary scattering amplitude. We are naturally led to consider the state dependence of the nuclear two-body correlation function.

The generalized derivation’of the equivalent coupling potential is studied in Section III. In the context of the approximations of Ref. [2], we show that this can still be achieved providing an additional local spin-orbit term is added to the former central coupling potential.

In Section IV, special attention is paid to the two-body correlation function. Starting from various nucleon-nucleon elementary interactions, we derive the corresponding two-body correlation functions in the framework of an approximate Bruecker-Hartree-Fock calculation for finite nuclei.

In the last section we present numerical results together with a comparison with experiment for 4He( p, p) 4He at 1.69 GeV/c laboratory proton incident momenta. We discuss the effects of several types of two-body correlations as well as the


influence of spin and of the lack of knowledge of the elementary nucleon-nucleon scattering amplitude.

A brief summary of our results was given in a recent publication [12].

II. THE OPTICAL POTENTIAL

A derivation of the optical potential [3] describing high-energy nucleon-nucleus elastic scattering has been given in Ref. [2]. Therefore we shall only state the main results and recall the assumptions involved.

In this formulation it is first supposed that the energy regime of the scattering is such that relativistic corrections will be essentially of kinematical nature [13] and that antisymmetrization of the projectile with the target nucleons may be neglected [14]. Then it is possible to describe the elastic scattering process in the projectile-target nucleus center-of-mass system (c.m.) by an equation of motion for a single channel wavefunction 1 $}

(E - K - Vopt)l d> = 0, (2.1)

where E and K are, respectively, the projectile energy and kinetic energy operator appropriately corrected for relativistic effects. Vopt is the optical potential approxi- mately given by

+ w - 1Y (Al I 1

- fi I $0) EC+’ - K - c - J,’

1 EC+’ _ K - z - v (2.2)

In this last expression N denotes the number of target nucleons, / $,) the ground state nuclear wavefunction and the ti’s the projectile-free target nucleon scattering matrices in the c.m. frame. There are two approximations leading to (2.2). The first has to do with the truncation of the optical potential’s expansion to terms up to second order in t. In this respect it is worth remembering that this is supposed to be a good approximation if three- and higher many-body nuclear correlations are small [4]. The second one refers to the introduction of the quantities Z and V, average values of the nuclear excitation energies and effective interactions between the projectile and the excited target. It is very much in the spirit of the definition of V to assume it is independent of spin and isospin. This assumption is, however, a matter of convenience, not of necessity.


The evaluation of the optical potential (2.2) in momentum space is simple in the weak binding approximation. In this case the individual scattering matrices are assumed to be independent of the internal motion of the target nucleons and (2.2) leads to

(k 1 T/opt 1 k’) = p$(k, k’) + @t(k, k’)

l%$(k, k’) = (N - 1) (4, 1 $ ,f eiq”Vi(k, k’) 1 4,) 2=1

(2.3)

(2.4)

vgt(k, k’) zz (N - 1)” 1 dk” dk”’ G(k”, k”‘)

x (4, I N(NI- 1) 2 ei(qz”j+ql’rf)tj(k, k”) t&“‘, k’) I 4,) 2#3=1

- (40 I+ 2 eiqz’rVj(k, k”) I c#& j=l

X <A / + .f eiql’riti(k”‘, k’) I +,,>I, (2.5) 2=1

G(k”, k”) z (k” / [E’+’ - K - 2 - VI-1 1 k”‘). (2.6)

In this expression ri represents the coordinate of target nucleon i in the nuclear center-of-mass system and

q = k - k’, q1 = k” - k’, qz = k - k”, (2.7)

k and k’ being projectile momenta in c.m. The last assumption implies that the scattering matrices t, in (2.4) and (2.5) act upon the target nucleons in spin- and isospin-space only.

We now relate fi to the experimentally accessible on-energy-shell nucleon- nucleon scattering amplitude M,,i in the two-body center-of-mass system (c.m.). For a given energy it may be written in the general form

M,,i(kcm , Km)

= h’(qcm) + &‘(qcm) 00 . oi + ci’(qcnl)(~o + 4 . R!m

+ &‘khn)(~o . Qcd~i . Qcm) + Ei’(qm)(oo . qcm)(oi * qcm), (2.8)

where the index zero refers to the projectile. D is the spin operator and qcm - - k cm - k&n, Qcm = (km + k&)/2, ncm = qcm x Qcm .

The five amplitudes Ai’,..., Ei' are still operators in isospin space. Their form is examplified by

Ai’&rJ = A+‘(q,In) + A-‘km) =o * Tci , (2.9)


where ‘c is the usual isospin operator. Following Ref. [3], we have

ri(k, k') = rl(~r') kr.

- Modkcm , kbm) kb

(2.10)

with

2N .sL 1 1

1+------

71(EL? = - (242 E; N2+1 m -_

N (

ELI m ’

>

(2.11)

l+FT-iT+y-

where kI,’ and Ed’ are the incident momentum and total energy of the projectile in the laboratory system.

The kinematics leading to (2.10) is strictly correct solely for vanishing momentum transfer. It is also the only situation for which both ti and Moi are simultaneously on-energy-shell. In the same approximation we have

4cm = k - k’ = q, k&c k + k’ Qcm = k’ 2 = ki&

TQ (2.12)

ncm = k&n k&n FqxQ=--p-n

which permits us to write

h(k, k’) = v(~L’)Mq) + h(q) a0 - oi + G(q)(o, + 4 * n

+ &(q)(ao * Q>(ai - Q) + MqXoo - s)(ui * s>L

with the definitions

(2.13)

Ai(q) = (kr’lkbm) A’(s), C,(q) = (k,‘lk’) G’(q), 4(q) = (k,‘Km) &‘(q),

4(q) = (k,‘kLn/k”) 4’(q), E,(q) = (k,‘l&m) G’(q). (2.14)

The first order optical potential (2.4) is readily evaluated in terms of these amplitudes. For a spin and isospin zero target nucleus one has

@t(k, W = (N - 1) r](~L)[A+(q) + C+(q) co - nl F(q), (2.15)

where

pi , 4,) (2.16)

is the form factor for the nuclear ground state.


