Nuclear matter calculations using separable potentials

Nuclear Physics A104 (1967) 283--288; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

NUCLEAR MATTER CALCULATIONS USING SEPARABLE POTENTIALS B. S. B H A K A R a n d R. J. M c C A R T H Y t

T. IV. Bonner Nuclear Laboratories tt, Rice University, Houston, Texas Received 8 J u n e 1967 Abstract: T h e binding energy per particle a n d saturation density o f nuclear m a t t e r are calculated using reaction matrix theory for a n u m b e r o f separable potentials in current use. N o n e o f the potentials tested yield satisfactory results b u t a separable potential which c o m b i n e s the desirable features o f the T a b a k i n a n d Puff potentials should be able to fit nuclear m a t t e r a n d nucleonnucleon scattering data. However, it appears doubtful that a separable potential can be f o u n d which can fit nuclear matter a n d nucleon-nucleon scattering data a n d still be suitable for nuclear Hartree-Fock calculation.

1. Introduction

Recent work on the nuclear many-body problem has shown the need for more accurate estimates of the contribution of three-body clusters to the binding energy of nuclear matter. The importance of the three-body clusters was first brought out by Rajaraman ~) and more recently by Bethe 2), who showed that an expansion for the energy should be made in the number of interacting nucleons rather than in the number of successive interactions. Bethe's treatment of the three-body clusters is based on the integral equation formalism of Faddeev 3) which has been applied so successfully to the three nucleon problem. However, applications 4-6) of the Bethe-Faddeev formalism have not made use of the mathematical simplicity offered by separable potentials. The Faddeev equations can be solved in an exact fashion using separable potentials. The nuclear many-body problem cannot, of course, be solved exactly but it is possible to treat the three-body geometry exactly by using separable potentials. We believe that a good estimate of the contribution of three-body clusters can be made using separable potentials provided a separable potential can be found which gives reasonable results in reaction matrix calculations including only two-particle correlations. There are many separable potentials in current use which yield approximately the correct binding energy per particle ( ~ 15 MeV) at a Fermi momentum k F ~ 1.4 f m - I which corresponds to the observed central density of large nuclei. However, most of these potentials, especially the S-state potentials which contain no repulsive term, do not yield the correct saturation properties in nuclear matter. In this paper, as a preliminary step to calculating three-body contributions, we present the binding t Present address: Carnegie Institute o f Technology, Pittsburg. tt W o r k s u p p o r t e d in part by the U.S. A t o m i c Energy C o m m i s s i o n . 283

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MCCARTHY

energy per particle and saturation curves for a number of separable potentials in current use. In these calculations, we set the potential energy of particle states equal to zero. In this way we include only those graphs corresponding to two-particle correlations. Our results show that it is possible to obtain a separable potential which can fit nucleon-nucleon scattering data and also yield the correct results in a nuclear matter calculation using reaction matrix theory. Therefore it should be possible to use separable potentials in calculating contribution of three-body clusters in nuclear matter. However, our results also show that it is unlikely that a separable potential can be found which can satisfy the above two conditions and still have small enough offdiagonal matrix elements to allow its use in nuclear Hartree-Fock calculations. Further discussion of this point will be given after the results are presented. We have carried out the calculations for six separable potentials found in the literature. These include the Tabakin v), Puff 8) and Yamaguchi 9) potentials, as well as three S-state potentials considered by Tabakin lo) in an investigation of the effect of short range correlations on the three-nucleon binding energy. 2. Theory The energy per particle of nuclear matter is given by (in units of h2/m)

E/A = 3ak~+½

~,, ( m n l G l m n - n m ) ,

(1)

tr/~ n < . k F

where kF, the Fermi momentum, is related to the density of nuclear matter by 2k 3

p = 3/r2 ,

(2)

and the reaction matrix G, is defined in operator form by G = v-v

Q- c .

(3)

e

The energy denominator, e, is defined as

e = E(i) + E(j) - E ( m ) - E(n),

(4)

where i a n d j represent intermediate (particle) states and m and n are occupied (hole) states. We approximate particle and hole energies by E(i) = ½kL

E(m) =

1. kZ,.+A.

