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Centre-of-mass and Pauli correlation corrections to proton-nucleus scattering at intermediate energies
Nuclear Physics
A352
(1981) 425-441
@ North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
CENTRE-OF-MASS AND PAUL1 CORRELATION CORRECTIONS TO PROTON-NUCLEUS SCATTERING AT INTERMEDIATE ENERGIES I. AHMAD* DPh-N/ME,
CEN Saclay, BP. 2, 91190,
Gif-sur- Yvette, France
and J.P.
AUGER
Laboratoire de Physique CAT, Universite’ d’OrlPans, 45045 O&fans, France Received 23 July 1979 (Revised 28 July 1980) Abstract: An effective profile hadron-nucleus scattering
function approach is proposed amplitude in Glauber theory.
for the correlation
expansion
of the
The problem of the c.m. and Pauli pair correlation corrections to medium-energy proton-nucleus scattering is studied. A simple expression for the c.m. pair correlation correction is derived and is shown to be more realistic than the previously proposed ones. Effects of both correlations on the elastic sections and their relative importance are studied and discussed.
scattering
differential
cross
1. Introduction Extensive experimental and theoretical made it amply clear that intermediate-energy tool for determining
nucleon
density
studies over the past several years have proton scattering is potentially a good
distributions
in nuclei (e.g. refs. ‘-“)I and special
efforts are now being made for studying various refinements to the theoretical analysis of the experimental data to enhance the credibility of the extracted information 4-6). 0 ne such refinement which has attracted closer scrutiny currently concerns the contribution of the two-body correlation to proton scattering which has hitherto been either neglected or treated in a very approximate manner. In a recent article Harrington and Varma ‘) have discussed this problem in some detail and have particularly studied the two-body correlation corrections arising from the translational invariance and antisymmetry of the target wave function (c.m. and Pauli pair correlations respectively) using the oscillator model for the target nucleus in which case the problem can be treated exactly. They find that these correlations are important enough to be included in any realistic analysis of the proton scattering data even for large-A nuclei. Since for a realistic description of the target it has not yet been possible to treat the problem of correlations exactly, these authors, working under the assumption that the total pair correlation is the sum of the c.m. and Pauli correlations, derive simple approximate expressions for accounting for them in the * Visiting scientist from Dept. of Physics, AMU, Aligarh-202001, 425
India.
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analysis of the scattering data using Glauber formalism “>. For the c.m. pair correlation they obtain the same expression as derived earlier by Alkhazov 5). In this work we look into the problem of the pair correlation correction somewhat differently, we treat the nuclear two-body correlation in the approach of Feshbach et ai. 9, according to which the total intrinsic pair correlation in general, is not simply the sum of the cm. and the Pauli correlations. Rather the Pauli (in general model) correlation, in contributing to the total intrinsic correlation is affected by the requirement of the translational invariance on the target wave function. This approach enables us to derive, rather simply, an expression for the c.m. pair correlation correction which we find to be more realistic than those proposed earlier 2,5*’). It also makes it possible to study the effect of the c.m. constraint on the contribution of the Pauli correlation which we find to be small. In sect. 2 we propose an effective profile function approach for the correlation expansion of the hadron-nucleus scattering amplitude in Glauber theory*). The approach is straightforward and does not invoke the ansatz ‘*lo) for the correlation expansion of the target A-body density. In sect. 3 we discuss the c.m. and Pauli pair correlations and derive expressions for accounting for them in hadron-nucleus scattering. In sect. 4 we compare the various previously proposed prescriptions for accounting for the c.m. pair correlation with the presently derived one and also study the relative importance of the c.m. and Pauli correlations on 1 GeV proton elastic scattering differential cross sections.
2. Correlation expansion for the scattering amplitude According to Glauber theory *) the amplitude describing the elastic scattering of a nucleon with momentum k from a target nucleus in the ground state (PO = IO)) is given by (2.la)
S(b) =
jijl[l--W
-Sill,
(2.lb)
where Q is the momentum transfer, A the target mass number, si are the projections of the target nucleon coordinates ri onto a plane perpendicular to k and r is the profile function related to the elementary NN amplitude f(q) as:
T(b)=+Jd2q e-iq'bf(q).
(2.2)
For simplicity of discussion we assume that all the target particles are identical and ignore the Coulomb scattering in eq. (2.la) which can be accounted for in the
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I. Ahmad, J.P. Auger / Proton-nucleus scattering
well-known manner [e.g. ref. “)I. Further it is assumed that the wave function of the target which depends only upon the intrinsic coordinates is normalized as:
I
r2,. . . ra)12S( r1 ‘“‘A
I+&,
* ‘+ “)
drr . . . drA = 1 .
(2.3)
To obtain the correlation expansion for the scattering amplitude F(q) we define an effective profile function y as: yj=To(b)-r(b-si)
9
(2.4)
ro(b) = (OJT(b - s)lO) .
(2.5)
where
Next we expand S as given by eq. (2.lb) in terms of yi as s=so+:
I=1
s,,
(2.6)
where so = (1 -r,)A
(2.7a)
)
SI = (1 -rlJ)A-’
C CC
3/ilYiz
’ *
’
Yir
*
(2.7b)
il
Substituting the expansion (2.6) in eq. (2.la) and noting that (OISr]O)= 0, one obtains an expansion for F(q) in which the successive terms depend upon the one-body density, two-body correlation function, three-body correlation function and so on. More explicitly evaluation of the ground-state expectation values for S2 and &, gives the following expressions:
(OIS210)
=$A(A - l)(l -r,)A-2
(oIs310) =$(A x
I
j C(rr,
r2)r(b
-sl)r(b
drl dr;? ,
G-1
- i)(A -2)(1 -+)A-3 c3(rl,
r2,
rJf(b
-sl)f(b
-sdf(b
where C and C, are the 2-body and 3-body correlation C(rl, C3h
-s2)
r2) = p2h
r2, r3) = p3h
with pi(r) as the i-body density.
(2.9)
functions:
r2) -plhh(r2)
r2, r3) -h(rdC(r2,
+m(r3Kk,
-SS) ch de t-b ,
(2.10)
, r3)+pl(r2)C(r3,
r2)1-pl(rl)pl(r2)pl(r3),
rd
(2.11)
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The expansion (2.6) admits a simple interpretation for the elastic scattering. The first term So which is similar to that obtained when one uses the independent-particle model describes the multiple scatterings of all orders (of course up to A-tuple scattering) in which the target always remains in the ground state, i.e. it represents a passive propagation of the projectile in the field of A nucleons. While the terms SI (I 2 2) describe a passive propagation of the projectile in the field of (A - f>nucleons and I-tuple scattering through the effective profile y which involve target excitation. A similar interpretation is obtained for the inelastic scattering case (#lo), where If) denotes an excited state of the target. In this case So does not contribute and the expansion starts from (f]Si]O). It is interesting to point out that the terms depending on products of the pair correlation function C(ri, r2) which appear when one uses ansatze for the correlation expansion for the A-body density 7,*2)do not appear explicitly in this approach. Such terms may however be extracted from SI for 1 even and greater than 2. For example the term quadratic in C may be obtained from S4 by applying the closure:
The first term on the r.h.s. of the above expression is the one under discussion. It is obvious that when such a term is considered together with the two-body correlation term the definition of the I-body correlation function as implied in eq. (2.7b) gets modified [it may be checked that the definition of the n-body correlation function for rz > 3 as implied in the present formulation coincides with that of ref. ‘)I. Although the 4-body correlation function in either case satisfies the generally required condition that the correlation function when integrated with respect to any one of the coordinates gives zero [because the same holds for C(rr, r2); since in practice the contribution of the terms containing higher powers in the pair correlation function is found to be small ‘“)I we will not discuss this point any further. We however, feel that it is better to regard such a term as forming part of the 4-body correlation term. It should also be stated that care must be exercised in expanding S(b) as in eq. (2.6) when the NN amplitude and hence r(b) is spin dependent; for the operators involved in the expansion generally do not commute and the expression (2.7b) for S,(b) is no longer valid. Fortunately at the energies of interest the spin dependence in the NN amplitude is fairly weak; hence for the purpose of the present study we would neglect it in all the terms in the expansion (2.6) except in So(b) which is the main contributor to the elastic scattering. Under this approximation the expression (2.7b) for S(b) is justified. In this work we restrict ourselves at Sz in the expansion for S with the aim of studying contributions of various two-body correlations to proton-nucleus
I. Ahmad, J.P. Auger / Proton-nucleus
429
scattering
scattering?. For future convenience we denote the integral on the r.h.s. of eq. (2.8) by I(b) which in the momentum space reads as:
I@)= where
C?(q,, q2) is
(&)’1e-“q1~qz”6~(ql)f(q2)~(q1, q2)d2q, d2qz
,
the double Fourier transform
of the pair correlation
(2.13) function
C(r1, Q). 3. Two-body
correlation
function
As already mentioned we intend to treat the pair correlation function C(rl, r2) in the approach of Feshbach et al. 9). These authors introduce a model wave function @‘“‘(rl,. . . r_,+)for the target such that the intrinsic one and the two-body densities &(q) and G2(q1, q2) in the momentum space may at least approximately be written as: (3.1)
L%(4)= K:(q)p’Y’(q) , 62(41,42)
=
K(q1+
42)L%“Y419
42)
(3.2)
9
where $M’(q) and jTcM)(ql, 42) are, respectively, the single and double Fourier transforms of the model one- and two-body densities, BUM’ and pi”‘(rt, Q), corresponding to the model wave function ecM). The quantity K(q) is the usual c.m. correlation correction factor. It is well known that the above expressions are exact if QcM) is taken as the harmonic oscillator shell-model wave function. In this case R(q) I=eq2’402A,
(3.3)
where cy2 is the oscillator constant. From eqs. (3.1) and (3.2) it follows that the intrinsic correlation function c(ql, corresponding to the expression (2.10) is of the form “) %42)
= K&l +q~)~(“)(q~,
42)
+
LrLhzI,
42)
,
42)
(3.4)
where (3.5) + It should be pointed out that truncating the presently proposed expansion at 522gives an expression for the pair correlation correction which differs from the one used in some earlier studies of the effect of the pair correlation on medium-energy proton-nucleus scattering in the Glauber model [e.g. ref. “)I. In these studies the expression for the pair correlation correction which is obtained using Foldy and Walecka’s expansion for the target A-body density contains some additional terms which are of higher order in the pair correlation function. As discussed before the reason behind the absence of such higher-order terms in the present formulation is that our definition of the n-body correlation function for n > 3 isdifferent from that adopted by Foldy and WaIecka lo). It however turnsout that the contributionof these higher-order terms is negligibly small ‘*). Therefore conclusions as regards the effect of the pair correlation in studies based on either approach should remain the same.
