Nuclear shell-model calculations and strong two-body correlations

ANNALS

OF PHYSICS

Nuclear

102,

Shell-Model

129-155 (1976.i

Calculations

and Strong

Two-Body

Correlations

J. M. IRVINE Daresbury Laboratory, Daresbury, Warrington WA4 4AD, England, and Department of Theoretical Physics, University of Manchester, Manchester Ml3 9PL, England AND G. S. MANI,

V. F. E. PUCKNELL,

M. VALLIERES,

AND F. YAZICI

Department of Theoreticat Physics, University of Manchester, Manchester Ml3

9PL, England

Received July 1975

Two-body Hamiltonians, like the Reid interaction, are derived by fitting the twonucleon data. It is an assumption that the many-body eigenstates of this Hamiltonian form a representation of the observed nuclear states. At best this has been demonstrated for the ground states of a few nuclei, e.g., the triton, and there the binding energy is off by 1%20x, this being attributed to uncertainties in the off-shell behavior of the interaction and to many-body forces. Since 15% of the total nuclear binding energy is much greater than the typical energy spacings observed in nuclear spectra, it is not at all clear that the calculable approximations to the many-body eigenstates of the N-N interaction can give useful information for nuclear spectroscopy. Using the method of correlated basis states coupled with an extremely large “no core” shell model basis as a set of trial variational functions, it is demonstrated that almost 100 levels in light nuclei can be identified with eigenstates of the Reid interaction. In so doing, a prescription is presented for defining effective operators in large shell-model calculations and the question of nuclear center of mass motion is reexamined.

1. INTRODUCTION

The advent of giant nuclear shell-model programs [I, 21 requires that we consider more carefully than before the input to such calculations. There are traditionally two approaches to the problem. First, the phenomenological approach, in which a model space is chosen and the input (matrix elements, single particle energies, effective charges, etc.) is treated as a set of fitting parameters. Second, an attempt at a microscopic calculations in which effective interactions, single particle states, etc. are calculated in a perturbation scheme based on the BruecknerGoldstone theory. The former approach suffers from the disadvantage that as the 129

Copyright 0 1976 by Academic Press,Inc. All rights of reproduction in any form reserved.

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size of the model space increases the number of fitting parameters also increases, and while this hopefully allows a more accurate representation of the data, it does not give much “insight” into the physics of the problem. The latter approach is suitable for describing the bulk properties of nondegenerate configurations, e.g., nuclear matter or double closed shell nuclei, however, it is extremely complicated to carry out spectra calculations in a fully self-consistent manner and the calculations have to be repeated anew for every different configuration space tested. For these reasons both approaches have notoriously failed to make any sensible statements about the convergence of shell-model calculations as the size of the configuration space is increased. One can view a shell-model calculation as a variational calculation, and thus expanding the configuration space merely serves to improve the trial wavefunction. However, this view can only be maintained if we deal with the bare Hamiltonian. Ho = 1 - (fi2/2m) Vi2 + 1 Vij , i i>j

where Vij is the best currently available nucleon-nucleon interaction (including the two-body coulomb potential) as deduced from the two-nucleon data. In the present work we shall base our calculations on the Reid [3] soft-core interaction. The traditional shell-model calculation involves trial variational wavefunctions which are linear combinations of Slater determinants. Each Slater determinant corresponds to a configuration of A particles distributed over A single-particle states. If we take any complete set of orthonormal single particle wavefunctions and consider all possible A-particle Slater determinants that can be formed from them, then these determinantal wavefunctions form a complete orthonormal set of wavefunctions spanning the A-particle Hilbert space. Thus an expansion in such Slater determinants is, in principle, capable of giving an exact representation of the eigenfunctions of Ho. The problem is that in practice we must truncate our configuration space in order to make the problem tractable and, within workable model spaces, we cannot produce a reliable representation of the eigenfunctions of Ho. The principal reason for the lack of convergence in the truncated spaces is the strong two-body correlations induced by the bare nucleon-nucleon interaction which cannot be adequately reproduced by the shell-model configuration mixing. Our philosophy will be to supplement the shell-model wavefunction by putting these strong two-body correlations in “by hand” via a Jastrow-type cluster expansion [4, 51. Unlike earlier calculations we do not aim to describe all the two-body correlations by our Jastrow function but only those which cannot be described by the shell-model configuration mixing. The larger the shell-model space, the less the need for Jastrow correlations, and hence in a large shell-model space we may hope that the cluster expansion will converge so rapidly that we may restrict

