Pauli correlations in heavy-ion collisions at high energies

Nuclear Physics A292 (1977) 506-522, © North-BoJlaard Pw~61WFhrp Co., Antrtsrdant Not to be raaroduoed UY Dhotoprlnt or mian®Im whhout writtm aermwtom from the publirL~

PAULI CORRELATIONS IN HEAVY-ION COLLISIONS AT HIGH ENERGIES f VICTOR FRANCO and W. T . NUTT Physics Department, Brooklyn College of the City University of New York, Brooklyn, NeK" York 1!1!0 Received 13 lune 1977 (Revised 28 July 1977) Ahstract : We calculate the effects of short-range correlations on the Glauber expansion for nucleusnucleus collisions using the Fermi gas model for nuclei . When we neglect the Pauli principle for collisions between heavy nuclei, calculation of the optical phase-shift function leads to non-unitary results and we cannot obtain cross sections . When we include Pauli correlations we find important cancellations in the optical phase-shift function, which make possible the calculation of total and differential cross sections for heavy nuclei .

l. Introduction

Glauber theory t " 2) has been very successful in describing the scattering of particles or composites of particles at medium and high energies . With the advent of many new facilities performing high energy scattering of nuclei from one another, a new field of study for Theorists was opened . Performing a Glauber expansion for a collision involving A t particles striking AZ particles involves an enormous amount of work even when At and A a are of modest size . A labor-saving approximation to the Glauber theory is the "optical limit" 3). This approximation yields a very simple means of calculating physical quantities. In effect, the usual optical limit is simply the first term in an expansion for the optical phase-shift function . It has been shown tha the reasons presented for this approximation are not valid a) and that succeeding ~ermsrin the expansion are significant s). In fact one finds that the series expansion does not converge at all for large values of A lA2 . Indeed the first correction to the usual first order optical limit result displays s) pathological behavior for large A lA2 . We show in this paper that the reasons for this behavior result from ignoring the short-range correlations and that, in fact, the prescription of ref.') when properly modified for c.m. corrections e) probably provides a qualitatively reliable calculation . Further, we show that the reasonably good agreement between the results obtained from the prescription of ref. 3), using a first order optical phase-shift function, and the results obtained by including the Pauli correlations in the expansion r Work supported in part by the Naàonal Science Foundation, the National Aeronautics and Space Administration, and the Qty University of New York Faculty Research Award Program. 506

PAULI CORRELATIONS

507

ofthe optical phase-shift function is a consequence of some remarkable cancellations brought about by short-range correlations . In sect . 2 we present a compact description of nucleus-nucleus scattering in the Glauber approximation and introduce short-range correlations in the expansion of the optical phase-shift function . In sect . 3 we introduce the short-range approximation. This approximation simply states that the correlation range andthe nucleonnucleon interaction range are both quite small compared to the size of a nucleus. In sect . 4 we approximâte a large nucleus by a Fermi gas in a square well. This approximation, which ought to be qualitatively reasonable for large nuclei, is used to calculate the expansion of the optical phase-shift function . Without the Pauli correlations of the Fermi gas, we obtain a correction term to the usual first order optical phase-shift function which is of high order in A, or AZ [ref. s)] . The contribution of this correction term to the imaginary part of the phase-shift function leads to non-unitary results for large A. This correction term is exactly cancelled when we introduce the Pauli correlations . We also show that the cancellation is exact for other short-range correlations as well, provided the density is uniform. The residual corrections from the Pauli correlations yield simply an effective nucleon-nucleon cross section in the first order optical phase-shift function, reminiscent of the result for nucleon-nucleus scattering') . In sects. 5 and 6 we summarize our results and present our conclusions. 2. Nucle~~s-nndeos scattering in the Glanber approximadon In this section we review the formalism for nucleus-nucleus scattering in the Glauber approximation, taking care to include the effects of short-range correlations in the nuclei. Nucleus-nucleus scattering is described by means of an optical phase-shift funotion, X(b) . The complete evaluation of X(b) is prohibitively difficult, except for the lightest of nuclei (A 1 ,AZ S 5). It is conventional to approximate X(b) by X,(b), the first order optical phase-shift function, which represents the first term in the expansion of X(b) in the number of nucleon-nucleon (NN) interactions . The iteration of X,(b) generates an infinite multiple scattering series . An the other hand, the Glauber approximation proscribes repeated interactions between a nucleôn in one nucleus and a nucleon in another nucleus . This proscription limits the multiple scattering series to A,A Z terms, and does not allow a class of terms present in the infinite multiple scattering series obtained from X,(b). An important modification arising from the succeeding terms in the expansion of X(b) is the elimination of those repeated NN interactions proscribed by the Glauber theory. In order to properly evaluate the succeeding terms in the expansion of X(b) the inclusion ofthe effects of short-range correlations is crucial. In the Glauber formalism we write the nucleus-nucleus scattering amplitude in