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I. Ahmad,
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scattering
and C’“‘(q~, 42) is the model pair correlation function in the momentum given by QcM’. Substituting eq. (3.4) in eq. (2.13) we have I(b) = &n(b) +Ln.(b)
f
space as
(3.6)
where i,,,(b) and &.,,,.(b) correspond to the first and the second terms of eq. (3.4) respectively. From eq. (3.4) it is clear that if the pair correlation in the model wave function is neglected one still has &,.(ql, 42) which is a consequence of requiring the translational invariance on the target wave function. We identify it with the c.m. pair correlation function’ and attempt, in the following, to obtain a simple expression for it, assuming that the oscillator model provides a good approximation to the realistic situation as far as the c.m. correlation correction is concerned. Substituting eq. (3.3) in eq. (3.5) we have CL&l,
92) =1%(4i)iil(q2) [e*l.q2’2Aa2-
11,
(3.7)
which is in general, not a very practicable expression for our purpose as it implies evaluation of a four-dimensional integration for getting Ic.n.(b), Thus some approximations need to be invoked. Fortunately a good approximation is indicated if we note that for medium and heavy nuclei the coefficient of qt - q2 in the argument of the exponential is quite small and moreover because of the sharp fall of p11(4)with 4 the integral f,.,.(b) gets contributions mainly from low q1 and q2 values. This consideration suggests that one expands the exponential in eq. (3.7) and retains only the first term to obtain* GXn.(q*l 42)
=
1 2Aa2 ~l(qt~~l(qz~ql’42
-
Now it is easy to see that the above expression expression for I,.,.(b):
f,.,.(b) =
-.A
*
(3.8)
leads to the following simple
D%@)12 7
(3.9)
where ro(b) is as defined before. This expression differs greatly from the corresponding expression of. Alkhazov et al, ‘) and Harrington and Varma 7). To facilitate comparison with previous works it is useful to go to the coordinate space. The c.m. correlation function C,.,. (rlt r2) which corresponds to the expression (3.8) can be easily shown to be
‘It should be noted that in ref. 9, &,,,,, is defined somewhat differently. * It is interesting to point out that small 41, q2 behavior of the expression (3.8) is the same as obtained ref. ‘) from model-independent considerations.
in
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431
which for central densities assumes the form
G.m.h, r2)=
-2A
yq;2)(;%).
(3.11)
Further if it is assumed that the nuclear density pr(r) is described by an appropriate single gaussian we have 3rl * r2 G.m,(~lt
r2)
=
-(A
_
I>(r2)
PlwPlt~2f
f
(3.12)
where (r2) is the mean square radius. The expression (3.12) is essentially the same as derived by Alkhazov et al. 2>and Harrington and Varma ‘) except for the fact that their expression contains (A - 2) instead of (A - 1) which is of little significance for medium and heavy nuclei. Thus we see that their expression is appropriate to a gaussian nuclear density which is not unexpected for the approximation used in their derivation of the expression for C_(rl, r2) imply such a density for the target [e.g. ref. ‘)I. Further it is worth mentioning that in deriving the expression (3.12) Warrington and Varma ‘) assume the intrinsic density to be the same as the model density whereas no such approximation is made in the present derivation. As a matter of fact according to their derivation the one-body density appearing in eq. (3.12) should be read as the model density. Next we study the contribution of the model correlation as described by e’“’ (qi, q2). For this we assume that it is mainly contributed by the Pauli correlation arising from the antisymmetry of the model wave function r#‘M’.The effect of the short-range (SR) correlation has been studied by Harrington and Varma ‘) and is found to be small. We therefore neglect it in the present study. Moreover it can be treated similarly. In the following we will not distinguish between the model and Pauli pair correlations and will denote the latter by Cp(rl, r2). From eq. (3.4) it follows that the total intrinsic pair correlation function is not simply the sum of the c.m. pair correlation function as defined by eq. (3.5) and the model pair correlation. Rather the model pair correlation function in the q-space appears as multiplied by the cm. correlation correction factor R(q, + q2) the effect of which may not, a priori, be said to be negligible. To investigate this point we, following Harrington and Varma ‘), take the Pauli correlation as:
Cdh r2) =
&+-f&i CNMhr
r2) p'")trl)p'M'(r2), 1
(3.13)
where CNM(r,
1*
(3.14)
’
with j1 as the spherical Bessel function of the first order and kFthe Fermi momentum.
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The quantity CNM(r, r’) is the pair correlation function appropriate to an infinite nuclear medium which is made appropriate to finite nuclei by invoking the local density approximation so that kF becomes a function of r according to the relation k,=(r) = [;~~Ap(~~(r)]~‘~.
(3.15)
The first term in eq. (3.13) is introduced to ensure that C,(r, r’) when integrated over any one of the coordinates becomes zero as it should be ‘). Further it is an excellent approximation to replace eq. (3.14) by the gaussian from ‘,i3) &r&r,
#%J= _$&)1/2
,-4&,-%l*
(j
(3.16)
Now using eqs. (3.13), (3.15) and (3.16) the Pauli correIation momentum space may be written as: w
3 - AT’ 2 (A-l)
1
CP(417 42) z -CA _ 1) &“’ (qI)&M’ (q*) --
function
in the
--5q’y/4k;(l)
dr e’q”[p’M’(r)J2 e kZ(r)
,
(3.17) where q = q1 + q2 and q’ = t(ql- 42) and c’l”‘(q) is the same as defined before. In obtaining the above expression we have invoked the short-range correlation approximation and the basic assumption of the Fermi gas model that the nuclear density is a slowly varying function of the space coordinate+ and hence within the integral we have replaced pcM)(rr) and p’“‘(rJ by ptM)(&rr + r2)). Next from eqs. (2.13), (3.4) and (3.17) we obtain for the model correlation part of I(b) the expression (3.18) where e-(b-sP/Sz
b(“‘(r)12 k&)[S +4#3’k;(r)]
d2
’
(3.19)
with S2 = p2 - 1/Aar2 and s as the projection of r on a plane perpendicular to k. In eq. (3.18) the term with the curly brackets corresponds to the first term in eq. (3.17) and is obtained under the same approximation [cf. eq. (3.8)] as used for evaluating I,.,.(b). The last term Y is obtained by using eq. (3.3) for K(ql +q2) and the usual parametrization for the NN scattering amplitude: ikg,(l fNN(q)
=
4n
-
ip) e_pzq2,2
,
(3.20)
’ Admittedly the approximation is questionable in the outer region of a nucleus where p(?r) and k&J become small. However, as pointed out in ref. ‘) the factor [pCM’(r)]’in the integrand should keep this region from contributing significantly.