F’$,\ consists of a central and spin-orbit local term. This is the result which is to be expected for the scattering of a spin l/2 particle on a spin zero target. In coordinate space we have

with

V$t(r, r’) = 6(r - d)[ V+‘(r) + iV$o . I], (2.17)

d%> = 470 - 1) T(ELI) J q2 4jdqr) A+(q) fTq), (2.18)

(2.19)

and I is the angular momentum operator. The second order optical potential (2.5) can be written in the form

@,$(k, k’) = (A’ - 1)” 1 dk” dk”’ G(k”, k”‘) X(Q2 , q, ; Q1 , ql) (2.20)

with q1 , q2 as defined by (2.7) and

Q1 = (k”’ + k’)/2 Q2 = Or + U/2 (2.21)

Jf(Q2 , qz ; Ql, a) = T~(EL’) c DYQ2 7 q, ; Ql> sd @T(s2 3 a) S*T

+ ZYQ2 9 qz ; Ql, a) F”(qz > sbl (2.22)

The operators YST and ZST are given in the appendix, they depend directly upon the nucleon-nucleon amplitudes A to E of Eq. (2.13).

PT(q2 , ql) and &(q, , qJ are, respectively, the nuclear ground state pair distribution function and correlation function for particles in a relative (S, T) state. They are defined as

where CST(q2 7 q1) = ~sT(Q2 3 a) - FST(Q2 7 0) FST(O, a), (2.24)

AST = 2s + 1 + (-)S+l Q~ . (I~ . 2T + 1 + (-)I+’ 7i . T< 32 4 4

(2.25)

is the usual projector onto a state of total spin S and total isospin T for particles j, i and

NST = (6, I f 4Y I 4,) (2.26) i#j=l

the normalization factor.


It is also shown in the appendix that the origin of the density dependent term ZsTFsT in (2.22) lies in double spin- or isospin-flip processes involving a single nuclear nucleon. Its order of magnitude is l/(N - 1) with respect to the pair correlation dependent one YSTcsT.

III. THE EQUIVALENT COUPLING POTENTIAL

As pointed out above, the complicated structure of the second order optical potential (2.20) prevents a simple solution of the scattering problem, i.e., of the Schroedinger equation (2.1). We shall use the coupled channel method to avoid this difficulty. This procedure is based on the assumption that the double scattering term (2.20) can be approximated by the factorized expression

f@(k, k’) N (N - 1)2 s dk” dk”’ G(k”, k”‘) @(Q2, q2) 6%‘(Q1 , a). (3.1)

In that event (2.1) reduces to the system

(E - K - V$tt> I y> = W - I) a I yl),

(E - K - 2 - V) j cpl) = (N - 1) 02 1 p), (3.2)

which is easy to solve if a local expression for the coupling potential 0L can be found. We postulate the form

@TQ, 4 = @c(q) + &o(q) uo - (q x Q> (3.3)

and perform this factorization in the eikonal approximation of the propagator G with Va constant. In that case the condition (2.20) = (3.1) becomes (see discussion in [2, Eel. (2.18) et seq.])

s ~~~ + irl x(Q2 ~ Q2 ’ Q1, a) = S -2k, :‘“x + i77 ~(Q2 ) 42) ~(Ql, a) 7 - Of (3.4)

with

Ql = @o + QY2 + G’n - d/4, Q2 = Ql + (q/2),

ql = ko - Q + (21 + W, Q2 = q - q1 7

(3.5)

where Q =&(k + k’) and q = k - k’. The direction of the vector k, is arbitrary and its magnitude is fixed by

k,2 = 2jL[E - E - V], (3.6)


where p represents the projectile-target nucleus reduced mass properly corrected for relativitic effects. In principle, condition (3.4) has to be fulfilled for all Q, q. However, we shall restrict ourselves to momenta for which both sides of this equation are supposed to take their maximum values. The structure of X (see Appendix) indicates that this function is essentially independent of the momenta Qz and Q1 . Furthermore, all ten high-energy amplitudes A* ,..., E* entering in its expression are known to be strongly suppressed for large arguments. Assuming the same features for 02 these considerations tell us that both members of (3.4) will be maximum if values of h can be found such that

k, - h and I ko - Q + VA f d/2 I

are small simultaneously. This condition is fulfilled if one chooses

$=Q (3.7)

and imposes

IQI =k0 and q-Q=O. (3.8)

As in the spinless case [2] it is for these particular momenta that the factorization will be performed.

To be consistent with the eikonal approximation for the propagator, terms of the order of 1 X l/k,, with respect to 1 in X and GYGZ are neglected. Using the symmetry of these quantities, (3.4) becomes in this approximation:

s -2k 0

f"x + irl [@ck72) ac(ql> + K&2) @so(41) ko2q2 . cl1

+ @e(q2) ~sokh) Qo * h1 x ko) + &hd f%o(qd uo * hz x ko)l

(3.9)

with

-K&Q2 , q, ; Ql , a> = -%(q2 , qd + %<a2 , sJ ~0 * (ql x ko)

+ %(q, > qz) ~0 * 6-h x ko). (3.10)

The functions x,,, are given in the appendix and may be expressed in the form

%A2 2 a> = 77YEL7 ,c, Ex-l2 9 Sl> m42 3 Ql) + -G%s2 , Sl> F”%l2 3 an (3.11)

These functions are invariant against rotation for a spherical symmetric nucleus.