50 8

V . FRANCO AND W. T. NUTT

terms of the optical phase-shift function X(b), as F(4) =

ik dz~~,b~l-etx(a1l 2~ ,1 '

where fiiq is the momentum transfer, 6 is the impact parameter vector, and fiik is the incident momentum. Similarly, the nucleon-nucleon elastic scattering amplitude may be written as

where fick,,, is the incident momentum of the nucleon and I',,(b) is the NN profile function. The optical phase-shift function is defined in terms of the nuclear wave functions, ~1 and ~z, and l',~n,,(b) by A, A= (2 .3) iX(b) = In «r,~z~ ~ ~ ~ 1-rNN(6-sr+s,)~I~Gi~z~, =i J-i

where the transverse coordinates in one nucleus (with A, nucleons) and those in the other nucleus (with A z nucleons) are denoted by s, and s~ respectively . The expansion of eq. (2.3) yields iX(b) = iX i (b)+iXz(b)+ . . . .

(2 .4)

The leading term in this expansion is given by iX i (b) _ -

AiAz

J

1- N~b - s i +si)P,(r,)P z (ri)d3r,d'Ti,

(2.5)

where the one-particle density functions, ~ p,(r), are defined by P~(ii) =

J ~~~(ri,

darAt. ~ . . rA~)~zd3rz . . .

(2.6)

The second order term in eq. (2.4) may be written as iXz(b) = i~-(iXl(b))z+iz(b)~~

(2 .7)

with fz(b) given by %(b)

° Ai(Ai

+A

-1 )Az(Az -1 ) JrNN( 6- si+si)l'NN( 6- sz+~z)P~i~(ri~rz)

IA z(Az -1)

+AZ A~(A 1 -1)

x Pi ~(ri~ ri)d s r i d s rzd s ~~d 3rz

I rNN(6- si +si)I'NN( 6-s~ +si)Pi(r,)Piz~(ri~ rz~ s rid srids>az ~

l'Nrv(b - si +si)I'N~(b - sz+si)Piz ~(rl~ rz)Pz(ri~ 3rid 3rid 3rv

(2.8)

509

PAULI CORRELATIONS

where piz ~ are the two-particle densities defined by pJZ~(rl, rz) =

(2.9) ~~G~dr i , rr . . . r~~)~ zd3r3 . . .d3rA ~ . J The expression (2.5) is the usual first order optical limit result for the phase-shift function 3). We note here that eqs . (2.5) and (2.8) are symmetric under the interchange of A, and A z (as is expected) and that they contain terms which vanish exactly in nucleonnucleus scattering (i.e., either A, or Az equal to unity) . We also note from eq. (2.9) that the two-particle density function p;z~ contains short-range correlation effects . The existence of terms of order AiA2 can lead to very large contributions to Xz when both A1 and A z become large. Thus a careful treatment of Xz is quite important for large A IA z . If one ignores the NN correlation effects and writes (2.10)

Piz~(ri~ rz) = P~(ri)P~~rz), one clearly satisfies the constraint J

a a Piz~(ri+ r z )d r i d r z = 1 .

To include correlation effects, we follow the treatment of Glauber for nucleonnucleus scattering and write A~Af- 1)P~ z ~(ri, rz) = AiLP~ri)P~rz~ri~ rz)+O(1/Ar)]~ x A ; P~ri)P~rz~ri~ rz)~

(2.12) (2.13)

The function g(r l ,rz) is the two-body correlation function which we will treat in greater detail in sect. 4 where we will also show that the replacement (2.13) in our approximation scheme satisfies the constraint (2.11). We may combine eqs. (2.5), (2.7), (2.8) and (2.12) to obtain iXz(6) = iAiAz ~ rNN(6-a~ +s'1)rNr~b-sz+sz)Pi(ri)Pi(rz)Pz(ri)Pz(rz) x {~ri~ rz)-1][9(ri,rz)-1] + ~9{r~, rz) -1] + ~ri+ r2) -1 ] }d3rid 3rid 3rzd 3ri +iAiAi ~ rNN(6-si +si)I'NN(b-si +ai)Pi(ri)Pz(ri)Pz(rz){~ri+ rz)-1] + 1 } x d3r ld3r'ld3rz +zAiAz ~rN~(b - si+ai)rN~b - sz+~1)Pi(ri)Pi(rz)Pz(ri) x{[g('l,rz) - 1]+1}d 3r,d 3rzd3r'1 , where we have written g(r  rz) _ [g(rl, rz)-1]+ 1 .