I. Ahmad,
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where c, is the NN total cross section, p the ratio of the real to the imaginary part of the forward scattering amplitude and p2 is the slope parameter. It may be pointed out that the above expression for Y is valid if S2 > 0 which is fortunately the case around 1 GeV for medium and heavy nuclei (in fact for A b 12) for which the Fermi gas model may be applied. The expression (3.19) for Y may be simplified further by invoking that 5’ is very small so that the expression within the curly brackets in eq. (3.19) may be replaced by ?j2(6 -s). This leads to m
Y=
1 I -cu dz[p’M’(r)12 k,(r)[5 +4p2&r)3
’
(3.21)
which is the same expression as obtained in ref. 7, in the zero-range approximation (/32-O). The present derivation, which takes into account the effect of the translational invariance on the Pauli correlation, however, shows that the expression (3.21) is more realistic than is implied by the zero-range approximation for now it is not p2 but S2( < p2) which is needed to be assumed as very small. Coming to the expression within the curly brackets in eq. (3.18) and comparing it with the corresponding term of ref. ‘), we note that the effect of multiplying (?‘iM’(ql,42) with K(qr +42) manifests itself in two ways. First in the appearance of the additional gradient term and second in the fact that rO(b) appearing in it should be evaluated with respect to the intrinsic ground-state density of the target and not the model density as implied by the derivation in ref. ‘)_ Calculations however, show that for medium and heavy nuclei both these effects are fairly small. Thus on the whole it may be concluded that the factor K(q, + q2) which multiplies ecM)(ql, q2) in eq. (3.4) does not modify the expression for the Pauli correction as derived by Harrington and Varma ‘) in any significant way. 4. Calculation
and discussion
In this section we present results of some calculations which test the reliability of the expression (3.9) for the c.m. pair correlation correction and demonstrate the importance of the c.m. and Pauli pair correlation corrections for p-nucleus scattering at medium energies. Since for the Pauli correlation correction the present approach leads to essentially the same expression as derived and studied in detail by Harrington and Varma 7), we in the following, pay more attention to the cm. pair correlation correction and compare predictions of the expression (3.9) with some recently proposed ones. To test the accuracy of eq. (3.9) we first consider the case of the target nucleus 160 for which the c.m. effect is relatively large and assume that its ground state is described by the oscillator shell model. It is well known that for a fully antisymmetrized oscillator model wave function the cm, effect can be accounted for exactly in a Glauber model calculation by multiplying the scattering amplitude as calculated
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I. Ahmad, J.F. Auger/
Proton-nucleus scattering
from the model wave function with the factor K(q) given by eq. (3.3). Obviously such a scattering amplitude also incorporates the Pauli correlation through the use of the antisymmetrical model wave function. Here we are interested in testing the reliability of the expression (3.9) for the c.m. pair correlation correction against the oscillator model calculation in which all orders of the c.m. correlation corrections are accounted for exactly. Since the cm. pair correlation as defined by eq. (3.5) is the total two-body correlation when the model correlation (in this case Pauli correlation) is neglected, we compare its predictions with those of the full oscillator model amplitude after neglecting the antisymmetry of the model wave function. In other words we compare the prediction of the amplitude as given by the correlation expansion approach when only the c.m. correlation is included with that of the full amplitude calculated using the product oscillator model wave functions after multiplying it with the factor+ K(q). For future convenience we denote the differential cross section and polarization as given by the latter by cro and PO and the corresponding quantities as calculated in the correlation expansion approach by cr and P. It may be clarified that in the correlation expansion approach the quantity fo(b), and hence SO@) as given by eq. (2.7a), is calculated using the intrinsic one-body density which is obtained from the model one-body density through eq. (3.1). The results for p-l60 differential cross sections are shown in fig. 1. The calculation has been performed with the oscillator wave function with the oscillator constant i4) w2 = 0.3226 fm-’ and the spin-independent part of the NN amplitude as given in ref. 3). The continuous curve shows CQ while the dotted one, which is almost indistinguishable from (TVup to 4 ~2.25 fm-‘, is the correlation expansion cal: culation with the c.m. pair correlation correction given by eq. (3.9). In order to be clearer on the significance of this good agreement we tested the accuracy of the approximation of replacing eq. (3.7) by eq. (3.8). For this we performed calculations of the elastic angular distributions with both the expressions (3.7) and (3.8) assuming, for simplicity, an appropriate gaussian density for the target. As expected, the two calculations were found to agree well up to about 4 = 2.5 fm-‘. Thus the good agreement between co and the correlation expansion calculation clearly shows that for 4 G 3 fm-r it is sufficient to go only up to the two-body correlation term in the correlation expansion. This is consistent with the finding of Harrington and Varma ‘) who made a similar calculation assuming a gaussian density for the target and using eq. (3.12) [which for a gaussian density is the same as eq. (3.11)] for the c.m. pair correlation function. Further the under’ It should be pointed out that since the expression (3.3) for K(q) is strictly valid for an antisymmetrized oscillator model wave function, in a sense, some effect of the antisymmetry tacitly persists even if the full amplitude is calculated by neglecting the antisymmetrization of the model wave function but multiplying the model amplitude with K(q). However, as follows from eqs. (3.1) and (3.2) since the same prescription is applied to obtain the one- and two-body densities involved in the correlation expansion calculation the above comparison is quite meaningful. The correspondence between the two calculations becomes clearer if one thinks of a situation where the Pauh correlation correction is negligibly small compared to the c.m. correlation effect.
I. Akmad, J.P. Auger / Proton-nucleus
.5
2
I
435
seatiering
3
q Cfm"') Fig. 1. Elastic diflerential cross sections for 1 GeV protons on r60. The continuous curve shows 00 as described in the text. The other curves represent correlation expansion calculations: the dashed curve is obtained when the c.m. pair correlation is neglected while the dotted and the dash-dotted curves are calculated, respectively, with the expressions for the cm. pair correlation correction as derived in this work and by Harrington and Varma ‘1.
of CYfor q 2 2.5 fm-l can be easily interpreted to have its origin largely in neglecting the higher-order correlations. The dashed curve shows the correlation expansion calculation when the pair correlation term is neglected. The difference between the solid and the dashed curve is thus the genuine c.m. pair correlation effect and is seen to be significant, being 18% and -50% in regions of the first and the second maxima. The dash-dotted curve is obtained by using the prescription of refs. 237)for the c-m. correlation correction. By comparison with the dashed curve, it is seen to highly overestimate the c.m. effect ( = 28% against the realistic 18%) at the first maximum’. estimation
f It must be pointed out that according to Harrington and Varma 7, their prescription for the cm. pair correlation correction is in error by about 6% at the first maximum in 160 which is close to = 8%, that we find using the same prescription. However these authors also state that in all the cases their expression underestimates the c.m. effect which is in contradiction with what we find here.
