Volume 77B, number 1 PHYSICS LETTERS 17 July 1978

MODEL ENERGY CALCULATIONS OF NUCLEAR MATTER

IN THE JASTROW VARIATIONAL APPROACH

Tapash CHAKRABORTY Tata Institute of Fundamental Research, Bombay 400005, lndia and Department of Physics, Dibrugarh University, Dibrugarh 786004, lndia

Received 10 April 1978

We calculate the model energy of nuclear matter within the FIY expansion scheme. Numerical results are presented for two semi-realistic central potentials and a state independent correlation function which is suitably constrained for good cluster convergence.

In recent years there has been an upsurge of activi- ties in the Jastrow variational calculations of nuclear matter [ 1 - 4 ] , in particular the approximate evalua- tion of the expectation value E v = (q~, H q O / ( ~ , ~ ) of the ground-state energy. However, Ktimmel has recent- ly proposed [5] the calculation of the model energy E m = (rb, Heg)/(cb, ~) , which, as he argues, would pro- vide a possible internal consistency check for the Jastrow theory. As pointed out by Zabolitzky [6] and discussed by several authors [1,7], the model kinetic energy can be evaluated in two ways. When the kinetic energy operator is applied to the left, one gets the ex- act result E F, and when the operator i s applied to the right one obtains the approximate result Tmo d. The difference between these two results is thus the error due to the omission of higher order terms (Ter r = Tmo d

-Ev). For a short-range correlation factor f(r), we develop

the factorized Iwamoto-Yamada (FLY) cluster expan- sion [1,2] for E m. We first define the generalized nor- malization integral,

J(/3) = (¢,, exp/3(H- EF)~ ) , (1)

corresponding to the quantity ( H - EF). E F = (qb, TqS) is the kinetic energy and the parameter/3 serves to de- fine E m in the form:

E m = E v + (O/a/3)lnJ(/3) 13 = o" (2)

Defining the subnormalization integrals, analogs of eq. (1) for subsystems of the A-nucleon system, and the cluster integrals as in refs. [1,2], we have the factor-cluster or van Kampen type expansion:

E m = E F +(AEm) 1 +(AEm) 2 +(AEm) 3 + . . . . + (AEm) A ,

(3) where the terms are arranged according to the number of bodies involved.

For a uniform exten~ted system like nuclear matter, the first few terms are:

= ~ 1-aj//" I (4) ( A g m ) l = 0 , (AEm) 2 i<]]i] Bfl 13=0'

[1 aj/,k [1 aj/~

+same for all other pairs {Lj, k }]]l I

31] 13=0

The "normalizing denominators" are first written as

J/i] = 1 +remainder, J,~kl 1 +remainder [8,91. t3=0 u 13=0

It was however observed that, if the general definition of the Pauli principle, (q~, (F-1)qb) = 0, as given by Ristig et al. [10] is applied "term by term" to the ex- pansion (3), the remainder terms vanish. The conse- quence of this effect shows up mostly in the three-body

Volume 77B, number 1 PHYSICS LETTERS 17 July 1978

case, where the two-body combination term vanishes• Considering only up to the two-body term, we now

have

(AEm) 2 = ~ (i/'J~22(12) It~)a, (5) i< j

where the effective two-body operator is

1 g22(12 ) = ~ [ ( t ( 1 ) + t ( Z ) , F ( 1 2 ) ] +F(12)v(12) , (6)

which for the choice of Jastrow function q~ = H/A</ f ( r i j ) ~ reduces to

~22(12) = f(rl2 ) o(12) (7)

- (h2/Zrn)[V2f(r l 2) + 2Vf(rl2)" V] .

It is to be noted that only the exchange part survives for the non-local term in eq. (7), with the gradient op- erator operating on l ( k F r ). We thus essentially have the Pandharipande-Bethe (PB) form or the I w a m o t o - Yamada (IY) form of the kinetic energy.

The terms in the square brackets are the two-body approximation of Tmo d, (cf. eq. (48) of Zabolitsky [6] ), and thus is a measure of the truncation error (Terr) of the approximation used•

For numerical study we employ two semi-realistic potentials (OMY and IY), which are used mostly as test potentials in the variational calculations of nuclear matter [4, 1 0 - 1 2 ] • The general form chosen for the correlation function f ( r ) is taken from refs. [4,12] :

f ( r ) = O, r <<. r c ,

= { 1 - exp [-/21 (r - re) ] }

X {1 + 7 e x p [ - u 2 ( r - rc)] }, r > r c ,

subject to the average Pauli condition expressed by

p f d r ( f ( r ) - 1)(1 - ¼l 2 (kFr) ) = 0 , (9)

where p is theparticle density. The main advantage o f using this type o f function and potentials is that in this case lowest order variational calculations o f E v [4,12] as well as FHNC calculations [11 ] are available. Some test calculations to check the efficacy o f this type of f ( r ) have also been performed [13,14] withthese potentials.