SHELL

MODEL

AND

TWO-BODY

CORRELATIONS

131

ourselves to two-body clusters. We should stress that this does not mean that we will ignore three- and four-body clustering but simply that they must be represented by the configuration mixing and not put in explicitly via the Jastrow functions. At this stage it is interesting to compare the philosophy of our approach with the very successful calculations of Pandharipande [6]. Pandharipande carries out a constrained variational calculation in an extremely small model space (a single nondegenerate fermi sea). The constraint is that on the average only one particle is within the range of the Jastrow function from an average particle in the system. If this constraint is met, then, clearly, the restriction to two-body clusters is valid. It is not at all clear why this procedure converges as rapidly as it is claimed. Our approach is to expand the shell-model space until the need for higher-order Jastrow clusters is reduced to an acceptable level. We shall be principally interested in finite nuclei and are thus faced with familiar problem of spurious center-of-mass motion associated with the localization of the nucleus [7]. The problem is further aggravated by the fact that in general we shall be considering multishell calculations. We may rewrite the Hamiltonian (1) in the form

where P is the center of mass momentum

and M is the total mass. M=Am.

p = CPi,

(3)

Throughout we shall define the relative and center-of-mass coordinates of a pair of particles by Tjj

= ( l/ d/z)(ri - r,),

Rij = CI/V’TQ(~~-I rj>,

and the corresponding conjugate momenta. eigenstates of H, are of the form 6~

(4)

From Eq. (2) we see that the exact

= e”K.R+,({r,j>),

(5)

with the corresponding eigenvalues E no =

K2/W4)

+

En .

(6)

In nuclear spectroscopy we are principally interested in the intrinsic energies E, and the corresponding functions Cp, . Clearly, with a trial wavefunctions of the

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form given in (5) the intrinsic trial wavefunctions projection procedure

are obtained by the familiar

cDnT({rij}) = j e-iK’R@;, dR.

(7)

However, most shell-model wavefunctions are not of the form (5) and in general are not even factorizable into center-of-mass and intrinsic functions. An exception much used in nuclear shell-model calculations is when the trial functions are constructed in a harmonic oscillator single particle basis. In general, any single such shell-model configuration will not factorize (the exception is a closed-shell configuration). However, provided that the configuration space consists of all configurations of a given intrinsic excitation, a new basis can be found which is a linear combination of the original shell model basis states and which factorizes in the form

If the configuration space is restricted to valence particles distributed over the lowest oscillator major shell then this is particularly simple in that only OScenterof-mass motion is present and hence the trial eigenenergies are &vm,j

= EnT + $fiu,

(9)

and hence the intrinsic excitation energies and the total excitation energies are the same. Clearly, in a multishell calculation the fact that the center-of-mass harmonic oscillator functions are not eigenfunctions of the center-of-mass operator in H, (Eq. (2)) means that, in general, the diagonalization of H,, will result in an optimum wavefunction of the form ‘i

=

C

‘i

n.NL

nNLY?NL

’

(10)

’

and the power of the variational principle will be dissipated in trying to achieve simultaneously a good representation of the center-of-mass function eiK.R

~ c ‘K, NL

NL@LcR)

(11)

and the intrinsic function

Since, as we have already said, we are principally interested in the intrinsic function, it is usual to project out of the factorizable basis those functions with a OScenter-

SHELL MODEL AND TWO-BODY

CORRELATIONS

133

of-mass motion and use the resulting subset of trial functions in the Hamiltonian diagonalization, i.e., +’ = / &+(R)

(13)

Y$,,,dR.