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From fig. 1 it is clear that the presently proposed prescription provides a much better estimate of the c.m. effect. Up to this point we have simply demonstrated that if the target nucleus is described by the oscillator model our expression (3.9) for the c.m. correlation correction gives much more accurate results than the corresponding one derived previously. As such this finding is not of much significance, for in general the oscillator model does not provide a very good description of the target nucleus. However the fact that for a realistic wave function for the target it is very difficult to find a practical expression for the c.m. correlation and that a fully antisymmetrized oscillator model wave function provides a better description of the nuclear target than those expressly or tacitly assumed in previous derivations of the correction term makes the presently derived expression (3.9) more reliable for estimating the c.m. effect in the analysis of p-nucleus scattering data for determining nuclear density distributions. A detailed comparison of some recently proposed prescriptions for accounting for the c.m. effect is presented if fig. 2 for 160 and two other target nuclei 40Ca and 208Pb, where we have plotted (o -LTO)x 100/a. with (+ and cro calculated in the same manner as stated before. However, unlike 160 the calculations for 40Ca and “‘Pb are made in terms of more realistic Hartree-Fock densities and the corresponding oscillator constant cy2(which is obtained from the maximum overlap of the HF wave function with the oscillator one) as given in ref. 14).Thus go for these two nuclei is not as exact as in the case of 160. The continuous and the dashed curves in fig. 2 correspond, respectively, to the presently proposed prescription and that of refs. 2*7). In an earlier paper Alkhazov 5, also proposed another approach for accounting for the cm. pair correlation which has been studied recently by Auger and Lombard. [This approach differs in the manner in which I,.,.(b) is used for calculating the elastic scattering amplitude; the expression for I,.,.(b) being the same as in refs. “*‘).I The predictions of this approach are shown by the dotted curves in the figure [unlike refs. ‘,r4) we do not put p2 = 0 in the calculation]. Finally the dash-dotted curve corresponds to the prescription for the c.m. correlation correction as used by Ray 15) in his KMT calculation. This curve is calculated by taking the second-order KMT potential arising from the c.m. pair correlation as given by Ray [this is essentially the same as proposed in ref. “)I and obtaining the corresponding expression for the Glauber model calculation following the approach of ref. ‘). Although Glauber model and KMT calculations are not identically the same still the comparison is meanin~ul at least qualitatively. A study of fig. 2 clearly shows that in all the three cases, light, medium and heavy nuclei, the expression for the c.m. pair correlation correction as derived in this work gives much better results. In this case the deviation is less than 10% for 160 and at 5% level for 4oCa and *‘*Pb up to q =3 fm-‘, while the other prescriptions have larger deviations. Further it is interesting to note that with our prescription the deviation is always negative, relatively large at the positions of minima and increases in magnitude with 4, This behaviour lends itself to a plausible interpretation as the
I. Ahmad, J.P. Auger / Proton-nucleus scattering
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437
I 2
q
effect of neglecting higher-order correlations in the correlation expansion calculation; for these higher-order terms are expected to manifest themselves more prominently in regions where lower-order calculations give smaller cross sections and to increase in importance with increasing q. Also it is important to note that with the presently proposed prescriptions the deviations are refatively much smaller in regions of maxima (up arrows) which play a relatively more important role in the analysis of the data.
I. Amman, J.P. Auger / Proton-nucleus scat fen‘ng
438
0
I
2
3
4
b Cfml Fig. 3. Real part of I,.,,(b) as defined by eq. (3.6). The description of the curves is the same as in fig. 2.
The distinction between the various approaches for accounting for the c.m. effect is made more transparent in fig. 3, where we have plotted the real part of I,.,.(b) as defined by eqs. (3.6) and (2.13). The continuous, dashed and dash-dotted curves correspond respectively to eqs. (3.1 l), (3.12) and that used by Ray 15). It is seen that the three approaches differ greatly. Most striking is the fact that while the first two signify the c.m. correction as a surface effect, the one of Ray “> shows it to be a volume effect. This qualitative difference may have its origin in the approximations involved in deriving the expressions as used by Ray 15). In its derivation one uses an expression for the c.m. pair correlation function in momentum space, which is valid only for small 4 -values 4). Next we take up the question of the importance of accounting for the two-body correlation terms in the analysis of p-nucleus scattering data. This question has been discussed recently by Auger and Lombard 14) for the c.m. correlation, and by Harrington and Varma’) for both the c.m. and the Pauli correlation within the framework of Glauber theory. Still it is desirable to discuss it briefly. A recent analysis using the KMT formalism by Ray 15) shows that for medium-weight nuclei the c.m. pair correlation is considerably less important than the Pauli correlation which is in disagreement with the findings of Harrington and Varma ‘) according to which both correlations are essentially equally important. In fig. 4 we show (a -al) X loo/at for 160, 40Ca and *‘*Pb, where u and ml are the differential cross sections with and without the two-body correlation terms as calculated in the correlation expansion approach. The dotted and dashed curves are
I. Ahmad, J.P. Auger / Proton-nucleus scattering
439
calculated, respectively, with the c.m. and the Pauli correlations only. The continuous ones are obtained when both the correlations are included. As expected the c.m. correlation effect decreases in importance with A while the Pauli one has the opposite trend. Taking the example of 40Ca we see that both the c.m. and the Pauli correlations are essentially equally important. This disagrees with Ray’s result “) that the Pauli correlation is about eight times as important as the c.m. correlation in this nucleus. The present calculation shows that the c.m. pair correlation contributes about 20%, 40% and 55% of the total effect at the first, second and the third maxima
q (fm-‘1 Fig. 4. Centre of mass and Pauli correlation effects on 1 GeV proton elastic scattering cross sections: dotted curve: c.m. correlation only; dashed curve: Pauli correlation only; continuous curve: both c.m. and Pauli correlations. The dash-dotted curve is for the Pauli correlation only neglecting the normalization of the corresponding two-body density.
440
I. Ahmad, f.P. Auger / Proton-nucleus scattering
(shown by arrows), respectively. There seem to be two main reasons for Ray’s findings. First as follows from the curves for 40Ca in fig. 2 his expression for the c.m. correlation correction in general underestimates the c.m. effect at the maxima and second while accounting for the Pauli correlation he disregards the first term in the expression (3.13), which is needed for the normalization of the two-body density. Neglecting this term has the effect of enhancing the Pauli effect as follows from the dash-dotted curve for 40Ca in fig. 4 which has been calculated with the Pauli correlation only but disregarding the normalization of the corresponding two-body density. We also studied the effect of the gradient term in the expression (3.18) for 1,(b) which arises because of the mixing of the c.m. constraint and the Pauli correlation. As expected its contribution was found to be quite small.
5. Conclusions In this work we have proposed an effective profile function approach for the correlation expansion of the hadron-nucleus scattering amplitude in the Glauber theory which does not invoke the ansatz for the correlation expansion of the A-body density of the target nucleus. Going only up to the two-body correlation term in the correlation expansion of the scattering amplitude we have studied the effects of the c.m. and Pauli pair correlations on 1 GeV proton scattering on nuclei by treating the problem of the pair correlation in the approach of Feshbach et al. 9). In particular we have derived a simple expression for the c.m. pair correlation correction to the Glauber amplitude and have shown it to be much more realistic than the previously proposed ones. Further, we have demonstrated that the c.m. correlation correction is predominantly a surface effect. We have also studied the effect of the c.m. constraint on the Pauli correlation correction and have shown it to be small. The present study shows that both the c.m. and the Pauli correlation corrections are important enough to be included in any detailed analysis of the elastic scattering data. In addition we fmd that for a medium-weight nucleus like 40Ca the c.m. correlation effect on the differential cross section is essentially as important as the Pauli correlation one and not much smaller as recently reported by Ray “). The authors are grateful to Drs. R.J. Lombard and A. Chaumeaux for discussions. One of us (I.A.) is grateful to Prof. J. Saudinos for providing him with excellent facilities for doing this work. References 1) G. Igo, Rev. Mod. Phys. 50 (1978) 523 2) G.D. Alkhazov, S.L. Betostotsky and A.A. Vorobyov, Phys. Reports C, 42 (1978) 89
I. Ahmad,
J.P. Auger / Proton-nucleus scattering
441
3) J.P. Auger and R.J. Lombard, Ann. of Phys. 115 (1978) 442 4) A. Chaumeaux, V. Layly and R. Schaeffer, Ann. of Phys. 116 (1978) 247
5) G.D. Alkhazov, Nucl. Phys. A280 (1977) 330 6) S.J. Wallace, Phys. Rev. Cl2 (1975) 179; C.W. Wong and SK. Young, Phys. Rev. Cl2 (1975) 1301; Cl5 (1977) 2146; D.R. Harrington and G.K. Varma, Phys. Lett. 74B (1978) 316 7) D.R. Harrington and G.K. Varma, Nucl. Phys. A306 (1878) 477 8) R.J. Glauber. in Lectures in theoretical physics, voi. 1, ed. W.E. Brittin and L.C. Dunham (Interscience, NY, 1959) p. 315 9) Ii. Feshbach, A. Gal and G. Hiifner, Ann. of Phys. 66 (1971) 20 10) L.L. Foldy and J.D. Walecka, Ann. of Phys. 54 (1969) 447 11) I. Ahmad, Nuci. Phys. A247 j1975) 418 12) Y. Abgrall et al., Nucl. Phys. A316 (1979) 389 13) V. Franc0 and W.T. Nutt, Nucl. Phys. A292 (1977) 506 14) J.P. Auger and R.J. Lombard, J. of Phys. G4 (1978) L261 15) L. Ray, Phys. Rev. Cl9 (1979) 1855
A352
(1981) 425-441
@ North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
CENTRE-OF-MASS AND PAUL1 CORRELATION CORRECTIONS TO PROTON-NUCLEUS SCATTERING AT INTERMEDIATE ENERGIES I. AHMAD* DPh-N/ME,
CEN Saclay, BP. 2, 91190,
Gif-sur- Yvette, France
and J.P.