(2.14)

51 0

V. FRANCO AND W . T . NUTT

To evaluate iX, and iXz by means of eqs. (2.5) and (2.14) we need to know the single particle densities, p,(r), the two-body correlation function, [g(n, rz)-1], and the NN profile function, l'ir(b). The profile function may be expressed in terms of the NN elastic scattering amplitude, f(q), by inverting eq. (2.2) to yield (2.15) We shall make use of eq. (2.14) for iXz(b) to examine the effects of short-range correlations. 3. Phase-shift functions for large uuadeI For large nuclei, the correlation function, g -1, and the NN profile function, l'N<,,, are both of such short range compared to the nuclear size that we can ignore the variations in density by making a peaking approximation when integrating over both of these functions. In order to verify this assumption, we have performed all the integrals which arise in Xl (b) and Xz(b) using Gàussian functions for pt, I',,m and g(r)-1 and have compared the exact result of such integrals with the approximate results obtained by ignoring the variations in the density over the range of the correlation function and the profile function . We found good agreement in this case between the two results. As expected, the larger the nucleus, the better the agreement. If we write g(n,rz)=g(~r,-rz~) and assume that the nuclear density varies slowly over distances of the order of the correlation length and the NN range of interaction, we may approximate eqs. (2.5) and (2 .14) by iX i (b) =

iXz(b)

= sAiAi

-2nj(0)A A z ik

N

'

P,(s,

z)Pz(s-b, z'kizsdzdz',

(3.1)

I rNN(si)rNN(~z - sz)~sr zz) - llf9{~~ - si, zi)-1]dzszdzsi

x dzs'zdz zdz'z ~ Pi(si+ zl)Pi(si - 6~ z'~kizsldz ld~l + 2 l AzAi

J

!'NN(~t)l'NN(s'i -sz)~sr zz)-1]dzszdzs'ldzz

P~(si - 6~ ~1)Pi(Ji+ z lkizs ldz ld~l +(2 H J 1)~ zf 2 C27ik (U)l + l AiAi [ Pi(sl, al)Pz(sl -6 + ~i)Pz(sl - 6, zikizsldaldzid~z+(2 `-" J 1)J lN x

r

51 1

PAULI CORRELATIONS

x f [g(r)-1]d3r+ CAiAz

J

p~(sl, zl)PZ(sl -6, z~~)Pz(si-6,

zi~ZSldzldzidzz

where we have used the result J

rNnr( 6~26 = 2n f(0)/~k N,

(3.3)

and where (2 .--. 1) denotes the interchange of the subscripts 1 and 2 in A, and p, . We should note that while eq . (3.1) is the same as the usual short-range approximation, eq., (3.2) is not since it does not assume l'HI,, is short range compared with g(r)-1 . It only assumes that l'Na and g(r)-1 are short range compared with the nuclear size. 4. Ferma gas model for nacle~s-nucleon scattering The evaluation of eq. (3.2) requires a consistent one-body density p,(r) and correlation function g(r)-1 . When the total number of nucleons is large we can approximate the density by a spherical square well and use the correlation function for a Fermi gas. This approximation ignores the dynamical short-range correlations which are present in real nuclei . As we shall see, the effects of this approximation are small since we always integrate over g(r)-1 and the volume integral of g(r)-1 is constrained by eqs. (2.11) and (2.13). The one-particle density is written as where

Here fiik F is the Fermi momentum, R, is the radius of the nucleus with A, nucleons, and B is the unit step function . Eqs. (4.2) and (4.3) define ßo . In writing eq . (4.2) we have used the fact that there are four different spin-isospin states for each eigenvalue of the wave vector of the individual-particle states . The two-body correlation function in the Fermi gas model is given by [see, for example, ref.')]

where J~ is a Bessel function of fractional order. A property of the Fermi gas correla-