In table 1, we present the model kinetic energy Tmod, and the truncation error Ter r for various values

"O o

6

° ~

O " O

o

6 ~

4 ~ Z

='7, aT~

o O

o

. g

"7

T

O

I I I I I I I I I

I i ~ T 7 I i i t i

Volume 77B, number 1 PHYSICS LETTERS 17 July 1978

of k F, calculated with f ( r ) of the form (8) and the two test potentials (OMY and IY). As evidence from this table, the deviation of Tmo d from E F (i.e. the truncation error) is quite large for all the density ranges considered. For both the potentials, at and around the respective equilibrium densities (in variational calculations) Ter r

"" E F. In the FHNC scheme, Tmo d - E F for nuclear matter has been calculated by Zaboli tzky [6]. He used, however, a different choice of functions (Pandharipande -Be the form [15] ) and the test potential o2(r )of ref. [16].

As expected, there is little difference between the variational and model potential energy values at all densities. The model energy E(n2) ( two-body) values are also given in table 1. The two-body variational energy E (2) values, obtained *1 in the manner discussed in ref. [4] are given in the last column of table 1. It is observed that, if we retain up to the two-body term of eq. (3) only, there is no minimum of the model energy with respect to density. It is to be recalled that this minimum for E o results from a large cancellation of the potential energy ( ~ 114 MeV for OMY at k F = 1.6 fm - 1 ) against the kinetic energy due to particle correlations ( ~ 70 MeV) (the (h2 /m [V f (12)] 2 term in W2(12 ) of ref. [4] ). The resulting value is labelled as (AE)2 in ref. [4]. The contribution from the counterpart of this kinetic energy in the model formalism is comparatively small ( ~ 41 MeV), and thus could not compensate for the extra binding. Incidentally, this term is called the " t runcat ion error" by Zaboli tzky [6].

Calculation of the three-body term of eq. (3) is not expected to bring any major change in our findings. The three-particle correlation which contributes purely to the kinetic energy (as three-body forces are not included) is expected to cancel more of the model potential energy

:~ 1 The E o tresult reported here is slightly different from that reported in ref. [4] where, besides the Pauli condition, other subsidiary conditions have been imposed on f(r). Our result in fact, agrees with that of ref. [11].

and might bring the model energy values close to the variational energy values. However, this will be done at the risk of getting larger values of Ter r and thus deviat- ing Tmo d further from E F. Nevertheless, three-body calculation of E m is interesting and will be reported elsewhere.

The author wishes to thank Dr. B. Banerjee and Dr. P. Mahanta for very useful conversations. He has benefited from correspondence with Dr. J. da Providencia.

References

[1] J.W.Clark, Crisis in nuclear matter theory (1976), un- published; Variational theory of nuclear matter (1976), unpublished.

[2] J.W.Clark and M.L. Ristig, in: The nuclear many-body problem, Vol. 2 (Editrice Compositori, Bologna, 1973).

[3] O. Benhar et al., Phys. Lett. 60B (1976) 129. [4] S.-O. Backman, D.A. Chakkalakal and J.W. Clark, Nucl.

Phys. A130 (1969) 635; S.-O. B~ckman et al., Phys. Lett. 41B (1972) 247.

[5] H. Kiimmel, Z. Phys. A279 (1976) 271. [6] J.G. Zabolitzky, Phys. Rev. A16 (1977) 1258. [7] B.D. Day, preprint (1977). [8] P. Westhaus and J.W. Clark, J. Math. Phys. 9 (1968) 149. [9] J.W. Clark and M.L. Ristig, Nuovo Cimento 70A (1970)

313. [10] M.L. Ristig, W.J. Ter Louw and J.W. Clark, Phys. Rev.

C5 (1972) 695. [11] E. Campani, S. Far, toni and S. Rosati, Nuovo Cimento

Lett. 15 (1976) 217. [12] D.A. Chakkalakal, C.-H. Yang and J.W. Clark, Nucl. Phys.

A271 (1976) 185; D.A. Chakkalakal, Ph.D. thesis, Washington Univ. (1968).

[13] J.W. Clark, P.M. Lain and W.J. Ter Louw, Nucl Phys. A255 (1975) 1.

[14] M.L. Ristig and P. Hecking, Phys. Lett. 65B (1976) 405. [15] V.R. Pandharipande and H.A. Bethe, Phys. Rev. C7

(1973) 1312. [16] V.R. Pandharipande, R.B. Wiringa and B.D. Day, Phys.

Lett. 57B (1975) 205.