AUGER
Laboratoire de Physique CAT, Universite’ d’OrlPans, 45045 O&fans, France Received 23 July 1979 (Revised 28 July 1980) Abstract: An effective profile hadron-nucleus scattering
function approach is proposed amplitude in Glauber theory.
for the correlation
expansion
of the
The problem of the c.m. and Pauli pair correlation corrections to medium-energy proton-nucleus scattering is studied. A simple expression for the c.m. pair correlation correction is derived and is shown to be more realistic than the previously proposed ones. Effects of both correlations on the elastic sections and their relative importance are studied and discussed.
scattering
differential
cross
1. Introduction Extensive experimental and theoretical made it amply clear that intermediate-energy tool for determining
nucleon
density
studies over the past several years have proton scattering is potentially a good
distributions
in nuclei (e.g. refs. ‘-“)I and special
efforts are now being made for studying various refinements to the theoretical analysis of the experimental data to enhance the credibility of the extracted information 4-6). 0 ne such refinement which has attracted closer scrutiny currently concerns the contribution of the two-body correlation to proton scattering which has hitherto been either neglected or treated in a very approximate manner. In a recent article Harrington and Varma ‘) have discussed this problem in some detail and have particularly studied the two-body correlation corrections arising from the translational invariance and antisymmetry of the target wave function (c.m. and Pauli pair correlations respectively) using the oscillator model for the target nucleus in which case the problem can be treated exactly. They find that these correlations are important enough to be included in any realistic analysis of the proton scattering data even for large-A nuclei. Since for a realistic description of the target it has not yet been possible to treat the problem of correlations exactly, these authors, working under the assumption that the total pair correlation is the sum of the c.m. and Pauli correlations, derive simple approximate expressions for accounting for them in the * Visiting scientist from Dept. of Physics, AMU, Aligarh-202001, 425
India.
426
I. Ah~u~,
J.P. Auger / Proton-nucleus scufter~ng
analysis of the scattering data using Glauber formalism “>. For the c.m. pair correlation they obtain the same expression as derived earlier by Alkhazov 5). In this work we look into the problem of the pair correlation correction somewhat differently, we treat the nuclear two-body correlation in the approach of Feshbach et ai. 9, according to which the total intrinsic pair correlation in general, is not simply the sum of the cm. and the Pauli correlations. Rather the Pauli (in general model) correlation, in contributing to the total intrinsic correlation is affected by the requirement of the translational invariance on the target wave function. This approach enables us to derive, rather simply, an expression for the c.m. pair correlation correction which we find to be more realistic than those proposed earlier 2,5*’). It also makes it possible to study the effect of the c.m. constraint on the contribution of the Pauli correlation which we find to be small. In sect. 2 we propose an effective profile function approach for the correlation expansion of the hadron-nucleus scattering amplitude in Glauber theory*). The approach is straightforward and does not invoke the ansatz ‘*lo) for the correlation expansion of the target A-body density. In sect. 3 we discuss the c.m. and Pauli pair correlations and derive expressions for accounting for them in hadron-nucleus scattering. In sect. 4 we compare the various previously proposed prescriptions for accounting for the c.m. pair correlation with the presently derived one and also study the relative importance of the c.m. and Pauli correlations on 1 GeV proton elastic scattering differential cross sections.
2. Correlation expansion for the scattering amplitude According to Glauber theory *) the amplitude describing the elastic scattering of a nucleon with momentum k from a target nucleus in the ground state (PO = IO)) is given by (2.la)
S(b) =
jijl[l--W
-Sill,
(2.lb)
where Q is the momentum transfer, A the target mass number, si are the projections of the target nucleon coordinates ri onto a plane perpendicular to k and r is the profile function related to the elementary NN amplitude f(q) as:
T(b)=+Jd2q e-iq'bf(q).
(2.2)
For simplicity of discussion we assume that all the target particles are identical and ignore the Coulomb scattering in eq. (2.la) which can be accounted for in the
427
I. Ahmad, J.P. Auger / Proton-nucleus scattering
well-known manner [e.g. ref. “)I. Further it is assumed that the wave function of the target which depends only upon the intrinsic coordinates is normalized as:
I
r2,. . . ra)12S( r1 ‘“‘A
I+&,
* ‘+ “)
drr . . . drA = 1 .
(2.3)
To obtain the correlation expansion for the scattering amplitude F(q) we define an effective profile function y as: yj=To(b)-r(b-si)
9
(2.4)
ro(b) = (OJT(b - s)lO) .
(2.5)
where
Next we expand S as given by eq. (2.lb) in terms of yi as s=so+:
I=1
s,,
(2.6)
where so = (1 -r,)A
(2.7a)
)
SI = (1 -rlJ)A-’
C CC
3/ilYiz
’ *
’
Yir
*
(2.7b)
il
Substituting the expansion (2.6) in eq. (2.la) and noting that (OISr]O)= 0, one obtains an expansion for F(q) in which the successive terms depend upon the one-body density, two-body correlation function, three-body correlation function and so on. More explicitly evaluation of the ground-state expectation values for S2 and &, gives the following expressions:
(OIS210)
=$A(A - l)(l -r,)A-2
(oIs310) =$(A x
I
j C(rr,
r2)r(b
-sl)r(b
drl dr;? ,
G-1
- i)(A -2)(1 -+)A-3 c3(rl,
r2,
rJf(b
-sl)f(b
-sdf(b
where C and C, are the 2-body and 3-body correlation C(rl, C3h
-s2)
r2) = p2h
r2, r3) = p3h
with pi(r) as the i-body density.