In terms of the Fourier transform X,,, and cpdC,SO of &,, and aCesO , respectively, and using the identity

1 i j dr e’“‘rS(2’(b) @c-z), -2k, . A + iv - - 2k,

(3.12)

where r = (b, z) and k, * r = k,z

Equation (3.9) becomes

s dz, dz, db ei”‘b@(z, - z2){Xf(b, z2 ; b, zJ + io, . (b x k,)

x [&(b, ~2 ; b, 4 + %(b, ~1 ; b, z2)II

= I

dz, dz, db ei”‘b@(z, - z2)[&(b, z2) + ia, * (b x k,) ds,,(b, z2)]

. IKUb, zl) + iao * @ x ko) @so@, dl, (3.13)

with

R,(r) = 5 $. R,(r) (3.14)

and

&,(b, z2 ; b, zl) = lim [

1 a a - ar + ~

b+b rl 1 ah1 . r2> 1 X,(b, , ~2 ; b, , ~1). (3.15)

The uniqueness of the two-dimensional Fourier transform and the symmetry in z1 , z2 of the integrands permit us to replace condition (3.13) by

I dz [&(b, z) + io, * (b x k,) &,(b, z)] = [f2(b2) + io, * (b x k,) g2(b2)]1/2

(3.16) with the definitions

f2(b2) = j 4 4 &(b, ~2 ; b, zd, (3.17a)

g2(b2) = j dz, dz, &,(b, z2 ; b, zl). (3.17b)

The central and spin-orbit parts of the coupling potential are readily obtained from (3.16) by an Abel transformation:

&(r2) = - $ jm du $ [h+([r2 + ~~1~‘~) + h-([r2 + u~]~/~)], (3.18a) 0

s&so(r2) = -!-- jm kor o

du a h+([r2 + 2~~1~‘~) - h-([r2 + UT’“> du2

[r2 + u2]l12 9 (3.18b)


with h*(b) = [f2(b2) * ik,bg”(b”)]1/2 (3.19)

and k,, given by (3.6). As can be easily checked, these results reduce to those of Ref. [2] when the spin and isospin dependence of the nucleon-nucleon scattering amplitude is omitted.

IV. THE CORRELATION FUNCTION

The structure of the target nucleus ground state enters into the expression for the optical potential through the four nuclear pair distribution functions FST(q, q’) or equivalently through the form factors Pr(q, 0) and the pair correlation functions &T(q, q’) (2.24). FsT(q, 0) . d is irectly related to the form factor F(q) (2.16) as follows:

1 Fcq)= N(N- l),,

c NTyq, 0). (4.1)

As was noted in Ref. [2], the main features of the elastic angular distribution are determined by the first term in the optical potential, the term which depends directly upon F(q). In order to obtain information on the correlation function csT it is thus necessary to employ reasonably accurate F(q). Although in principle it might be possible to obtain both from these experiments this does not look like a promising approach. Rather, data which is available from other studies of nuclear structure is employed. For example, electron scattering determines the charge form factor for the protons., In the case of 4He this is sufficient since the form factor for the neutrons can be expected to be the same. The situation is not so simple for heavier nuclei. In those cases whatever information one has from the many studies of the target nucleus both theoretical and experimental must be brought to bear. However, and this is our main point, the correlation functions can be obtained from the high energy experiments only if the distribution functions are known with sufficient accuracy.

We now refer to the description of these pair distribution and correlation functions. Following Ref. [2] we represent the nuclear ground state by a model wavefunction 1 I$,“). As it is usually done, 1 4,“) will be written in terms of coordinates ri’ referring to a fixed center

ri’=R+ri, (4.2)

where R represents the nuclear center-of-mass coordinate in that system and {ri} the intrinsic coordinates.

The motion of the center-of-mass of the nucleus is not properly described by ] +,“) which th ere ore f cannot be employed directly when evaluating intrinsic


nuclear quantities. This problem is solved by assuming that this wavefunction does factorize

(rl’,..., h’ I 9oM> = <R I &.“.Xr, ,..., fN I 4,). (4.3)

This is realized if 1 #,,“) is constructed with harmonic oscillator singIe particle wavefunctions (one frequency) with Jastrow’s correlation factors. The model wavefunctions we are going to use will not be necessary of that type. However, we will still assume such a factorization neglecting any further correlation between the center-of-mass motion and the intrinsic ones introduced by our model.

We define the model pair distribution functions

(4.4)

in terms of which (2.23) reads

FSTh, $1 = FiiTCq, q’)lFcdq + 41, (4.5) where

Fcdq) = (q4,” I @J’~ I +,“). (4.6)

One is led to consider, besides the intrinsic correlation functions (2.24), the model correlation functions describing the Pauli and dynamical correlations

‘?%q, s’> = F%q, $1 - Fiti%> 0) Fii-(0, $1, (4.7)

and the so-called center-of-mass or kinematical correlation functions

The effect of this last term is for the case of 4He of great importance as noted in [2]. A simple phenomenological description of the dynamical correlations is given

in Ref. [2]. This approach starts from a Slater determinant model wavefunction from which nondynamically correlated (NDC) distribution functions can be calculated

NDCF.S$(~, qJ) = xi+? 6j I e i(e.r,‘+e’.r,‘)&T I ij)

C,+j Gj I AfzT I ij> ’ (4.9)

antisymmetrization of the two-body wavefunction being understood. In coordinate space, these functions correspond to the two-body densities NDCpF(r, r’) from which the correlated ones pF(r, r’) are obtained by the ansatz

pZ(r, r’) = .CST(NDCpZ(r, r’))[l - gsT(I r - r’ I)], (4.10)


CsT being a normalization factor and gST(r) a short-ranged function. A possible parametrization is

gST(r) = hSTe-(~I~$T)2; (4.11)

rST and hS* < 1 represent, respectively, the range and the strength of the dynamical &reIation.

We now present a more fundamental determination of the pair distribution functions (4.4). It is based on an approximate Brueckner-Hartree-Fock calculation for finite nuclei whose description may be found in Ref. [8].

In Brueckner’s theory the nuclear structure is, in principle, entirely described by the reaction matrix G, which is related to an underlying two-body elementary interaction V by

G(w) = V + V@/(w+ - Ho)] G(w). (4.12)

In this definition & denotes the Pauli projector onto unoccupied two-body states

(4.13)

where 1 i ) or lj ) are occupied states and H,, is the unperturbed hamiltonian which may contain, in addition to the kinetic energy operator K, a single particle potential U

H,, = K + U. (4.14)

It defines the single particle wavefunctions 1 i) and spectrum {E<}:

H, 1 i) = Ei / i). (4.15)

We assume a Brueckner-Hartree-Fock choice for the hole-hole and particle-hole matrix elements of U whereas its matrix elements between unoccupied states will be taken to be zero.