This projection procedure is rather cumbersome and an alternative procedure is simply to add to the Hamiltonian H,, an auxiliary oscillator Hamiltonian acting on the center-of-mass of the nucleus + (l/2) Mm2R2).

(14)

ez,.wL = EnT + X(2N + L + $) T&J,

(15)

H’ = H,, + h((P/2M)

The corresponding eigenenergies are then

and by increasing h (an arbitrary constant) we can effectively remove all states from the low energy spectrum which do not correspond to a zero point oscillatory center-of-mass motion. We find that this procedure for handling the center-of-mass motion is deficient in two respects: First, it is only valid for an oscillator basis, and second, even in an oscillator basis it is only valid in a very restricted class of configuration spaces, the so-called nfiw spaces. For these reasons we shall reexamine the question of center-of-mass motion in Section 3.

2. TWO-BODY

CORRELATION

FUNCTIONS

Setting aside the problem of center-of-mass motion for the present, our trial wavefunctions are of the form ‘P=F@,

where @ is a multiconfiguration two-body correlation functions

shell-model wavefunction F=

H&j.

i>i

(16)

and F is a product of (17)

Early calculations were restricted to spherically symmetric spatial functions A, = the boundary conditions being that f must suppress the two-body wavefunction wherever the internucleon potential is strongly repulsive (i.e., at short range) and tend to unity at large distances corresponding to the “healing” of the nuclear wavefunction. Calculations with such correlation functions and using “realistic” nucleon-nucleon interactions consistently underbind nuclear matter

f(rii),

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and finite nuclei by -50 %, even with interactions which, when used in BruecknerGoldstone calculations, underbind by only 10-20 %. The principal source of this discrepancy are the noncentral correlations induced by the tensor force. This is well known in Brueckner-Golsdtone calculations where the very important ?S,-%S, interaction matrix element receives a large boost from the tensor force acting in second order; schematically [8], (3% I ve,e I “&> = C3S, I Vcentral I 3S,> - Weti

I<“& I it,,,,,

I 3&)l2.

(18)

Such correlations will then show up in first-order modifications of the wavefunctions, e.g., the D-state admixture in the deuteron. We shall, until it proves necessary to do otherwise, restrict our noncentral correlations to the 3Sl-3D1 channels and we seek a correlation operator linear in the tensor operator (19) i.e., where II,, is a projection operator onto the space of all two-body functions with the exception of the “S, and 3D1 states, f0 and f3 both must tend to unity at large distances, A, projects onto 3S, and 3D, states, and (1 = tp + c&j)

(21)

(13

is a projection operator, i.e., A2 = (1, or (/3” + (201/3- 2a?2)$j + W) A, = (/3 + “$j) A3 . This has the solutions 01 = 0, /3 = 1, i.e., no tensor correlations, or CL= or 01 = -9, /3 = 6. Thus the most general correlation function of the prescribed form is fij = Adrid Al + f+kii> A+ + f-h>

l-l- 9

(22) $, p

=

9,

(23)

where f. , f+ , and f- all tend to unity for large rij and (1, = (% + @I,) A3 9 A- = (* - QS,,) A, .

(24)

We can rewrite (23) in the form A5 = Jxrd

4l + hh)(l

+ 4%) &I

A3

9

(25)

SHELL MODEL AND TWO-BODY

wheref,(rij)

= $f+(rij)

+ $f-(rii)

CORRELATIONS

135

has the required asymptotic behavior and

a,r.j) Lf+(riJ- f-(rid1 2 =1 6 f&-id ’ which tends to zero at large distances. We observe that if the spatial correlations induced in the two orthogonal states #+ defined by A$*

= $L

(27)

are the same then there are no tensor correlations, a fact already noted by Clark [9]. In Sections 5 and 6 we shall present results of calculations in which& ,f+ , andfare parameterized in the form .f;r(r) = 0,

r < rci,

= 1 - exp[-&(r

i=O,+,-

- rJ2]

r 3 rci

(28)

for various rei , fli , and also for calculations in which f0 andf, are assumed to have the form (28) and in which a(rij) is replaced by a constant G representing some average of 01over the volume of the nucleus without specifically specifying the form off+ and f- separately.