(2.9)
functions:
r2) -plhh(r2)
r2, r3) -h(rdC(r2,
+m(r3Kk,
-SS) ch de t-b ,
(2.10)
, r3)+pl(r2)C(r3,
r2)1-pl(rl)pl(r2)pl(r3),
rd
(2.11)
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J.P. Auger / Proton-nucleus
scattering
The expansion (2.6) admits a simple interpretation for the elastic scattering. The first term So which is similar to that obtained when one uses the independent-particle model describes the multiple scatterings of all orders (of course up to A-tuple scattering) in which the target always remains in the ground state, i.e. it represents a passive propagation of the projectile in the field of A nucleons. While the terms SI (I 2 2) describe a passive propagation of the projectile in the field of (A - f>nucleons and I-tuple scattering through the effective profile y which involve target excitation. A similar interpretation is obtained for the inelastic scattering case (#lo), where If) denotes an excited state of the target. In this case So does not contribute and the expansion starts from (f]Si]O). It is interesting to point out that the terms depending on products of the pair correlation function C(ri, r2) which appear when one uses ansatze for the correlation expansion for the A-body density 7,*2)do not appear explicitly in this approach. Such terms may however be extracted from SI for 1 even and greater than 2. For example the term quadratic in C may be obtained from S4 by applying the closure:
The first term on the r.h.s. of the above expression is the one under discussion. It is obvious that when such a term is considered together with the two-body correlation term the definition of the I-body correlation function as implied in eq. (2.7b) gets modified [it may be checked that the definition of the n-body correlation function for rz > 3 as implied in the present formulation coincides with that of ref. ‘)I. Although the 4-body correlation function in either case satisfies the generally required condition that the correlation function when integrated with respect to any one of the coordinates gives zero [because the same holds for C(rr, r2); since in practice the contribution of the terms containing higher powers in the pair correlation function is found to be small ‘“)I we will not discuss this point any further. We however, feel that it is better to regard such a term as forming part of the 4-body correlation term. It should also be stated that care must be exercised in expanding S(b) as in eq. (2.6) when the NN amplitude and hence r(b) is spin dependent; for the operators involved in the expansion generally do not commute and the expression (2.7b) for S,(b) is no longer valid. Fortunately at the energies of interest the spin dependence in the NN amplitude is fairly weak; hence for the purpose of the present study we would neglect it in all the terms in the expansion (2.6) except in So(b) which is the main contributor to the elastic scattering. Under this approximation the expression (2.7b) for S(b) is justified. In this work we restrict ourselves at Sz in the expansion for S with the aim of studying contributions of various two-body correlations to proton-nucleus
I. Ahmad, J.P. Auger / Proton-nucleus
429
scattering
scattering?. For future convenience we denote the integral on the r.h.s. of eq. (2.8) by I(b) which in the momentum space reads as:
I@)= where
C?(q,, q2) is
(&)’1e-“q1~qz”6~(ql)f(q2)~(q1, q2)d2q, d2qz
,
the double Fourier transform
of the pair correlation
(2.13) function
C(r1, Q). 3. Two-body
correlation
function
As already mentioned we intend to treat the pair correlation function C(rl, r2) in the approach of Feshbach et al. 9). These authors introduce a model wave function @‘“‘(rl,. . . r_,+)for the target such that the intrinsic one and the two-body densities &(q) and G2(q1, q2) in the momentum space may at least approximately be written as: (3.1)
L%(4)= K:(q)p’Y’(q) , 62(41,42)
=
K(q1+
42)L%“Y419
42)
(3.2)
9
where $M’(q) and jTcM)(ql, 42) are, respectively, the single and double Fourier transforms of the model one- and two-body densities, BUM’ and pi”‘(rt, Q), corresponding to the model wave function ecM). The quantity K(q) is the usual c.m. correlation correction factor. It is well known that the above expressions are exact if QcM) is taken as the harmonic oscillator shell-model wave function. In this case R(q) I=eq2’402A,
(3.3)
where cy2 is the oscillator constant. From eqs. (3.1) and (3.2) it follows that the intrinsic correlation function c(ql, corresponding to the expression (2.10) is of the form “) %42)
= K&l +q~)~(“)(q~,
42)
+
LrLhzI,
42)
,
42)
(3.4)
where (3.5) + It should be pointed out that truncating the presently proposed expansion at 522gives an expression for the pair correlation correction which differs from the one used in some earlier studies of the effect of the pair correlation on medium-energy proton-nucleus scattering in the Glauber model [e.g. ref. “)I. In these studies the expression for the pair correlation correction which is obtained using Foldy and Walecka’s expansion for the target A-body density contains some additional terms which are of higher order in the pair correlation function. As discussed before the reason behind the absence of such higher-order terms in the present formulation is that our definition of the n-body correlation function for n > 3 isdifferent from that adopted by Foldy and WaIecka lo). It however turnsout that the contributionof these higher-order terms is negligibly small ‘*). Therefore conclusions as regards the effect of the pair correlation in studies based on either approach should remain the same.
430
I. Ahmad,
J.P. Auger f Proton-nucleus
scattering
and C’“‘(q~, 42) is the model pair correlation function in the momentum given by QcM’. Substituting eq. (3.4) in eq. (2.13) we have I(b) = &n(b) +Ln.(b)
f
space as
(3.6)
where i,,,(b) and &.,,,.(b) correspond to the first and the second terms of eq. (3.4) respectively. From eq. (3.4) it is clear that if the pair correlation in the model wave function is neglected one still has &,.(ql, 42) which is a consequence of requiring the translational invariance on the target wave function. We identify it with the c.m. pair correlation function’ and attempt, in the following, to obtain a simple expression for it, assuming that the oscillator model provides a good approximation to the realistic situation as far as the c.m. correlation correction is concerned. Substituting eq. (3.3) in eq. (3.5) we have CL&l,
92) =1%(4i)iil(q2) [e*l.q2’2Aa2-
11,
(3.7)
which is in general, not a very practicable expression for our purpose as it implies evaluation of a four-dimensional integration for getting Ic.n.(b), Thus some approximations need to be invoked. Fortunately a good approximation is indicated if we note that for medium and heavy nuclei the coefficient of qt - q2 in the argument of the exponential is quite small and moreover because of the sharp fall of p11(4)with 4 the integral f,.,.(b) gets contributions mainly from low q1 and q2 values. This consideration suggests that one expands the exponential in eq. (3.7) and retains only the first term to obtain* GXn.(q*l 42)
=
1 2Aa2 ~l(qt~~l(qz~ql’42
-
Now it is easy to see that the above expression expression for I,.,.(b):
f,.,.(b) =
-.A
*
(3.8)
leads to the following simple
D%@)12 7
(3.9)
where ro(b) is as defined before. This expression differs greatly from the corresponding expression of. Alkhazov et al, ‘) and Harrington and Varma 7). To facilitate comparison with previous works it is useful to go to the coordinate space. The c.m. correlation function C,.,. (rlt r2) which corresponds to the expression (3.8) can be easily shown to be
‘It should be noted that in ref. 9, &,,,,, is defined somewhat differently. * It is interesting to point out that small 41, q2 behavior of the expression (3.8) is the same as obtained ref. ‘) from model-independent considerations.
in
I. Algae,
J.P. Auger / Proton-nucleus
scattering
431
which for central densities assumes the form
G.m.h, r2)=
-2A
yq;2)(;%).
(3.11)
Further if it is assumed that the nuclear density pr(r) is described by an appropriate single gaussian we have 3rl * r2 G.m,(~lt
r2)
=
-(A
_
I>(r2)
PlwPlt~2f
f
(3.12)
where (r2) is the mean square radius. The expression (3.12) is essentially the same as derived by Alkhazov et al. 2>and Harrington and Varma ‘) except for the fact that their expression contains (A - 2) instead of (A - 1) which is of little significance for medium and heavy nuclei. Thus we see that their expression is appropriate to a gaussian nuclear density which is not unexpected for the approximation used in their derivation of the expression for C_(rl, r2) imply such a density for the target [e.g. ref. ‘)I. Further it is worth mentioning that in deriving the expression (3.12) Warrington and Varma ‘) assume the intrinsic density to be the same as the model density whereas no such approximation is made in the present derivation. As a matter of fact according to their derivation the one-body density appearing in eq. (3.12) should be read as the model density. Next we study the contribution of the model correlation as described by e’“’ (qi, q2). For this we assume that it is mainly contributed by the Pauli correlation arising from the antisymmetry of the model wave function r#‘M’.The effect of the short-range (SR) correlation has been studied by Harrington and Varma ‘) and is found to be small. We therefore neglect it in the present study. Moreover it can be treated similarly. In the following we will not distinguish between the model and Pauli pair correlations and will denote the latter by Cp(rl, r2). From eq. (3.4) it follows that the total intrinsic pair correlation function is not simply the sum of the c.m. pair correlation function as defined by eq. (3.5) and the model pair correlation. Rather the model pair correlation function in the q-space appears as multiplied by the cm. correlation correction factor R(q, + q2) the effect of which may not, a priori, be said to be negligible. To investigate this point we, following Harrington and Varma ‘), take the Pauli correlation as:
Cdh r2) =
&+-f&i CNMhr
r2) p'")trl)p'M'(r2), 1
(3.13)
where CNM(r,
1*
(3.14)
’
with j1 as the spherical Bessel function of the first order and kFthe Fermi momentum.
432
I. Ahmad,
J.P. Auger / Proton-nucleus
scattering
The quantity CNM(r, r’) is the pair correlation function appropriate to an infinite nuclear medium which is made appropriate to finite nuclei by invoking the local density approximation so that kF becomes a function of r according to the relation k,=(r) = [;~~Ap(~~(r)]~‘~.
(3.15)
The first term in eq. (3.13) is introduced to ensure that C,(r, r’) when integrated over any one of the coordinates becomes zero as it should be ‘). Further it is an excellent approximation to replace eq. (3.14) by the gaussian from ‘,i3) &r&r,
#%J= _$&)1/2
,-4&,-%l*
(j
(3.16)
Now using eqs. (3.13), (3.15) and (3.16) the Pauli correIation momentum space may be written as: w
3 - AT’ 2 (A-l)
1
CP(417 42) z -CA _ 1) &“’ (qI)&M’ (q*) --
function
in the
--5q’y/4k;(l)
dr e’q”[p’M’(r)J2 e kZ(r)
,
(3.17) where q = q1 + q2 and q’ = t(ql- 42) and c’l”‘(q) is the same as defined before. In obtaining the above expression we have invoked the short-range correlation approximation and the basic assumption of the Fermi gas model that the nuclear density is a slowly varying function of the space coordinate+ and hence within the integral we have replaced pcM)(rr) and p’“‘(rJ by ptM)(&rr + r2)). Next from eqs. (2.13), (3.4) and (3.17) we obtain for the model correlation part of I(b) the expression (3.18) where e-(b-sP/Sz
b(“‘(r)12 k&)[S +4#3’k;(r)]
d2
’
(3.19)
with S2 = p2 - 1/Aar2 and s as the projection of r on a plane perpendicular to k. In eq. (3.18) the term with the curly brackets corresponds to the first term in eq. (3.17) and is obtained under the same approximation [cf. eq. (3.8)] as used for evaluating I,.,.(b). The last term Y is obtained by using eq. (3.3) for K(ql +q2) and the usual parametrization for the NN scattering amplitude: ikg,(l fNN(q)
=
4n
-
ip) e_pzq2,2
,
(3.20)
’ Admittedly the approximation is questionable in the outer region of a nucleus where p(?r) and k&J become small. However, as pointed out in ref. ‘) the factor [pCM’(r)]’in the integrand should keep this region from contributing significantly.