In terms of the reaction matrix (4.12) the model two-body correlated wavefunction is written

I &?‘%j = I ij> + @/@~ij - f&,)1 WJ~J I ij>

= I ij> - I X&i&, (4.16)

where we introduced the defect wavefunction x and where wij = l i + 6j .


Further discussion is facilitated by the introduction of the two-body density operators

diT(r, r’) = 6(r - ri) 6(r’ - ri) Ar’,

dST(r, r’) = 5 dy(r, r’) i#j=l

(4.17)

(4.18)

in terms of which the Fourier transform of (4.4) reads

From the Goldstone’s theorem this matrix element is given by the sum of all linked diagrams containing only one dST insertion. We shall approximate this series by

= & $ (ij - Xii(%) I r, r’) AST(r, r’ I ij - xij(wij)), 2#3=1

where the antisymmetrization of the two-body wavefunction is understood and NST the normalization factor corresponding to (2.26) in our approximation is:

NsT = g (ij - xij(wii) / flST j ij - x~~(w,~)) i#j=l

(4.21)

= i+$zl [Gj I AsT I ij> + (xd4 I AST I xd4X

Because of the assumed self-consistent choice of the single-particle potential this approximation corresponds, in the Goldstone expansion of (4.19), to the sum of all diagrams up to first order in G and one additional important second order diagram

+ exch

The first diagram generates the Pauli correlations and the others describe the effects of the dynamics.


For light doubly closed-shell nuclei, it is reasonable to approximate the self- consistent occupied states by appropriate oscillator wavefunctions. The oscillator length b is chosen to reproduce, in the absence of dynamics, the experimental nuclear rms radius. In such an oscillator basis, actual reaction matrix and defect wavefunction [8] calculations are best performed in the two-body relative center- of-mass (r.c.m) representation. In that event the Pauli operator & must be “angle- averaged” and the G matrix elements assumed diagonal in the r.c.m. quantum numbers. One can thus write the expansions:

(r, r’ 1 ij) = c (NLM, n(l,S) JmJ ; Tmr 1 ij)(xl, / nl,)(XL / NL)

. ~~sJ(3 Y,“@> I TM,), (4.23)

0, r’ I X&J& = c WLM; M&S) Jw ; TmT I ijXx& 1 X;&?(Q) &>

* (XL I NL) ?VysJ(%) Y,“(g) I Tm,). (4.24)

The quantum numbers NLM, n(Z,S) JmJ describe the states associated with the motion of the center of mass of the two particles and their relative motion respectively.

The (vector) spherical harmonics are written in the usual way and

r - r’ X=T---’

r + r’

rel Jbb’

rel

<XL, I NJ%) = &,.L, f 3i2 &,&I ( 1 rel

(4.25)

(4.26)

(4.27)

R,l being the conventional oscillator radial wavefunctions and

Introducing these partial wave expansions into (4.20), one obtains

= -p i j=l c Gj I NLM l 5 n(K) JmJ ; T’mT)@Li@, 1?(P) JriiJ ; i+#iT I ij)

* [(nl 1 xl’) - (nzx;y(wjj) I xl’)][(xl’ 1 El) - (XZ’ I &y(ctJjj) El)]

- (NL 1 XL)(Xz 1 mL>

. Y,“(2)* Y,-“<r2) ??$!J(j2) (T’m, 1 flsT / Pfi,) CV%,l(j;), (4.28)


the secon.d sum being over all quantum numbers except S and T. We write

I ij) = I flil.$?li ;iSi$ti) / njlimj ; &S&tj). (4.29)

Because of the relatively weak dependence of the G matrix elements on the starting energies Q we assume that they can be taken constant for given shells

In that event and for a doubly closed-shell nucleus the summation over misiti and mjsjti can be independently carried out in (4.28)

,zti (ij 1 NLM, n(W) JmJ ; T’mT)(iifLi@, i(k?‘) J?FzJ ; 5?“fi, 1 ij)

nzl-v5

where we used the standard notation for the Clebsch-Gordan coefficients and Moshinsky brackets.

Using the identities

(4.31)

= ; (2J + 1)(2r + 1)(2X + l)(-)v+“+m’+z+i+h

and

= & ~(21' + 1)(2P + 1)(2~ + 1)(2E + i)y

(4.33)


one is finally led to

= & 1 (-Y+A El - C-1 S’“‘] (27- + 1N-4 + l)@ + l)@J + wf + 1) (4r)2

. [(2Z’ + 1)(22’ + 1)(2L + 1)(2E + l)]“”

* (nJinjlj ; A j NLnl; h)(N%i; A / nil& ; A)

.I; ; “xl/; / ;jj; ; “,I(; ; “,)(k k “,)

* (NL I XL)(XL / NE)[(nZ ( xl’) - (nZxi”L’((wij) 1 xl’)]

. [(xl’ / 7ii) - (XP 1 &QJij) id)] P,(si * !I), (4.34)

where the 3 -j and 6 - j symbols as well as the Legendre polynomials are written in the usual way.2

IV.1 Correlation functions for 4He

We now illustrate the preceding section in the case of the 4He nucleus for which harmonic oscillator wavefunctions are known to be a good representation of the single-particle states. The spatial part of the model wavefunction for 4He is then just a sample product of four (Is) harmonic oscillator wavefunctions, i.e., four gaussians. We choose the oscillator length to be b = 1.36 fm. This value corresponds to a rms radius of 3b/&? = 1.43 fm., which is close to the experimental value.

In a first approach we use the simple phenomenological description (4.10) of the dynamics. We assume state independence, and therefore we omit, for the time being, the superscripts S and T. Straightforward algebra lead to the following expressions for the various correlation functions defined by (2.24), (4.7) and (4.8)

where

C(q, , q2) = Q(p) e-*p’ - !&I~) L&p,) e-+(%‘+%‘),

CM(ql , q2) = Q(p) e&p2 - Q(pl) i&p,) e-f(P12+%2),

G4@l1 7 Q2) = e-t(D12+Pa2qetP2 _ etP12etP22],

pi = (b/d3 qi >

(4.35)

(4.36)

(4.37)

2 In the absence of dynamic correlations (X = 0), formula (4.34) gives the two-body densities generated by the Pauli correlations. The corresponding correlation function averaged over S and T and expressed in momentum space takes the form (3.31) in Ref. [2] for l&O. However, the factor A, given in (3.32) in the same reference is l/960 instead of the value quoted. For 12C the same factor is also incorrect, its value being l/3564.