3. CENTER-OF-MASS MOTION Consider a shell-model calculation in a Wood-Saxon single particle base. The resulting eigenfunctions will not factorize cleanly into a center-of-mass function and an intrinsic function. Nor can we expect the expectation value of P2/2M to be exactly the same for all the eigenfunctions or even for any subset of the eigenfunctions. However, this does not mean that we must discard such calculations completely, but simply that we must interpret the resulting spectra with care. As far as energy levels are concerned, what we wish to know are the intrinsic excitations relative to the ground state. Hence the ground state expectation of (P2/2kQ, defines the basis for the interpretation of the spectra and all states for which (P2/2M) differs from the ground state expectation value by an energy which is much smaller than the intrinsic excitation energy can still give us useful information about the energy spectrum. Consider a shell-model calculation for 160 in a harmonic oscillator single particle basis and in a complete 2& configuration space, i.e., the positive parity trial functions consist of the closed core configuration plus all (Os-l(lsOd)l), (Op-l(lpOf)l), and (Op-2(lsOd)2) configurations. Such a calculation will fail badly to reproduce certain low lying states in the 160 spectrum, in particular, the 6.04 MeV 0+ first

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excited state. However, if the configuration space is increased to include the (OP-~, (1~04~) configuration then we do obtain a 0+ level at approximately 6 MeV [lo]. Unfortunately, since we are not using a complete 4&.0 space, i.e., we have not included configurations like (OS-~,(l.~Od)~),(OP-~,(l~Of)~), etc., the four particle-four hole states do not factorize exactly. However, the 6 MeV 0+ state differs in its center-of-mass kinetic energy compared with the ground state by a fraction of 1 MeV. Indeed, a recent calculation [I l] reports that, using the hamiltonian (14), a value of h = lo5 is required to raise the energy of this configuration by 20 MeV. This suggests that while the center-of-mass motion is not compZeteZy degenerate with the ground state, the 6 MeV Of configuration provides a reasonable representation of the excited state and should not be discarded completely just because it is not entirely pure. Indeed, there is an additional advantage in keeping these states, for we know that when the model space is large enough the center-of-mass functions will converge to the same function for all states in the spectrum (there, of course, will be a number of degenerate spectra differing in their center-of-mass motion). If we assume that the ground state converges most rapidly then the difference in the center-of-mass motion of the excited states is an indication of the relative rates of convergence in different model spaces. In a traditional nfiw shell model calculation all the states in the spectrum are presented as being sharp, i.e., presumably equally well converge, which is obviously nonsense. We intend to present a prescription which will allow us to live with small discrepancies in the center-of-mass motion. Since we are particularly interested in the intrinsic states, we shall begin by projecting out from our trial functions (16) the translationally invariant component YYT= UF@.

(29)

There have been a number of prescriptions for the operator U. Perhaps the best known is that due to Gartenhaus and Schwartz [12] U,, = =.5$,, exp{-(ih/2)(P

* R + R . P)}.

This particular form of the operator has been criticized in the literature [13] recently. However, as we shall see, we do not require the specific form, only the assumption that such an operator exists. Note also that U and F commute since F is a function only of relative coordinates. Our eigenenergies for comparison with experimental spectra are now obtained by making stationary the functional (F 1H,, 1p)/(‘r/’ 1YIT). We shall discard as not capable of providing useful information all eigenfunctions for which the difference in the expectation value of the center-of-mass kinetic energy compared with that of the ground state is greater than or comparable to the intrinsic excitation energy. We shall retain all other states, but assign a theoretical width to excited

SHELL MODEL AND TWO-BODY

137

CORRELATIONS

states representing the difference in center-of-mass motion of these states compared with the ground state, and compare the resulting spectrum with experiment. While this makes sense when discussing energy levels, we must clearly treat calculated transition rates, etc. involving states with substantial theoretical widths with caution. However, it is perhaps time that attention was pair to theoretical “error bars” in large calculations.