I. Ahmad,
433
J.P. Auger / Proton-nucleus scattering
where c, is the NN total cross section, p the ratio of the real to the imaginary part of the forward scattering amplitude and p2 is the slope parameter. It may be pointed out that the above expression for Y is valid if S2 > 0 which is fortunately the case around 1 GeV for medium and heavy nuclei (in fact for A b 12) for which the Fermi gas model may be applied. The expression (3.19) for Y may be simplified further by invoking that 5’ is very small so that the expression within the curly brackets in eq. (3.19) may be replaced by ?j2(6 -s). This leads to m
Y=
1 I -cu dz[p’M’(r)12 k,(r)[5 +4p2&r)3
’
(3.21)
which is the same expression as obtained in ref. 7, in the zero-range approximation (/32-O). The present derivation, which takes into account the effect of the translational invariance on the Pauli correlation, however, shows that the expression (3.21) is more realistic than is implied by the zero-range approximation for now it is not p2 but S2( < p2) which is needed to be assumed as very small. Coming to the expression within the curly brackets in eq. (3.18) and comparing it with the corresponding term of ref. ‘), we note that the effect of multiplying (?‘iM’(ql,42) with K(qr +42) manifests itself in two ways. First in the appearance of the additional gradient term and second in the fact that rO(b) appearing in it should be evaluated with respect to the intrinsic ground-state density of the target and not the model density as implied by the derivation in ref. ‘)_ Calculations however, show that for medium and heavy nuclei both these effects are fairly small. Thus on the whole it may be concluded that the factor K(q, + q2) which multiplies ecM)(ql, q2) in eq. (3.4) does not modify the expression for the Pauli correction as derived by Harrington and Varma ‘) in any significant way. 4. Calculation
and discussion
In this section we present results of some calculations which test the reliability of the expression (3.9) for the c.m. pair correlation correction and demonstrate the importance of the c.m. and Pauli pair correlation corrections for p-nucleus scattering at medium energies. Since for the Pauli correlation correction the present approach leads to essentially the same expression as derived and studied in detail by Harrington and Varma 7), we in the following, pay more attention to the cm. pair correlation correction and compare predictions of the expression (3.9) with some recently proposed ones. To test the accuracy of eq. (3.9) we first consider the case of the target nucleus 160 for which the c.m. effect is relatively large and assume that its ground state is described by the oscillator shell model. It is well known that for a fully antisymmetrized oscillator model wave function the cm, effect can be accounted for exactly in a Glauber model calculation by multiplying the scattering amplitude as calculated
434
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Proton-nucleus scattering
from the model wave function with the factor K(q) given by eq. (3.3). Obviously such a scattering amplitude also incorporates the Pauli correlation through the use of the antisymmetrical model wave function. Here we are interested in testing the reliability of the expression (3.9) for the c.m. pair correlation correction against the oscillator model calculation in which all orders of the c.m. correlation corrections are accounted for exactly. Since the cm. pair correlation as defined by eq. (3.5) is the total two-body correlation when the model correlation (in this case Pauli correlation) is neglected, we compare its predictions with those of the full oscillator model amplitude after neglecting the antisymmetry of the model wave function. In other words we compare the prediction of the amplitude as given by the correlation expansion approach when only the c.m. correlation is included with that of the full amplitude calculated using the product oscillator model wave functions after multiplying it with the factor+ K(q). For future convenience we denote the differential cross section and polarization as given by the latter by cro and PO and the corresponding quantities as calculated in the correlation expansion approach by cr and P. It may be clarified that in the correlation expansion approach the quantity fo(b), and hence SO@) as given by eq. (2.7a), is calculated using the intrinsic one-body density which is obtained from the model one-body density through eq. (3.1). The results for p-l60 differential cross sections are shown in fig. 1. The calculation has been performed with the oscillator wave function with the oscillator constant i4) w2 = 0.3226 fm-’ and the spin-independent part of the NN amplitude as given in ref. 3). The continuous curve shows CQ while the dotted one, which is almost indistinguishable from (TVup to 4 ~2.25 fm-‘, is the correlation expansion cal: culation with the c.m. pair correlation correction given by eq. (3.9). In order to be clearer on the significance of this good agreement we tested the accuracy of the approximation of replacing eq. (3.7) by eq. (3.8). For this we performed calculations of the elastic angular distributions with both the expressions (3.7) and (3.8) assuming, for simplicity, an appropriate gaussian density for the target. As expected, the two calculations were found to agree well up to about 4 = 2.5 fm-‘. Thus the good agreement between co and the correlation expansion calculation clearly shows that for 4 G 3 fm-r it is sufficient to go only up to the two-body correlation term in the correlation expansion. This is consistent with the finding of Harrington and Varma ‘) who made a similar calculation assuming a gaussian density for the target and using eq. (3.12) [which for a gaussian density is the same as eq. (3.11)] for the c.m. pair correlation function. Further the under’ It should be pointed out that since the expression (3.3) for K(q) is strictly valid for an antisymmetrized oscillator model wave function, in a sense, some effect of the antisymmetry tacitly persists even if the full amplitude is calculated by neglecting the antisymmetrization of the model wave function but multiplying the model amplitude with K(q). However, as follows from eqs. (3.1) and (3.2) since the same prescription is applied to obtain the one- and two-body densities involved in the correlation expansion calculation the above comparison is quite meaningful. The correspondence between the two calculations becomes clearer if one thinks of a situation where the Pauh correlation correction is negligibly small compared to the c.m. correlation effect.
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q Cfm"') Fig. 1. Elastic diflerential cross sections for 1 GeV protons on r60. The continuous curve shows 00 as described in the text. The other curves represent correlation expansion calculations: the dashed curve is obtained when the c.m. pair correlation is neglected while the dotted and the dash-dotted curves are calculated, respectively, with the expressions for the cm. pair correlation correction as derived in this work and by Harrington and Varma ‘1.
of CYfor q 2 2.5 fm-l can be easily interpreted to have its origin largely in neglecting the higher-order correlations. The dashed curve shows the correlation expansion calculation when the pair correlation term is neglected. The difference between the solid and the dashed curve is thus the genuine c.m. pair correlation effect and is seen to be significant, being 18% and -50% in regions of the first and the second maxima. The dash-dotted curve is obtained by using the prescription of refs. 237)for the c-m. correlation correction. By comparison with the dashed curve, it is seen to highly overestimate the c.m. effect ( = 28% against the realistic 18%) at the first maximum’. estimation
f It must be pointed out that according to Harrington and Varma 7, their prescription for the cm. pair correlation correction is in error by about 6% at the first maximum in 160 which is close to = 8%, that we find using the same prescription. However these authors also state that in all the cases their expression underestimates the c.m. effect which is in contradiction with what we find here.