Sl2

V . FRANCO AND W. T . NUTT

tion function is the constraint

which is interpreted in the Fermi gas model as introducing a vacancy in the neighborhood of each fermion ("exchange hole") to describe the blocking caused by the Pauli principle. From eqs. (2.11) and (2 .13) one can show that for large nuclei the constraint (4 .5) obtains for any short-range correlation if the one-particle density is uniform. When eqs. (4.1) and (4.5) are used in eq . (3.2), the coefficient of [2nf(0)/ikN] z vanishes identically ! This is a most dramatic result and completely removes the uncorrelated scattering terms which contribute large positive real parts to iX for large nuclei . This cancellation allows one to obtain meaningful results for cross sections involving large nuclei . If correlations were neglected in this model, highly nonunitary expressions would result, and rather nonsensical values for cross sections would be obtained. If we define J(R,, Rz , b) by .t(R 1, Rz, b) =

1

4

f g(R1-

sl +zl~(Rz- (s, - b) +zi ~zs ldz idzi,

(4.6)

the function iXz may be written as iXz (b) = 2aô.f(R 1 , Rz , b) I

rNN(si)rNr~sz - sz)[9(sz~ zz) -1][9(si - sr zâ) -1 ]d zszdzsi

x dzs'zdzzdzz+4~ô.~(R 1, Rz, b) ~ rNN(si)r(si - sz)[9(sz+ zz )-1]dzszdzsidz z , and iXl may be written as iXi(b)

z _ - C2~i r,(0)J 4AôJ(R>> Rz, b).

(4.7)

(4.8)

The evaluation of J(Rl , Rz , b) in eqs. (4.7) and (4.8) is described in the appendix . At high energies the NN elastic scattering amplitude may be represented by the form

where QK,, is the total cross section for NN collisions, a is a parameter which gives the real part of the NN amplitude and a is a parameter which yields a good fit to NN elastic scattering at small q. The NN profile function is then found from eqs.

51 3

PAULI CORRELATIONS

(2 .5) and (4 .9) to be rNN(b)

-

vN 1-ia e -6 4na )

/2

(4.10)

For g(r)-1 it is convenient to use as an approximation to eq . (4.4) the expression 9(r)-1 = -

3

(ßk ) e-wkFi2~~, ~ ~o n

with ß = ~s. This produces an excellent fit to the Fermi gas correlation function given by eq . (4.4), as shown in fig. 1 . This approximation also exactly satisfies the relation (4.12) obtained by integrating g(r)-1 given by eq . (4.4). It should be noted that eq . (4 .12) yields a very reasonable value for the correlation length. o.ss o.so o.ls o.~o o.os 0

Fig . 1 . Fermi gas correlation function, g(r)-1 : the solid curve is exact and the dashed curve is a Gaussian fit.

By means of eqs. (4.9}{4.11) we may express X, and z QNr>(1

- ia)

-4t3ô

Q~(1 2 C

- ta) Js

~kF)z n

~

1 1+4aß Z kFZ

in the rather simple forms (4 .13)

2

tX2(b) _

XZ

-

1 1 Z 2 } .~(R R 1 , Z, 6). (4.14) 4 1+2aß k F)

This becomes the usual short-range approximation in the limit a ~ 0. We see from eq. (4.14) that iX2 always has a negative real part provided a2 < 1 (which is the case at high energies). We also note that X2(b) has the same b-dependence as Xt(b).

514

V . FRANCO AND W. T. NLJTT

Hence, the effect of Xz may be reproduced simply by an effective NN total cross section, vN,,r, and ratio of real to imaginary part of the NN amplitude, a', in the first order optical phase-shift function . These effective NN parameters are given by z ( / ) ~NN~ 1 -ia) { 1 ) + ~') = aNN( 1QNN(1 2n ~kF [1 +4aßzkF 4 1 +2aßzkF]}' (4.15 which becomes, for a = 0,

dNN = ~NN {1 +

~NMI'kp)2

2rz

1 1 _1 [1 +4aßz kF 4 1 +2aßzkF]}

(4.16)

At high energies, the effective NN cross section dam,, is, according to eq . (4.16), approximately 14 ~ larger than the free NN total cross section QM,,. Before discussing the calculated results of these approximations, we note that the integral of the right-hand side of the approximation (2.13) over r, and rz is Ai ~9(r~>rz)P~(ri)P~rz)d3r~d3rz = Ai

I Pr(r~)P~rz~3rid3rz I

+Ar p~(ri)Pr(rz)[9(rl~rz)-1]d3rld3rz .