(5) (6)

2m* The parameters A and m* are determined self-consistently for each value ofkv. If we

NUCLEAR MATTERCALCULATIONS

285

further define the initial relative m o m e n t u m k o = ½(kin- k~), the intermediate relative m o m e n t u m p = ½(k~- k~), and the centre-of-mass m o m e n t u m K = k s + k,,, then the energy denominator can be written as e

= p2+72

(7)

,

where .m .* -.- . [ . K2 . . . ko2 , 4m* m*

y2 = - 2 A +

(8)

Finally, a typical G-matrix element is given, in terms of relative coordinate, by

(klGlko) = ( k l v l k o ) -

f d~p (klv[p)Q(p,p2-t-^?2 K, k~)(plalko)

(9)

We use the angle averaged Pauli operator, Q(p, K, kF), which has been shown 11) to be sufficiently accurate in nuclear matter calculations. All of the potentials tested can be written in the general form as given by T a b a k i n 7)

(klvlk') = -2

~

~ ., ).~UL(I¢).~ML,(k a-~, ~TJ*~ ", ), f;L,(klk

(10)

~t, M, L, L"

where • represents the q u a n t u m numbers J, S and T and we are using the following definitions of Tabakin: f,~. r,(kl k') = iL- r'(_ g~L(k)g=L,(k') + h=L(k)h=r,(k')),

W~(Y¢) = E

(ML MslCLsIJM) YLML(li)XJu. PT.

(11)

(12)

MS'ML.

Substitution of eqs. (10) and (11) into eq. (9) yields 2

~

'

-~

0Z *~

(klGIko) = n E GLL,(klko).WLM(Ii)21r,u(~o).

(13)

o:.M LL"

where we made use of the following five definitions

G~t,(kl ko) = [D'(',,)]-' [ - g~t(k)g=L,(ko){1 + H=(y)} + h,L(k)h,r.(ko){1 - G'(?)} + {g~L(k)h=L.(ko)+ h,L(k)g,L.(ko)}M'(7)], (14) (with O = 1)

~(y) = 2

k~ dk(k~ + ~ ) - , I-g~,.(k)y,

(15)

kZdk(k2+72)_,[h~L(k)y '

(16)

k 2 dk(k 2 +7 2) - 1g~L(k)h~L(k),

(17)

°

2 W(7) = 7 °

M~(Y) = ~2

D~(7) = {1 + m(7)}{ l - a~(7)} + {M~(7)}2.

(18)

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BHAKAR AND R.J.

MCCARTHY

F o r a given value of k v we evaluated the integrals for G'(y), H*(7), and M ' ( 7 ) only once using average values for K and 7 2. This averaging process we found to be accurate to within 0.5 MeV in the binding energy per particle. When doing calculations with the Puff potential it is necessary to take the limit of eq. (14) as the strength of the repulsive shell goes to infinity. In these calculations we are particularly interested in the shape and strength of the repulsive parts of the separable potentials. The repulsive terms are given by h ( k ) h ( k ' ) in eq. (11). The Yamaguchi potential sets these terms equal to zero and thus contains no repulsion. A repulsive term defined by h(k) -

ot k 2+ 3 2

(19)

is called a soft core repulsion while a repulsive term defined by h(k) = ~ sin krc

(20)

k is a hard shell repulsion of strength a and hard shell radius r c. A repulsive potential defined by eq. (20) is the Fourier transform of a delta function potential in coordinate space 8). 3. Results and discussion