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From fig. 1 it is clear that the presently proposed prescription provides a much better estimate of the c.m. effect. Up to this point we have simply demonstrated that if the target nucleus is described by the oscillator model our expression (3.9) for the c.m. correlation correction gives much more accurate results than the corresponding one derived previously. As such this finding is not of much significance, for in general the oscillator model does not provide a very good description of the target nucleus. However the fact that for a realistic wave function for the target it is very difficult to find a practical expression for the c.m. correlation and that a fully antisymmetrized oscillator model wave function provides a better description of the nuclear target than those expressly or tacitly assumed in previous derivations of the correction term makes the presently derived expression (3.9) more reliable for estimating the c.m. effect in the analysis of p-nucleus scattering data for determining nuclear density distributions. A detailed comparison of some recently proposed prescriptions for accounting for the c.m. effect is presented if fig. 2 for 160 and two other target nuclei 40Ca and 208Pb, where we have plotted (o -LTO)x 100/a. with (+ and cro calculated in the same manner as stated before. However, unlike 160 the calculations for 40Ca and “‘Pb are made in terms of more realistic Hartree-Fock densities and the corresponding oscillator constant cy2(which is obtained from the maximum overlap of the HF wave function with the oscillator one) as given in ref. 14).Thus go for these two nuclei is not as exact as in the case of 160. The continuous and the dashed curves in fig. 2 correspond, respectively, to the presently proposed prescription and that of refs. 2*7). In an earlier paper Alkhazov 5, also proposed another approach for accounting for the cm. pair correlation which has been studied recently by Auger and Lombard. [This approach differs in the manner in which I,.,.(b) is used for calculating the elastic scattering amplitude; the expression for I,.,.(b) being the same as in refs. “*‘).I The predictions of this approach are shown by the dotted curves in the figure [unlike refs. ‘,r4) we do not put p2 = 0 in the calculation]. Finally the dash-dotted curve corresponds to the prescription for the c.m. correlation correction as used by Ray 15) in his KMT calculation. This curve is calculated by taking the second-order KMT potential arising from the c.m. pair correlation as given by Ray [this is essentially the same as proposed in ref. “)I and obtaining the corresponding expression for the Glauber model calculation following the approach of ref. ‘). Although Glauber model and KMT calculations are not identically the same still the comparison is meanin~ul at least qualitatively. A study of fig. 2 clearly shows that in all the three cases, light, medium and heavy nuclei, the expression for the c.m. pair correlation correction as derived in this work gives much better results. In this case the deviation is less than 10% for 160 and at 5% level for 4oCa and *‘*Pb up to q =3 fm-‘, while the other prescriptions have larger deviations. Further it is interesting to note that with our prescription the deviation is always negative, relatively large at the positions of minima and increases in magnitude with 4, This behaviour lends itself to a plausible interpretation as the
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effect of neglecting higher-order correlations in the correlation expansion calculation; for these higher-order terms are expected to manifest themselves more prominently in regions where lower-order calculations give smaller cross sections and to increase in importance with increasing q. Also it is important to note that with the presently proposed prescriptions the deviations are refatively much smaller in regions of maxima (up arrows) which play a relatively more important role in the analysis of the data.
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b Cfml Fig. 3. Real part of I,.,,(b) as defined by eq. (3.6). The description of the curves is the same as in fig. 2.
The distinction between the various approaches for accounting for the c.m. effect is made more transparent in fig. 3, where we have plotted the real part of I,.,.(b) as defined by eqs. (3.6) and (2.13). The continuous, dashed and dash-dotted curves correspond respectively to eqs. (3.1 l), (3.12) and that used by Ray 15). It is seen that the three approaches differ greatly. Most striking is the fact that while the first two signify the c.m. correction as a surface effect, the one of Ray “> shows it to be a volume effect. This qualitative difference may have its origin in the approximations involved in deriving the expressions as used by Ray 15). In its derivation one uses an expression for the c.m. pair correlation function in momentum space, which is valid only for small 4 -values 4). Next we take up the question of the importance of accounting for the two-body correlation terms in the analysis of p-nucleus scattering data. This question has been discussed recently by Auger and Lombard 14) for the c.m. correlation, and by Harrington and Varma’) for both the c.m. and the Pauli correlation within the framework of Glauber theory. Still it is desirable to discuss it briefly. A recent analysis using the KMT formalism by Ray 15) shows that for medium-weight nuclei the c.m. pair correlation is considerably less important than the Pauli correlation which is in disagreement with the findings of Harrington and Varma ‘) according to which both correlations are essentially equally important. In fig. 4 we show (a -al) X loo/at for 160, 40Ca and *‘*Pb, where u and ml are the differential cross sections with and without the two-body correlation terms as calculated in the correlation expansion approach. The dotted and dashed curves are
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calculated, respectively, with the c.m. and the Pauli correlations only. The continuous ones are obtained when both the correlations are included. As expected the c.m. correlation effect decreases in importance with A while the Pauli one has the opposite trend. Taking the example of 40Ca we see that both the c.m. and the Pauli correlations are essentially equally important. This disagrees with Ray’s result “) that the Pauli correlation is about eight times as important as the c.m. correlation in this nucleus. The present calculation shows that the c.m. pair correlation contributes about 20%, 40% and 55% of the total effect at the first, second and the third maxima
q (fm-‘1 Fig. 4. Centre of mass and Pauli correlation effects on 1 GeV proton elastic scattering cross sections: dotted curve: c.m. correlation only; dashed curve: Pauli correlation only; continuous curve: both c.m. and Pauli correlations. The dash-dotted curve is for the Pauli correlation only neglecting the normalization of the corresponding two-body density.
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(shown by arrows), respectively. There seem to be two main reasons for Ray’s findings. First as follows from the curves for 40Ca in fig. 2 his expression for the c.m. correlation correction in general underestimates the c.m. effect at the maxima and second while accounting for the Pauli correlation he disregards the first term in the expression (3.13), which is needed for the normalization of the two-body density. Neglecting this term has the effect of enhancing the Pauli effect as follows from the dash-dotted curve for 40Ca in fig. 4 which has been calculated with the Pauli correlation only but disregarding the normalization of the corresponding two-body density. We also studied the effect of the gradient term in the expression (3.18) for 1,(b) which arises because of the mixing of the c.m. constraint and the Pauli correlation. As expected its contribution was found to be quite small.
5. Conclusions In this work we have proposed an effective profile function approach for the correlation expansion of the hadron-nucleus scattering amplitude in the Glauber theory which does not invoke the ansatz for the correlation expansion of the A-body density of the target nucleus. Going only up to the two-body correlation term in the correlation expansion of the scattering amplitude we have studied the effects of the c.m. and Pauli pair correlations on 1 GeV proton scattering on nuclei by treating the problem of the pair correlation in the approach of Feshbach et al. 9). In particular we have derived a simple expression for the c.m. pair correlation correction to the Glauber amplitude and have shown it to be much more realistic than the previously proposed ones. Further, we have demonstrated that the c.m. correlation correction is predominantly a surface effect. We have also studied the effect of the c.m. constraint on the Pauli correlation correction and have shown it to be small. The present study shows that both the c.m. and the Pauli correlation corrections are important enough to be included in any detailed analysis of the elastic scattering data. In addition we fmd that for a medium-weight nucleus like 40Ca the c.m. correlation effect on the differential cross section is essentially as important as the Pauli correlation one and not much smaller as recently reported by Ray “). The authors are grateful to Drs. R.J. Lombard and A. Chaumeaux for discussions. One of us (I.A.) is grateful to Prof. J. Saudinos for providing him with excellent facilities for doing this work. References 1) G. Igo, Rev. Mod. Phys. 50 (1978) 523 2) G.D. Alkhazov, S.L. Betostotsky and A.A. Vorobyov, Phys. Reports C, 42 (1978) 89
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3) J.P. Auger and R.J. Lombard, Ann. of Phys. 115 (1978) 442 4) A. Chaumeaux, V. Layly and R. Schaeffer, Ann. of Phys. 116 (1978) 247
5) G.D. Alkhazov, Nucl. Phys. A280 (1977) 330 6) S.J. Wallace, Phys. Rev. Cl2 (1975) 179; C.W. Wong and SK. Young, Phys. Rev. Cl2 (1975) 1301; Cl5 (1977) 2146; D.R. Harrington and G.K. Varma, Phys. Lett. 74B (1978) 316 7) D.R. Harrington and G.K. Varma, Nucl. Phys. A306 (1878) 477 8) R.J. Glauber. in Lectures in theoretical physics, voi. 1, ed. W.E. Brittin and L.C. Dunham (Interscience, NY, 1959) p. 315 9) Ii. Feshbach, A. Gal and G. Hiifner, Ann. of Phys. 66 (1971) 20 10) L.L. Foldy and J.D. Walecka, Ann. of Phys. 54 (1969) 447 11) I. Ahmad, Nuci. Phys. A247 j1975) 418 12) Y. Abgrall et al., Nucl. Phys. A316 (1979) 389 13) V. Franc0 and W.T. Nutt, Nucl. Phys. A292 (1977) 506 14) J.P. Auger and R.J. Lombard, J. of Phys. G4 (1978) L261 15) L. Ray, Phys. Rev. Cl9 (1979) 1855