P = p1 -+ pz and p = p1 - pz are the conjugate variables of x and X defined by (4.25). The function B may be written

with

and

Gyp) = [e-)B2 - 4PNl[~ -- 431, (4.38)

u(p) = hK3P-w (4.39)

K = rc2/(b2 + rc2). (4.40)

As pointed out before, the rms radius is slightly modified by the correlations. From the pair distribution function leading to (4.35) one finds

112

1 . (4.41)

0

He Correlation FlJncttons

-O.Z- T 4

‘” / --- CM Correlations Only

Only (rc=.4fm)

~ CM +Dynom~coI (rc =.4fm)

-06 .-

-

FIG. 1. Influence of the kinematics (center-of-mass correlations) on the 4He phenomenological correlation function corresponding to the parameters h = 1 and r, = 0.4 fm.


Considering a typical case of repulsive dynamical correlations (see Eq. 4.11)

h = 1, rc = 0,4 fm,

(4.41) gives a value which is within less than 1 o/O in agreement with the non- correlated one.

For these values of the parameters, Fig. 1 displays, in coordinate space, the correlation functions (4.35)-(4.37) with one particle fixed at the center of the nucleus. For the purpose of this figure these functions were divided by the square of the nuclear center-of-mass density so that dimensionless quantities are plotted. Pauli correlations being absent for 4He, (4.35) involves the center of mass and dynamical correlations only. The attractive center-of-mass correlations, whose range is of the order of the nuclear radius, clearly manifest themselves in that figure.

Now we address ourselves to the more fundamental description of the dynamics as presented in the preceding Section.

F--I 02

Defect Wove Functions

‘S, -

3 { :-- % Component

-- ‘D, Component x 0.5

“1”,.05-

-0.1 -

0.2 -

o-

-0.2 -

8 5 m

-0.6 -

Correloltlon F”nct,ons c (r0)

9 (r-0) = pop(r))

Dynamical correiat~on~ only

(S,T) =lO,l) -

(S,Tl =(I,O) ---

FIG. 2. ‘He defect wavefunction and dynamical correlation functions generated by the Reid soft core potential.

ON HIGH-ENERGY ELASTIC SCATTERING OF PROTONS BY NUCLEI

In the case of 4He, (4.34) reduces to

99

&T(l- 3 r’) = (2s + 1)(2T + lx1 - (-Fl R&(X) c [&&&) - &Q)]” Ns’(47r)2 b6 7

1 (4.42)

where we introduced the notation

The factor [ 1 - (- l)S+T] expresses the fact that only space-symmetric pairs are found in *He. Restricting ourselves to such pairs, the calculations lead to expressions similar to (4.35)-(4.37) for the correlation functions, the function o being replaced by

\ 1.2 ‘\

i

\ 1 \

:“:p&---;- -0.4r 2

g(r.0) = * (S.Tl= (I,O) ---

Dynamlcol ‘orrelaflons only

I I I I

0 1 2 r Ifml

3 J

4

FIG. 3. 4He defect wavefunction and dynamical correlation function generated by the Tabakin potential.


Starting from defect wavefunctions in momentum space [15] calculated as in Ref. [8] for the Reid soft-core and Tabakin potentials, we generated the corresponding model correlation functions. Their normalized version together with the defect wavefunctions xx;‘(x) are given in Figs. 2 and 3. The value of the starting energy w = -25 MeV was chosen but results were found rather insensitive for values of w between -10 and -40 MeV.

For the partial waves in which we are interested, the strong short-ranged repulsive behavior of the Reid potential manifests itself by the suppression of the two-body correlated wavefunction at short distances. In this region the ‘S, and “S, defect wavefunctions closely follow the oscillator wavefunctions. This leads to vanishing two-body densities near the origin. A long-ranged (1 fm < r < 2 fm) attractive behavior of the correlation functions is also clearly seen in Fig. 2.

0.2

0

-0.2

;:

s cu

-0-E

4.C

1 I I

\ \

4He Correlation Functions

\ \

;.(o,r,=LmL

\ [PCO)l’

‘\ \

0 0.5 1.0 1.5 r [fml

i IV) I

:

FIG. 4. Comparison between the 4He dynamical correlation functions employed in calculations of p-4He scattering. Curves corresponding to the Reid soft-core and Tabakin potentials represent functions averaged over spin and isospin.


For the Tabakin potential (Fig. 3) the situation is entirely different and the correlation functions manifest a dominant attractive behavior.

The correlated rms radius may be easily written in terms of the function W(P) = (wlO(p) t w01(p))/2

(r2)1/2 = g [l - ; &j(O) + f &y))]l’z.

Both for the Reid and Tabakin potentials this radius has been found to be in close agreement with its experimental value.

In Fig. 4 we give a comparison of the spin- and isospin-averaged correlation functions obtained by the two methods described above. We think they offer a fair choice for the study of the effects of the correlations in proton-4He scattering analysis.

Finally we comment that in our second approach, with approximations leading to (4.20) and (4.24), the effective model 4He wavefunction does factorize into a center OF mass component and an intrinsic component in agreement with (4.3).

V. NUMERICAL RESULTS FOR p”He ELASTIC SCATTERING

In high-energy proton-nucleus elastic scattering the effects of the dynamical correlations are known to be enhanced for light target nucleus [2]. For this reason we will restrict ourselves to the study ofp-4He scattering in our numerical analysis.

In this section we shall first discuss the data which, besides the correlation functions studied in the preceding section, are needed to solve the p4He elastic scattering problem according to Sections II and III. Then, together with the experimental observations [7], we shall present the numerical results of our analysis for incident protons of 1.69 GeV/c momentum in the laboratory.

Electron scattering experiments [16] yield the 4He charge form factor which may be accurately parametrized by

with

P(q) = [I - (2q2)6] e-@q2 (5.1)

a = 0.316 f 0.001 fm,

,4 = 0.681 i 0.002 fm.