(4.17)

The first term is simply Ai. We evaluate the second term in the short-range approximation in which the nuclear sizes are large compared to the correlation length . Eq. (4.17) may then be written as Ai f9(rl, rz)P~(ri)P~{rz~ 3 rid3rz = Ar +Ai fPi(rib 3ri f~r) - l~d3r~

(4.18)

We recall the constraint of eq. (4.5) and the form of the single particle density eq. (4.1) to obtain (4.19) P~rb3r+ Ai 9(r i , rz)P~ri)P1(rz~3rld3rz = Ai -A~ J f~o ~ where po,/po = 1/A, [see eqs . (4.2) and (4.3)] . From the normalization of the single particle densities, eq . (4.19) becomes Ai f9(rvrz)P~ri)P~rz~ 3rld3 rz = A~At-1)~

(4.20)

On the other hand, the normalization of the two-particle density leads, for the integral of the left-hand side of approximation (2.13) over rl and rz , to the result

I

A~Ai - 1) Piz~(ri+ rz~sridarz = Ai(A~-1) .

(4.21)

Thus the integrals of the two sides of approximation (2.13) over rl and rz are iden-

PAULI CORRELATIONS

51 5

tical, and therefore this approximation does not alter the norm of the nuclear wave function . An important point to note here is that because of this constraint, which must be satisfied by all short-range correlation functions in order to conserve the norm of the nuclear wave function, we always obtain the cancellations in eq . (3.2) for nuclei with uniform density! However e4. (4.12) would no longer be satisfied if we included the short-range dynamical oorrelations . Thus the results presented in the next section are representative of the scattering of nuclei with uniform densities whether one demands dynamical correlations or not. S. Results

We have calculated both the total cross sections and the elastic scattering differential cross sections for a variety of incident and target nuclei with 1 S A 5 208 . In our calculations the parameters for the NN scattering amplitude are taken to be a) vt,,N = 42.7 mb, a = 0.241 fm2, and a = -0.28 corresponding to a nucleon incident energy of 2.1 GeV. Thus our results correspond to an incident nucleus of energy 2.1 GeV/nucleon. We have taken the nuclear radii to be given by R~ = 1 .1A} fm.

(5.1)

The corresponding value of the Fermi momentum is kF = 1 .385 fm- '. This reprodaces the measured value of-the nuclear density at the center of the nucleus . (On the other hand, the resulting nuclear rms radii are generally smallerthan the measured

Fig. 2. Real part of tX for "G`=C elastic scattering at 2.1 GeV/nucleon. The short-dashed curve corresponds to X = Xi . The solid curve corresponds to X = Xi +Xs with Pauli oornlations included . The long-dashed curve corresponds to X = Xi +Xs with g(r) ' 1 . The dotted curve corresponds to X - Xi +Xz with no correlations.

516

V. FRANCO AND W . T. NUTT r.

v L Fu r W W

0 C

Fig. 3 . Real part of the profile function, r(b), for ' =G I= C elastic scattering at 2 .1 GeV/nucleon . Notation is the same as in fig. 2 .

1 -tlMÂRY . X

0 _ 100 _ 200

^~ 0.75

v L i m

300

Z

500

u

0.5

- 600

m

W W

000

6

0.25

C -1000

0

2

4

6

8

10

12

Fig . 4. Real part of IX for =°sPb-sospb a~~ scattering at 2 .1 GeV/nucleon . Notation is the same as in fig . 2.

0

L ~

L

1

1

1

2

6

8

10

12

4

Wi g . 5 . peal part of the profile function, I'(b), for ~ °s Pb-=ospb elastic scattering at 2 .1 GeV/ nucleon. The dashed curve wrreapond to X ~ XI . The solid curve oomsponda to X = X, +Xs ~~ Pauli oorrehitions.