4. THE CLUSTER EXPANSION

AND EFFECTIVE SHELL-MODEL

Our first task is to optimize the functional (YT 1I&, 1YT)/(YT we do in two stages, writing E = (‘YT j Ho / YT)/(YT

/ YT), and this

1!f’“) = (@ 1F+CJ+HJJF 1@‘)I(@ / F+U+iJF I @) (31)

= (@ / F+i?iF I @)/(@ j F+F I@).

The transformed

OPERATORS

Hamiltonian B = iJ+HJJ

(32)

is obtained by rij +

rij

,

ri --f ri - R, ri - R -+ ri - R,

For the Hamiltonian

Pij + Pii 7 Pi + Pi - h/WP,

(33)

P + 0.

(2), then we have simply

B = Ho - (P2/2M) = C ((pfj/M) i>j

+ Vii).

(34)

We may now make a cluster expansion of (31) E = E'2' + E(3) . . . .

(35)

where the two-body energy is simply (36)

Note that there is no normalization correction in Ef2), since this corresponds to the unlinked clusters, i.e., in the Goldstone expansion the uncorrelated wavefunctions are normalized to unity (@ / @) = 1 and the removal of the unlinked clusters is achieved with the normalization (@ 1Y) = 1, where Y is the correlated function [14]. Thus, if the higher-order clusters are negligible, we have reduced

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our problem to a standard shell-model calculation with an effective Hamiltonian (37) Because we have truncated expansion (35), we cannot subject (36) to an unconstrained variation, since (36) is only valid if the cluster expansion converges, i.e., if (( 1 - f;J”)

rfj drij <

1,

where r. is the mean nearest neighbor distance, defined from the nuclear radius R by R = roA1J3,

(39)

and ( ) implies channel average, including the tensor operator. Our procedure will be to choose some LX?< 1 and carry out a constrained variational calculation of Ef2) subject to fixed X. We then search for the minimum S, So , for which, in a given model space ,??A’)agrees with the experimental ground state binding energy of the nucleus under consideration. Our theoretical prediction is then that E(2) Theory

E

(2) Gxpt

*

%E&,t

.

(40)

If we can achieve this with small X0 then we can claim that to this accuracy the eigenstates of Ho are a good representation of the energy levels of the observed nuclear states. It is perhaps interesting to review what we already know about the eigen energies of Ho (using the Reid soft-core interaction). We know that the calculated two-body ground state agrees with experiment to five significant figures [3] (it is fitted to do so). We also know that the calculated binding energy for the three-body state [15-171 and nuclear matter [18] is ~15 o/o less than the experimental value. It is of interest to see if this discrepancy of -15 % is independent of nuclear mass number and whether 15 % uncertainty in the total binding energy is sufficient to destroy any worthwhile comparison between calculated and experimental spectra where the relevant energy spacings are more frequently of order 1 ‘A of the ground state binding energy. The numerical results presented in this paper will almost entirely be restricted to the calculation of energy levels. However, we note that we have in fact presented prescription for defining a consistent set of effective operators for use in large shell-model programs. By analogy with (37), we define the effective operator Oerr associated with a bare operator 0 by Oefl .= F+ U+O UF.

(41)

SHELL

MODEL

AND

TWO-BODY

139

CORRELATIONS

The effect of the unitary transformation U+OU is contained in (33), while for consistency we should again expand F and retain only the two-body cluster terms. By analogy with (32) and (37) we have for a bare two-body operator

Similarly for one-body we may write

operators,

e.g. electromagnetic

transition

operators,

(43) rf [2(A -

I)]-’

c .&[u+(oy’ i+j

+ OI”) U]fij

.