The saturation curves obtained using five of the potentials tested are shown in fig. 1, where the energy per particle is plotted versus the Fermi momentum. We have not included the results obtained with the Yamaguchi potential. This potential yields a binding energy per particle of 16 MeV at k r = 1.4 f m - 1 but shows no sign of saturation since it contains no repulsive term. We also include for comparison the energy per particle calculated by Tabakin up to second order in v using an effective mass approximation. This curve is labelled TO and should be compared to curve T which includes scattering to all orders in v using the same potential. The Tabakin potential includes S, P and D partial waves, a soft core repulsion, and a form of tensor coupling. The potential used to obtain curve TA also has a soft core repulsion and separate singlet and triplet-S state potentials. This potential yields a saturation curve of the same sample as T but displaced downward in energy. The extra repulsion in T comes mainly from the inclusion of the higher partial waves. Potential TB has an average S-state force designed to be an average of the singlet and triplet terms of potential TA. TB also contains a soft core repulsion. Potential TC is an average S-state potential defined to produce the same phase shifts as potential TB while containing a stronger repulsive term corresponding to a "hard shell" potential. Potentials TA, TB and TC are described in detail by Tabakin but we would like to point out here that the parameters of the hard shell repulsion of TC correspond to a hard shell strength of 44 BeV and a hard shell radius of 0.182 fm.

287

NUCLEAR MATTER CALCULATIONS

The saturation curve obtained with the Puff potential is most realistic. This potential contains separate singlet and triplet S-state potentials both of which have a hard shell repulsion with infinite strength and radius 0.45 fm. This hard shell radius is consistent with the hard core radii used in realistic local potentials such as the HamadaJohnston or Yale. By combining the desirable features of the Puff potential with the Tabakin potential one could undoubtedly find a non-local separable potential which would fit nucleon-nucleon scattering data and yield reasonable values for both binding energy per particle and saturation density in nuclear matter. ,

,

,

,

,

-4 -8

\

\\

TO

-12 -16 (D

2; -20

,.< kJ

-24

-28 TB -52 -56 -4(? ;.0

i

1.z

I

1.4

i

i

1.6

KF

,

I

i

2'.0 2.z z.4 ( f m -1 )

1.8

2.6

Fig. 1. Saturation curves for five potentials.

The curves TO and T are both calculated using the Tabakin potential but curve T takes into account scattering to all orders in v while curve TO is obtained by considering scattering only to second order in v. The difference between these two curves shows that the off-energy shell matrix elements of the Tabakin potential are not small enough to neglect in nuclear Hartree-Fock calculations. Moreover, the soft repulsive core term in the Tabakin potential is not strong enough to yield decent saturation properties in nuclear matter. The introduction of a very strong hard shell repulsion with a reasonable radius would result in even larger off-energy-shell matrix elements and an even slower convergence rate in Hartree-Fock calculations. Thus, from the results obtained here it appears doubtful that a separable potential can be found which will simultaneously fit nucleon-nucleon scattering data, yield the correct bind-

288

B . S . BHAKAR AND R . J . MCCARTHY

ing energy and saturation density in nuclear matter, and still be useful in HartreeFock calculations in nuclei. Hartree-Fock calculations can, of course, be performed with effective potentials or, possibly, with realistic velocity dependent potentials. However, the most realistic approach at the moment is probably the use of effective interactions in shell-model calculations using the method of Kuo and Brown 12). The authors would like to express their gratitude to Dr. H. S. K6hler for many helpful discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

R. Rajaraman, Phys. Rev. 129 (1963) 265 H. A. Bethe, Phys. Rev. 138 (1965) B804 L. D. Faddeev, JETP (Sov. Phys.) 12 (1961) 1014 H. A. Bethe, preprint Ben Day, Phys. Rev. 151 (1966) 826 P. C. Bhargava and D. W. L. Sprung, Ann. of Phys. 42 (1967) 222 Frank Tabakin, Ann. of Phys. 30 (1964) 51 R. D. Puff, Ann. of Phys. 13 (1961) 317 Y. Yamaguchi, Phys. Rev. 95 (1954) 1628, 1635 Frank Tabakin, Phys. Rev. 137 (1965) B75 G. E. Brown, G. T. Shappert and C. W. Wong, Nuclcar Physics 56 (1964) 191 T. T. S. Kuo and G. E. Brown, Nuclear Physics 85 (1966) 40