PAULI CORRELATTONS

51 7

values . Snce our aim is to compare theoretical results, this discrepancy is not significant.) In fig. 2 we present the real part of iX for' zC- 1zC collisions at 2.1 GeV/nucleon. The curve labelled X(1) denotes Re(iXl), and is obtained from eq. (4.13). It corresponds to the usual first order optical phase-shift function 3). The curve labelled X(2) denotes Re(iXl +iX z ), and is obtained from eqs. (4.13) and (4.14). It contains the effects of Pauli correlations. It has the same b-dependence as curve X(1), but is larger in magnitude. Ifwe use the approximation given by eq. (2.10) to calculate Xz, the resulting values for Re(iXl +iXz)would be those shown by the curve labelled X(2) . This approximation ignores the effects of Pauli correlations, but does conserve the norm of the nuclear wave functions. It produces a significant increase in Re(iX), but for this case does not violate the unitary bound Re(iX) 5 0. If wrrelations were ignored in our calculation of Xz by arbitrarily setting g(r) equal to unity, the curve labelled z(2) would be obtained for Re(iXl ±iXz). It badly violates unitarity. In fig. 3 we show the corresponding real parts of I'(b) _- 1- exp (iX) . The effect of Xz with Pauli correlations is to slightly increase Re T in the peripheral region of the interaction. Ignoring Pauli correlations but conserving the norm of the nuclear wave functions [curve labelled X(2)] leads to a Re T which is greatly suppressed beyond x 3 fm. Since the result corresponding to setting g(r) = 1 takes on extremely large values, it cannot be shown. In fig. 4 we show the real part of iX for z°ePb-zoaPb ~llisions at 2.1 GeV/nucleon. Again the effects of including Xz with Pauli correlations is to make Re(iX) more negative when compared with Re(iXl), as is seen by comparing the curves labelled X(2) and X(1) . Calculating Xz without Pauli correlations leads to such a large increase in Re(iX) that the result becomes badly non-unitary for b > 3 fm [see curve labelled X(2)]. The result obtained by setting g(r) equal to unity is so badly non-unitary for most of the range b < R 1 +Rz = 2R that it is not shown. In fig. 5 we present Re T for z °e Pl~ zospb collisions at 2.1 GeV/nucleon. Since Re(iXl) and Re(iXl +iXz) are quite negative except near b x Rl +Rz = 2R, Re T is very close to unity except near 6 x 2R. Again, the effects of Pauli correlations is to slightly increase Re Tin this region . Since the values of Re T are badly non-unitary when Pauli correlations are ignored, the two corresponding curves are not shown. We note from figs . 2-5 that for medium sized nuclei (A 1 x A z x 12, for example), ignoring Pauli oorrelations yields results for Xi +Xz and T which are rather different when compared with the results obtained with Pauli correlations included. Furthermore, the results obtained with Pauli oorrelations do not differ qualitatively from the results of the usual first order optical phase-shift function Xi . For heavy nuclei, ignoring Pauli correlations in calculating Xi +Xz leads to nonsense (highly nonunitary results), whereas including Pauli coirelations leads to a profile function which is almost identical to that obtained from Xi alone. In table 1 we present total cross sections for collisions between nuclei A, and Az at 2.1 GeV/nucleon. The rows labelled Az (a) contain cxoss sections calculated with

V . FRANCO AND W. T. NUTT

51 8

T~a~ 1 Total aces sections in mb for collision in the Ferrai gas model between nuclei of mass numbers A, and A s at an incâdent energy of 2.1 GeV/nucleon A'

As 1(a) (b) (c) (d) 2(a) (b) (c) (d) 4(a) (b) (c) (d) 12(a) (b) (d) 16(a) (b) (d) 24(a) (b) (d) 40(a) (b) 58(a) (b) 116(a) (b) 208(a) ro)

1

2

4

12

16

24

40

58

116

208

37.9 42.0 17.1 42.5

70 .4 77 .4 27 .8 71 .9 124 135 43 .8 112

128 140 41 .5 124 211 227 65.0 178 332 351 111 256

319 342 55 .8 296 467 490 174 377 646 670 980 472 1064 1090 731

401 428 57 .0 371 569 595 342 460 766 791 1897 558 1215 1242 868 1376 1403 1087

552 584 86.6 509 749 778 1244 608 972 999 -4110 712 1470 1498 1199 1646 1675 1694 1940 1969 2488

815 856 516 752 1052 1085 6254 863 1311 1341

1076 1123 2443 993 1343 1378 - 4391 1112 1631 1664

1781 1838 28803 1650 2107 2148

2689 2754

1775 2460 2497

2624 3489 3531

974 1880 1909 2049 2078 2108 2706 2406 2437 - 716 2921 2953

1230 2258 2289 2969 2474 2506 1127 2831 2863 28947 3387 3420 3886 3920

1924 3213 3248 1288 3470 3505 25891 3888 3923

2847 4372 4411 - 691 4670 4708

4534 4570 5109 5145 6496 6534

5891 5929 6542 6581 8100 8139 9879

2500 3071 3117

5152 5191

(a) X = x~ . ro) x = Xi +Xs with Pauli correlations. (d) X = Xi +Xs with no Pauli correlations.