In general the transformed operator U+Ofl)U will not be a one-body operator. In fact, and Eh or M(h - 1) transition operator will transform into a sum of 1,2,..., X body operators. However, successive multiparticle operators appear with additional coefficients in l/A. As specific examples we note the results of El and Ml operators. (i)

The electric dipole operator transforms as 7 eiri -

T ei(r, - R) = 7 (ei - Ze/A) ri .

(4.4

We note that it remains a one-body operator but that it acquires the familiar effective charge. (ii) The magnetic dipole operator transforms as T &

(rj * pi) =

7 $ c

(ej

I

(ri - R) * (pi - (l/A) -

(244

+

2n1,c

CWA2))

r.

PI

z

h p,

I

+ 1 ((pei/A)(-,(e;lA))+ (ZelA2)) (r. * p. - r. h p.) (45) t z. 3 3 4 m3 c id

The electric quadrupole operator transforms as the magnetic dipole operator with ri A pj replaced by the quadrupole tensor rirj - $ri . rj . We note that the Ml operator transforms into a one-body plus two-body operator and that the two-body operator has as a coefficient an extra factor l/A. We note in passing that it would be extremely fortuitous if the two-body component of the electric

140

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quadrupole transition operator could be successfully represented by an effective charge supplement of the one-body component as is usually assumed. Higher multipole operators are usually only measurable in heavier nuclei; if one assumes that the charge density fluctuations corresponding to the transition involve a limited number of “valence” particles A, (an assumption essential in the shell-model) then the importance of the multiparticle operator would be expected to fall off as (&/A) and hence, hopefully, we would restrict our interests to twobody operators. Finally, we note that the presence of tensor correlations can dramatically effect the spin dependence of the effective operators and, in particular, can lead to a substantial enhancement of certain electric quadrupole transitions. Thus if our calculations have converged, then we have eliminated the need for effective charges and other such fudge factors and we have replaced them with a specific prescription for calculating effective operators.

5. GROUND

STATE BINDING

ENERGIES AND CORRELATION

PARAMETERS

The form of correlation function under consideration is presented in (25) and (28). The choice of rCi is dictated by the bare potential. For a hard core interaction rC. would be equal to the*hard core radius. For the Reid soft-core interaction r,: Gill be chosen between 0.2 and 0.5 F. ,!I;l” F is a measure of the healing distance and, typically, we shall consider pi 2 100 F-2. A greater rate of healing introduces high momentum components into the wavefunction and drives up the kinetic energy. The cluster convergence parameter X is of the form /lrn (1 -f,(r))”

r2 dr + 8 jom a2(r)(l -f,(r))”

r2 dri.

(46)

The first integral is the familiar excluded volume contribution to the cluster convergence parameter. Forf,(r) of the form (28) we have explicitly Xl = -$

jom (1 - f3(r))2 r2 dr (47)

czzp,”

+ -$

I&

($’

The second integral represents the contribution is approximately a2(r)(l -f3(r))2

+ $

(+J2

+ +I.

from the tensor correlations and r2 dr N

(

%

3 82, >

(48)

SHELL

MODEL

AND

TWO-BODY

141

CORRELATIONS

where E2 is the mean value of a2(r) over the volume of the wound in the two-body wavefunction. From Eqs. (26) and (28) it is clear that ol < 0.25 and we note that a deuteron wavefunction of the form~tij#ii(3S1) would contain an 800~? % D-state admixture, i.e., suggesting that E N 0.1 might be more appropriate. In this section we shall calculate the ground state binding energy of some light nuclei A < 14 and nuclear matter using the Hamiltonian of Eq. (37). We shall use the Glasgow-Manchester shell-model code [I] with as oscillator basis to investigate the model space convergence properties of the ground states of the nuclei 3 < A < 14. The oscillator parameter hw is treated variationally and the correlation function parameters are used to minimize the binding energies subject to the constraint of constant X. We then search for the smallest X such that we can reproduce the observed ground state binding energy. We shall consider two parameterizations of the correlation function; the first will correspond to Eq. (23) where each of the spatial functions is of the form (28); there are thus 6 variational parameters rCLand /$ (i = 0, +, -). We shall refer to these results as set I. The second parameterization will be in the form (25) where we shall assume f0 = f3 and is of the form (28) and where we replace iy by C% and treat it as a single parameter without inquiring in detail about the form off+ andf_ . There is, from Eq. (26) the constraint that ol < 0.25. In this case there are only three parameters rC , p, and ol, and we shall refer to these results as set II. We shall, in a few cases, demonstrate that both parameterizations are capable of giving essentially similar results for the same value