By assuming equal proton and neutron distributions and a gaussian proton form factor

F,(q) = &<r,W (5.2)


with [17]

one obtains (r,2)1/2 = 0.83 fm

F(q) = [l - (012q2)6] &3-Cr,2>/W (5.3)

for the intrinsic ground state 4He form factor (2.16). The parameter 01 does not affect the 4He rms radius. If one neglects CY, (5.3) is

compatible with an harmonic oscillator model for *He whose oscillator length parameter

b = -& [f12 - i (r,2)]1’2 = 1.36 fm (5.4)

is in agreement with the value employed in the preceding Section.

100 01 01 c5

4Hetp.p) 4tia

= 1.69 GeV/c

u,,,“;~ = 152 f 8 mb

FIG. 5. Comparison of calculated p4He scattering with experiment [7]. The effects of spin and the influence of the density-dependent term of the second order potential are also represented.


The complete determination of the first order potential (2.17) and coupling potential (3.18) requires the knowledge of the elementary scattering amplitudes (2.14). At the present time very few nucleon-nucleon scattering data are available at the energy we are interested in. The knowledge of these amplitudes is therefore far from satisfactory and this will be a main source of uncertainty in our analysis.

With France [ 181 we shall assume an exponential parametrization for all proton- proton and neutron-proton amplitudes; for example,

A,,(q) = A+(q) + A-(q) = ADDe-i%Qe (5.5)

and we a.llow complex values for all parameters. At the energy under consideration the nucleon-nucleon scattering data and some

sensible assumptions [18, 191 determine all parameters except for three magnitudes, namely,

lff I n!o 9 I G, I> and I En, I. (5.6)

These will be considered as free parameters in our analysis.

0.01 01 0.5 IO, ,,

1.0 -t[(GeV/c)*] I I I

PLpIB = 1.69 GeV/c

c+~+~P = 15228mb

No dynamical correlations

- E,p = 0.9 E:znd. cr+tot = 146mb

--- Enp= /I E:pnd’

000iI 1 I I II I I 0 20 40 60

em, [De-d

FIG. 6. Effects of changes in the nucleon-nucleon scattering amplitude parameter LX-~.

-


TABLE 1

Parameters of the standard NN scattering amplitude 1.69 GeVk laboratory incident momentum employed in calculation of @He elastic scattering. The parametrization is exemplified by (5.5)

and it refers to the amplitudes given in (2.8) multiplied by kL’/k:,

I PP np

ANN [fml IO.201 + i3 431 (I 34 + 12.681

I& [fml -10.671 + iO.1531 0

CNN [frn3] (0.0545 + 10.0702) (0.234 + iO.301)

D,JN [ftdl (0.00765 + 10000266) 0

ENN [fdl j (3.12-100270) (4.83 -iO.O419)

PP

(9.83 -12 41) I3 20 10 806)

(25.9 + i 4.72)

(787 l ,0.8461

(13.1 tl50.0)

(44.0 + Il.21 I

0.1 0.5 1.0 -t[tGeV/c12]

No dynamlcol correlations

- cnp = 0.9 C;y’ IT+~+ = 147mb

--_ c

0.001 I 0 20 40 60

e,, [WI

FIG. 7. Effects of changes in the nucleon-nucleon scattering amplitude parameter C,, .


As in Ref. [2] we shall not attempt a determination of the average effective potential I? One expects it is comparable to the first order optical potential. Throughout our numerical analysis we have therefore chosen

V(r) = VP’(r), (5.7)

with VA”(r) as given by (2.18). The re;sults of our calculations are shown in Figs. 5-13. In Fig. 5 we present a

comparison between experiment [7] and theory in the absence of dynamical correlations. The full line is the result obtained by adjusting the parameters (5.6) to the values listed in Table I. From now on we shall refer to the nucleon-nucleon scattering amplitude corresponding to these values of the parameters as “standard” amplitude. In the same figure the dashed line represents the result of a calculation using the spin-averaged standard amplitude (B+ = C* = D+ = & = O).The dot- dashed line corresponds to an analysis with no density-dependent term in the coupling potential (ZST = 0 in (2.22)).

1.0 -t[(GeV/c?]

+-+-

4He (p,p) 4He

PM3 = 1.69 GeV/c

exp mlor

= 152 t 8 mb

No dynamlcol correlations

csnp =09 o,Sdond. CJ,~,= 146mb

I- - CX,,~ = I.1 c~~~““~. IT+,,+= 146mb

O.OOl~ I 1 I _I---/1 0 20 40 60

@CM [WI FIG. 8. Effects of changes in the nucleon-nucleon scattering amplitude parameter E,,


The influence of spin-effects is clearly demonstrated by this figure. As expected they substantially increase the cross section in the angular range where double scattering is preponderant.

Because of the low mass number of 4He the double spin- and isospin-flip mechanisms contribute appreciably to the scattering process as shown by the dot-dashed line.

Still in the absence of dynamical correlations, Figs. 6-8 represent a test of the sensitivity of our calculations to a &IO% variation of the values of Table I for the adjusted quantities (5.6). First note the insensitivity of the angular distribution in the small angle region occupied by the first maximum centered at 0” to variations in these parameters. This angular region depends only upon the densities and not the correlations. It depends primarily on the spin averaged cross section and is thus insensitive to variations in / C,, 1, / En, I and / anp /. The results show more dependence on I C,, /. By increasing its value one augments the effects of spin which, as pointed out above, tends to increase the cross section for large angles.

0 0.01 IO 7

I 0.1 0.5 - I I\

1.0 -t [(GeV/c12] I I ,

plab = I. 69 GeV/c

I CT,,,‘:” = 152 + 8 mb

i Standard N-N

i Scotterlng Amplitude

0 20 4.2 %l[W 6o

1.0 -

t-l - 2 E

- O.l-

-3 - b 72 -

FIG. 9. Effects of the dynamical correlation on the calculated p4He scattering.


Variations of / E,, 1 only affect the scattering pattern for very high momentum transfers. This is easily understood if one remembers the q2 dependence of the corresponding amplitude.