X = Xt (the usual optical phase-shift function). The rows labelled A2 (b) contain cross sections calculated with X = Xi +Xz+ ~~B «ls~ (4.13) and (4.14), and include the ~auli oornlations. The dit%rence between these two sets of cross sections is not very large, being about 10 ~ for A1 = A2 = 1 and decreasing to x 0.4 ~ for A t = A Z = 208. The rows labelled A 2 (c) correspond to X2 calculated with g(r) set equal to unity (Pauli oornlation ignored). The rows labelled AZ (d) correspond to X 2 calculated with no Pauli eornlations but with the norm of the nuclear wave functions preserved [eq. (2.10)] . The calculations corresponding to the rows labelled A2 (c) are non-unitary for all but the smallest values of A l A 2 , and are certainly

PAULI CORRELATIONS

51 9

unreliable for A IA Z > 60 . (In some cases they are negative . In some cases they are very large in magnitude and are not shown.) The same remarks hold for the rows labelled AZ (d), except that now the results become unreliable for A lA2 Z 400, reflecting the improvement of the approximation for p ;2 ' when compared to the case in which g = 1 . We note that for heavy nuclei (A t ,A Z X40), the effect of X2 (calculated with Pauli oorrelations) on the total cross section is less than z 1 ~. This is in contrast to the results of ref. 9) where for heavy nuclei the effect of XZ calculated with the approximation (2.10), which ignores Pauli correlations, was as large as x 11 %. In fig. 6 we present the differential cross section dQ/d~t~ for tsGtzC elastic scattering at 2.1 GeV/nucleon. The results obtained from the second order optical phaseshift function Xz, with Pauli correlations included, differ only slightly from the results obtained from the usual first order optical phase-shift function, even out near the fourth maximum. On the other hand, the results obtained from the second order optical phase-shift function with Pauli correlations neglected [curve labelled

.

r~ a

.

10 4

~) X (2)

10 3

l0 0

n

10~

a

n

~9

xiß

10 1

10~

1~1

0

1

0.1

1

0.2

.3 0

0.4

~(ß " Y/e) 3 Fg . 6. Ditïbrential coca actions, do/d~t~, for '=G'=C elastic scattering at 2.1 OeV/nucleon . Notation is the name as in fig. 2.

ld~

0

OA1

0.02 .OA3

.04 0

-rCß "V/e)~ Fg. 7 . DitFerential aosa actions, da/d~t~, for

2°°Pb-1° 'Pb elastic scattering at 2.1 GeV/nucleon.

The dashed curve corresponds to X = XI . The solid curve oorr+espond to X ~ X l + X3 with Pauli correlations.

520

V . FRANCO AND W. T. NLJTT

but with the correct norm for the nuclear wave functions are quite different from those shown in the curves labelled X(1) and X(2), even in the forward direction. In fig. 7 we present the differential cross section dQ/d~t~ for z°BPb-2°BPb elastic scattering at 2.1 GeV/nucleon. The results obtained from the second order optical phase-shift function, Xz, with Pauli correlations included, differ negligibly from the results obtained from the usual first order optical phase-shift function out to at least the fourth minimum. The results obtained by ignoring Pauli conelations are not shown in this figure because they are badly non-unitary in this case and give nonsense . These calculations demonstrate how, when Pauli conelations are included in the evaluation of the optical phase-shift function, total and differential cross sections for heavy-ion collisions may be calculated and how the results are not qualitatively different from the results obtained with the first order optical phase-shift function . Thus for heavy nuclei the results obtained from a correct first order phase-shift function Xi are likely to be more reliable than the results obtained from a second order phase-shift calculation which ignores Pauli conelations. A method for calculating the first order optical phase-shift function, correctly taking into account c.m. conelations, has recently been given 6). X(2)]

6. Dis~ion and conclaeiona The inclusion of Pauli oonelations in the expansion of X removes certain double scattering processes which tend to make the nucleus more transparent. The sequential double scattering terms have been accounted for in the exponentiation of iXi . Thus the correction term, iXz, is made up of processes in which one nucleon is scattered by two others simultaneously . Because the contributions of these processes to iXs have positive real parts they tend, in effect, to increase flux. That is to say, they allow the transmission of nucleons from one nucleus through a second via double scattering processes in which the nucleons of the second nucleus are required to be very close to one another. Hence the sensitivity of these processes to short-range conelations. Any short-range repulsive correlation would produce a similar effect by preventing the nucleons in a nucleus from being very close to one another. We expect that other calculations with different densities and correlation functions would suppress the double scattering processes represented by XZ . We have calculated the corrections to the usual first order optical limit (X = Xi) arising from the next term in the expansion, Xz, taking into àocount Pauli conelations. Using the Ferrai gas model we have exhibited a remarkable result for Xs. Although Xa has several terms which are of higher order in A 1 and AZ than Xi , all of these terms cancel when the Pauli conelations are treated consistently. If one performs a calculation of the usual optical phase-shift function, Xi, using realistic nuclear wave functions, any corrections to Xi must be evaluated with these same nuclear wave functions and care must be taken to ensure proper treatment of

