Volume 87B, number 4 PHYSICS LETTERS 19 November 1979

MODEL ENERGY - A STUDY OF THE JASTROW CORRELATION FACTOR

Tapash CHAKRABORTY 1 Tata Institute o f Fundamental Research, Bombay 400 005, India

Received 17 May 1979 Revised manuscript received 4 September 1979

We calculate the model energy of nuclear matter for two semi-realistic potentials with state-independent hard cores. Here we have included the delta-function contributions arising from the hard cores, which have been omitted in a previously published paper.

Our paper on the subject o f the two-body model energy of nuclear matter [1 ] contains some inconsis- tencies, essentially due to the omission of an impor- tant contribution.

As demonstrated by Zabolitzky [2], the efficacy of the Jastrow wavefunction • = F ~ , where the spa- tial correlations are embodied in the factor F = I I i< j f ( r i j ) , can be checked by calculating the model energy E m = (~ IHI ~)/(~l ~ ) . As usual, • represents the ground-state wavefunction of a noninteracting Fermi gas. If the variational energy obtained for a suitably chosen correlation function f ( r ) is very close to the true ground-state energy, then the function f ( r ) could be considered to be not very far from the ground-state wavefunction. In this case, we expect also good agree- ment between the values of the variational energy and the model energy evaluated with this choice of f ( r ) . (For a discussion on the limitation of this method, see Day [3].)

As in ref. [1 ] we define the generalized normaliza- tion integral

J03) = (q~, exp/~(H - EF) ~ ) , (1)

and the model energy as

E m = E F + (a/a/3) In J(/3)It~= 0 . (2)

The factor-cluster expansion o f E m is

E m = E F + (z2iEm) 2 +.. . + (AEm) A , (3)

I Present address: Institut far Theoretische Physik, Universit/it zu K/51n, D-5000 K~51n 41, Germany.

where the terms are arranged according to the number of correlated bodies. The two-body term of (3) is

(AEm) 2 = ~ (i/1122(I 2)Ii/) a , (4) i</

where the appropriate spin (isospin) summation is im- plied, and the effective two-body operator is

~22(12 ) = I t ( l ) + t (2) ,F(12)] +F(12 ) o(12) . (5)

For the choice of Jastrow function, as defined above,

~2(12) = f ( r l2 ) o(12)

- ( l i 2 / m ) [ V 2 f + 2Vf(rl2) • V] . (6)

For a numerical study, we employ two semi-realistic hard-core (central) potentials, OMY and IY, and use the opt imal function of BCC [4]:

f ( r ) = O, r <~rc,

f ( r ) = { 1 - exp [-/al(r - re) ] ) (7)

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

The basic difference between the present work and the calculation of Zabolitzky is the presence of a hard core in the potentials adopted. For these two" potentials, the slope o f f ( r ) is discontinuous at the hard-core radius (r = rc), so that V2f(r) contains a term proportional to

~ ( r - r c ) ,

Tdelt a = --or [1 -- 112 (kFrc) ] ,

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19 November 1979 Volume 87B, number 4 PHYSICS LETTERS

Table 1 Model energy (E m) and variational energy (Ev) for nuclear matter (OMY and IY) with BCC correlation function.

Pot. k F E F Tmo d Ter r E m E v (fm -I ) (MeV) (MeV) (MeV) (MeV) (MeV)

OMY 1.2 17.91 16.32 -1.59 -10.96 -10.7 1.3 21.02 18.53 -2.49 -15.02 -12.8 1.4 24.38 20.61 -3.77 -20.69 -14.6 1.5 27.99 22.54 -5.45 -27.12 -15.9 1.54 29.5 23.42 -6.09 -28.81 -16.1 1.6 31.85 25.08 -6.77 -28.64 -15.3 1.7 35.95 27.76 -8.19 -24.87 -10.3

IY 1.1 15.05 14.13 -0.92 - 6.58 -5.0 1.2 17.91 16.08 -1.83 - 9.39 -6.2 1.3 21.02 18.04 -2.98 -12.9 -7.2 1.36 23.01 19.14 -3.87 -15.99 -7.5 1.4 24.38 19.88 -4.5 - I 8.18 -7.5 1.5 27.99 22.33 -5.66 -17.07 -6.1 1.6 31.85 25.04 -6.8 -13.71 -1.2

where ot = 21rp(~2/m) t t l (1 + 3') r 2. In order to deter-

mine the cont r ibut ion o f f ( r ) o(r) at and inside the

hard core, we adopt a procedure suggested by Iwamoto

and Yamada [5]: the two-body problem is solved for

a core potent ia l o f finite height and then we let t h e

height tend to infini ty. In that case, considering that

this holds in the case o f many-body problems [5], one

finds that , while f ( r ) o(r) is zero inside the core, at the

hard-core radius (r = rc) a second 8-funct ion contr ibu- t ion is obta ined which is similar to that arising f rom

35

30

25

~ 2o

15

10 ~0 1.2 Z~ ~6

kFt~O Fig. 1. Model kinetic energies for nuclear matter with OMY and IY potentials.

0 -10

-20

"25

"30

1.3 1.'~ 1.'s 1.'6 17 kFtf~,lJ

Fig. 2. Model energy (E m) and variational energy (Ev) of nuclear matter for OMY potential.

3 2 2

Volume 87B, number 4 PHYSICS LETTERS 19 November 1979

V2f(r) but o f opposite sign and different magnitude,

Vdelt a = ~a[1 + 12(kFrc) ] .

These kinetic and potential contributions have been omitted in ref. [1].

In table 1, we present the numerical results o f the adopted procedure for the model energy with the optimal f ( r ) of BCC corresponding to the two hard- core potentials. As seen in table 1 and fig. 1, for both potentials at low density, Tmo d is reasonably close to EF, rendering Ter r small. The model energy values for both potentials saturate at densities very close to those at which the variational energy values saturate (fig. 2). The minimum o f E m is, however, far from the mini- mum o f E v. Some improvement could presumably be

obtained if we include the next higher-order term of (3), i.e. (AEm) 3 . In fact, the E v values quoted in table 1 are the sums of two- and three-body terms of the FlY expansion.

The author wishes to thank John Zabolitzky and B.D. Day for their comments and suggestions.

References

[1 ] T. Chakraborty, Phys. Lett. 77B (1978) 6. [2] J.G. Zabolitzky, Phys. Rev. A16 (1977) 1258. [3] B.D. Day, Rev. Mod. Phys. 50 (1978) 495. [4] S.-O. B/ickman et al., Phys. Lett. 41B (1972) 247. [5] F. Iwamoto and M. Yamada, Prog. Theor. Phys. 18 (1957)

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