Figures 9 and 10 illustrate the effects on the scattering of the dynamical correlations described in Section IV. The repulsive (hST = h = 1) correlations with I, = 0.4 and 0.6 fm increase the secondary diffraction maximum of the cross section in a characteristic way already recognized in spin-independent analysis [2]. On the other hand, the correlations generated by the Tabakin potential, which have been identified as typically attractive (Fig. 3) lead to a pronounced reduction of that diffraction maximum.

20

04 -t [(G&//c)‘] 0.6

4He (p,p14He

Plob = 1.69 GeV/c

Standard N-N Scattering Amplitude

--- rc = 0.4 fm

-- rc=0.6fm

- Reid soft-core

‘\

I

--- Tabakin

T T / T \

FIG. 10. Effects of the dynamical correlations in the region of the second maximum of the cross-section. Note the scales are linear.

The case of the Reid soft-core potential requires more attention. We already noticed that the corresponding correlation functions (Fig. 2) exhibit a short- ranged repulsive behavior and a long-ranged attractive one. It has been demon-


FIG. 11. scattering.

Influence of the dynamical correlations on the calculated polarization for p-4He

FIG. 12. Central and spin-orbit part of the first order optical potential (2.18) and (2.19) with the standard NN scattering amplitude.

as calculated from


strated [;!I that, in terms of the parametrization (4.1 I), the effects of the dynamical correlations on the scattering are governed by the parameter

(Xr,lb)Cl + 2(~,/~cW, (5.8)

where r, represents a measure of the range of the high-energy nucleon-nucleon interaction. The effects of long-ranged dynamical correlations are therefore strongly enhanced and they may well dominate those of stronger correlations which are of shorter range. According to our results this indeed seems the case for the two kinds of correlations generated by the Reid potential.

Note that the effects of correlations (see Fig. 10) are not small. However, because of the uncertainties in the nucleon-nucleon amplitude it is not possible to draw any definite conclusions. It does indicate that the demand on experimental accuracy required to distinguish among the various models for nuclear pair correlations is not great. But it should be emphasized that good nucleon-nucleon date is also needed.

In Fig. 11 we give results for the polarization as obtained by using the standard amplitude. Again the dynamical correlations affect the results beyond the single scattering region. Effects of the same order of magnitude were observed by varying

FIG. 13. Coupling potentials (3.18) calculated with the standard NN scattering amplitude.


the nucleon-nucleon scattering parameters as mentioned before. These results are in qualitative agreement with previous analysis [ 18,201 but the lack of experimental data prevents further discussion.

Figure 12 represents the central and spin orbit parts of the first order optical potential employed in our calculations with the standard N-N scattering amplitude. Im P’:‘(r) shows much less strength than the result given by Fig. 2 in Ref. [2] where the scale is incorrect. In comparison with this reference we obtain an opposite sign for Re V~)(Y). This is due to a difference in the sign of the real part of the non spin-isospin-flip amplitude A+(q) employed in this work.

In Fig. 13 we display the results for the coupling potential in the absence of dynamics as well as for two types of dynamical correlations. In comparison with the above mentioned reference we observe a much weaker effect of the dynamics on this potential.

VI. CONCLUDING REMARKS

In this work we have generalized the coupled channel description of the high- energy scattering of protons by nuclei so as to include the spin and isospin dependence of the elementary nucleon-nucleon interaction.

We have not mentioned the effects of three- and higher many-body nuclear correlations. Their study is now being carried out [4]. These results will be required in order to have an insight on the angular range of validity of the present analysis.

We did not comment on the eikonal approximation involved in the factorization of the second order optical potential. At least in the context of a simple model calculation a test of its accuracy should be made beyond that already discussed in [2]. As far as the solution of the actual proton-nucleus elastic scattering problem is concerned, any improvement on this approximation would involve expanding X into a set of factorable terms leading finally to additional channels. Furthermore, it should be emphasized that the unknown off-energy shell behavior of the elementary scattering amplitude does not justify, a priori, such an improvement.

In this paper we have also given a method of constructing nuclear pair correlation functions in the framework of a many-body theory. The effects of these correlations on p-4He elastic scattering were shown to be substantial. However, the present poor knowledge of the elementary nucleon-nucleon scattering amplitude as well as the present accuracy of thep-4He elastic scattering data does not allow definitive conclusions about their nature. But it does seem well within the range of experiments to be performed in the near future.


APPENDIX

In this appendix we shall give first the expressions of the operators YST and ZST appearing in

J’tQz 3 qz ; Ql, ad = r12(~~) c F’(Qz 7 qz ; Ql, ad @Th 3 qd S.T

+ ZsT(Q2 > qz ; Ql , a) f’ST(q, 3 d- (2.22)

From (2.5) and (2.20) we have the definition

Jf(Q2 , qr: ; Ql , a>

= ($, 1 5 eirq~‘rj+ql’riltj(k, k”) t#“, k’) 1 4,) i#j=l

eiqz.‘Vj(k, k”) ) bo) (4, j G& 2 eiql”iti(k”‘, k’) 1 +o>, (A-1) 1=1

with the ti’s given by (2.13) together with (2.9) for the isospin dependence. Before giving the general result consider the simple case

tik k’) = rl(EL’)V+(q) + C+(q)(ao + 4 . nl 64.2)

to illustrate the algebra required in the derivation of the complete formula. We shall restrict ourselves to nucleus with zero spin (in an L - S representation, for example). In that event nuclear matrix elements containing only one spin operator ui vanish and the introduction of (A.2) into (A. 1) leads to

[

1 = q2(cL’) ($0 ’ N(N - 1) izj=l

i ei[qz.rj+ql.ri’([A*(q2) + C+(q,) Qo . n21

. [A+(q,) + C+(q,) 00 . %I + C+(q,) C+(qJ(aj * $)(Gi * nl)) 1 do)

-- ($0 I $ i eiqY4(q2) + C+(q,) cl0 f nz) I 4,) 34

. <do 1; 5 eiqlW+(qd + C+(q,) no * n,) I q5,) . i=l 1

For a spin zero nucleus the scalar part of (Do * n2)(cri * n,)