52 1

PAULI CORRELATIONS

the Pauli conelations. Indeed one must consider the effects ofall short-range conelations which differ somewhat in a real nucleus from the Fenni gas model because of the dynamical conelations and because of the varying density, effects which we have not treated here . In general g(r)-1 is negative and will produce a suppression of Xz . Thus we conclude that in the case of heavy nuclei care should be exercised in treating the effects of short-range conelations on Xz. Appendix

In this appendix we describe the evaluation of the integrals for the optical phaseshift function . We have made use of the short-range approximation (i.e., the oonelation length and the NN interaction range small compared to the nuclear size) to reduce our integrals to the general type, I~k(Ri+Rr b)

= I Pi(s+z~)Pi(s-b~ z z ~ zsdz l dz z .

(A.1)

Analytical evaluation of these integrals in closed form for the case of step function densities is probably not possible . Additionally, accurate numerical evaluation of the four-dimensional integral is very costly. We have used an algorithm for the evaluation of such integrals which involves describing a spherical distribution by a large number of cylindrical segments . Thus p! may be approximated by P!(s~ Z) where

N - Po! ~

q=1

ie~(Ziq-ZX2--Z1 .9-1)~+e~-(Z+ZlgxZ+Zl .q-1)JI`~1r9-sZ)~

(A.2)

This approximation introduces an immediate simplification of e9. (A.1): ( k 2R1 2Rz sz~rzzp _ (a-b)z~ (A.5) I R , R , b) = P~olPoz ~ ~ dzs~rz q=1 p=1 J where R, and Rz are the radii of nuclei 1 and 2 respectively . The subscripts on the small r denote nuclei 1 and 2 also. Thus, ri = Ri(1-p z lM z) for example. [It is the simplification exhibited by eq. (A.5) that allows us to write the integrals remaining in X, + Xz in terms of the single integral, J, of eq . (4.6).] Since the integral appearing in e4. (A.5) is symmetric under the interchange of rl and rz, we need only consider the integral

NM

1,(r~, r b) =

J

lq -

dz sB(r~ -az)8(r; -(s-6)z).

(A.6)

The integral in e4 . (A .6) is simply the overlap of two circles of radii r~ and

522

V. FRANCO AND W. T. NUTT

r, separated by a distance b. There are three possibilities (i) b > r, +r< in which case Is = 0.

(A.~

(ü) r, - r < < b < r, + r< , in which case

I. = rC~n+sin-t(~}z>)-z>

where

1-z>7,

(A.8)

(iii) b < r, -r5 , in which case I s = nr~ .

(A.11)

The integral in eq . (A .5) is then evaluated by substituting either rtq or rZp for r, and rZp or rtq for r< in the appropriate case. For j = k = 1, we have from eqs. (4.6) and (A.5) the result .f(R t , R Z , b) =

x

~r

('

~ ~ dZS9(r~q -sZ )8(r2p -(s-b)Z ). MN q=1 p=1 J

(A.12)

For each value of p and q, the integral in eq . (A.12) may be evaluated by means of eqs. (A.7}{A.11). The double sum converges quite well for N and M x50-100. The integrals in eq. (3.2) which cancel one another were also evaluated using similar techniques . We thank Dr. Crirish K. Varma for a useful discussion. Refereaces 1) R. J. Glauber, in Lectures in theoretical physics, vol. 1, ed . W. E. Briton and L. Duuham (Interacience, New York, 1959) p. 315 2) R. J. Glauber, in High energy physics ând nuclear structure, ed . S. Devons (Plenum, New York, 1970) p. 207 3) W. Czyt and L. C. Maximon, Ann. of Phys . 32 (1969) S9 4) V. Franco, Phys. Lett . 64B (1976) 13 5) V. Franco and G. K. Varna, Phys. Rev. C1S (1977) 1375 6) V. Franco and A. Tekou, Phys . Rev. C16 (1977) 658 7) A. Bohr and B. R Mottelson, Nuclearstructure, vol. 1 (Benjamin, New York, 1969) p. 150 8) O. Henri et al., University of California Radiation Laboratory report UCRL"20000-NN (1970) ; D. V. Bugg et al., Phyâ. Rev. 116 (1966) 980; T. J. Devlm et d., Phys . Rev. D8 (1973) 136 9) V. Franco and G. K. Varna, Phys. Rev., to be published