Kuj . n2)(oi . s)lo = in2 . n,(aj * 4

(A.31

64.4)

595/76/I-S


will be the only contribution of the corresponding term in (A.3). On the other hand one may write

lji = 1 njsir, (A.5a) ST

1 - clj 3

. cri = g (-)s+l & Jr (A.5b)

so that, with definitions (2.16) and (2.23), (A.3) takes the form

x(Qz 5 qz ; QI 3 sJ

= r12(EL) [z N(;: 1) i [A+(q,) + C+(q,) 00 . %lbf+Gh) + C+(d Qo . sl

+ a C+k2) C+kd n2 . nl FST(q2 y ad 1 - W+(q,) + C+(q2) a0 4 n21[A+(qd + C+(a) =. . n,l F(qJ %h) G4.6)

Finally with definition (2.24) and assuming the form factors to be state independent (A.6) may be rewritten as

-VQ2 , qz ; Q1 , a) = ~YEL’) [z ,,,“T 1) P+(q,) + C+(q,) a0 . nzl

x iA+ + C+(qJ m. - n,l @Vh ,a)

C+(q2) C+(qd n2 . nlWa2 9 a)], 64.7)

which corresponds to (2.22) with

WQ2 , q2 ; Ql , a>

= Ncr: 1) [A+(a) + C+(q2) o. * n,l[A+(sd + C+(q,) o. - n,], (A.8a)

-WQ2 3 qz ; QI 9 sd NS’ (-)s+l

~ C+(q2) C+(d n2 * nl . = N(N - 1) 2s + 1 (A.8b)

When all components of the amplitude (2.13) are considered and assuming zero isospin nucleus, the same algebra leads to

NST Ysr(Q2 3 % i QI 3 %> = N(N _ 1) Y+(QB > qz ; QI 9 n3, (A-9)


NST ZST(Q, 3 ‘I2 ; QI 3 %> = N(N _ 1> [

(-)S'l

2s + 1 z+(Q2 3 Qz ; Ql , ‘41)

+ & (-lT+l v-(Q2 3 qz ; Ql , q,)

3( -)S+T

+ 0s + 1)GT + 1) dQ2 3 qz ; QI 9 a)]

where

y-k = LMq2) + G4q2)~o . nzlKk(41) + G&h)(oo . s)l

and

z+ = &(q2)9 W&d + C&J =. . nl + Nql)Q12 + -WA q1211 + [Gk2) n2 + k(q2>(~o . Q2) Q2 + E4q2)(ao * q2) s21 . [C&d nl + R&)(~o . Q1) Q1 + Mql)(oo . a) aI

with

and

P’(Q2 9 a), G(Q, 3 a)> = FG + GF

nls2 = q1.2 x Ql.2 .

(A.lO)

(A. 11)

(A.12)

(A.13)

(A.14)

By inspection of (A.lO)-(A.12) one sees that the density-dependent term ZsTFsT in (2.22) only appears because double spin- or isospin-flip processes involving one nuclear nucleon are forbidden by (A. 1). These terms can appear only in the double scattering term because the single scattering term in which the nucleus remains in the ground state vanishes. The order of magnitude of this term can be evaluated by assuming spin agd isospin independent pair distribution function F and correlation function C. Then the sum over S and T can be performed in (2.22) using the result [12]

NsT = (1/16)[N(N - 1)(2S + 1)(2T + 1) - 3N(2T + l)(-)s+l

- 3N(-)T+‘(2S + 1) - 9N(-)s+T] (A.15)

for a spin and isospin zero nucleus. One obtains

+ P/W - I)l[z+(Q2 T q2 ; Q1 p a) + W(Q2 y q2 ; Q1, sd + WQ2 , q2 ; Q1 , a)1 F(s2 , sd>. (A.16)

This relation shows that it may be a good approximation to neglect these double (iso) spin-flip terms for heavier nuclei.


We now give the eikonal approximation of X for the momenta defined by (3.7) and (3.8). Neglecting terms of the order of I A l/k,, and quantities vanishing after integration (3.4) we obtain to a good approximation

= A&2) A(q1) + C&32) G(q1) ko%, * 91 + &?2) ctm Qo * (a x ko) + C&d 4ql) co * (9, x ko), (A.17)

= w&2)> W&l) + c + I 00 . h x ko) + hc(q,) ko2 + -Wd q1211 (4 1

+ C&2) GM ko2q, . q1 + Mq,) MsA ko4 + &(q,) Wd(q2 - qJ2.

(A.18)

In this approximation X may therefore be written as given by (3.10) and (3.11) with

NsT YfT = jjT(N - 1) [A+(%) A+(a) + C+(qa) C+(q,) ko2q, - a], (A.19)

yfT- N(N- 1) NST A+(q,) C+(q,), (A.20)

NST “’ - N(N - 1) [ (-)‘+’ [{~+(%A W+(qd + D+(q,) ko2 + E+(q,) q12)} 2S + 1

+ c+(qa) c+(qd ko2q2 . q1 + D+(q,) D+(q,) ko4 + E+(q,) E+(q&qz . qI)“]

+ 3(-)‘f’ 2T + , [A-&J A-(q,) + C-h) C-k,) ko2q, . a]

3( -)S+r + (2s + 1)(2T + 1) [ @-(q,), (W-h) + D-h,) k,2 + E-(q,) q2):

+ C&2) C-(e) ko2qz * q1 + D-k,) DA,) ko4

+ Jw2) UqJGI, . a,q]v (A.21)

ZST zzz NST (-)Sfl 3( -)Tfl B N(N - 1) 2s + , B+(q2) ‘+(ql) + 2~ + , A-(q2) c-(ql)

+ (2s :($kT+ 1) WI,) C&l)]. (A.22)


ACKNOWLEDGMENTS

The authors wish to thank Professor V. Franc0 for making available to them the values of the nucleon-nucleon scattering parameters and Dr. P. U. Sauer for providing the 4He defect wavefunctions. One of the authors (E. L.) acknowledges the warm hospitality extended to him by the Center for Theoretical Physics at M.I.T.

REFERENCES

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on high-energy elastic scattering of protons by nuclei

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