of X.

5.1. The Deuteron

The deuteron is unique amongst the nuclei that we shall study in that we already know the exact eigenvalue and eigenfunction of Ho [3]. These have been calculated by Reid and reproduce all the known deuteron properties to extremely high accuracy (see Table I). Further, since we now have only two particles, we can treat the correlation parameters as unrestricted variational parameters without concern for the value of X (in set II, & < 0.25). TABLE

I

Properties of the Deuteron as Calculated by Reid [3] Compared with Experiment

Binding energy Magnetic moment Quadrupole moment Deduced D-state admixture (%)

Experiment

Reid

2.22452 zt 0.00010 0.85741 & O.ooool 0.278 f 0.008 4-8

2.22 0.85 0.28 6.47

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We consider the intrinsic wavefunction series of relative oscillator states

AL.

for the deuteron as an expansion in a

In Fig. 1 we plot the calculated binding energy as a function of N for three cases: 51

No correlation

function

Best spatial correlation function E =O. p = IOfm-*.r, = 0.1 fm Best set I and set II rc+ = r,- = 0.1fm. h = 0.25 p+= IOfm+, rc =O.lfm,p=5ff17-~, p= IOfm-z.?w = 50MeV

-3 J

FIG. 1. The calculated binding energy of the deuteron: (1) with no correlation function, (2) with the best spatial correlation function (LX= 0), and (3) with the best correlation function of set I (indistinguishable from best correlation function of set II).

(1) no correlation function, (2) the optimum spatial correlation function calculated for N = 10, and (3) the optimum correlation function of set I calculated for N = 10. First, we note that there is essentially no difference between the results for set I and set II. Second, we note that with no correlation function the binding energy converges for N E 30 while with a correlation function the same level of convergences can be obtained with about half this number of terms. Without a correlation function the binding energy falls quite rapidly for N rising to -20 at which point it is within -0.3 MeV of the exact value. This last piece of binding accumulates very slowly as N increases from 20 to 40. These latter terms contribute to the gradual buildup of the tail of the wavefunction (see Fig. 2), which is important in the deuteron with its relatively low binding energy. In Table II we compare the calculated properties of the deuteron with their experimental values. We see

SHELL

4-

-

MODEL

AND

TWO-BODY

143

CORRELATIONS

Reid

------ 20quanta -10 quanta

3-

fm

FIG. 2. The calculated deuteron wavefunctions corresponding of Fig. 1.

TABLE

to the set II parameterization

II

Calculated Properties of the Deutero@

N

BE(MeV)

D-state (%I

,.2WF

Q(F3

No correlation function

10 20

1.43 1.98

7.07 6.58

3.08 3.54

0.239 0.259

Set I

10 20

1.78 2.14

7.28 6.71

3.05 3.49

0.236 0.257

Set II

10 20

1.86 2.13 2.22

7.14 6.70 6.47

3.15 3.52 3.70

0.252 0.266 0.280

Reid

0 Set I parameterization r,, = r,- = 0.1 F, /3+ = 10 F-2, ,9- = 5 F-2. Set II parameterization = 0.1 F, ,B = 10 F*; in all states G = 0.25. In both cases the optimum oscillator constant is plw = 50 MeV.

r,

595/102/I-10

144

IRVINE ET AL.

that a 10 term expansion with either of the parameterizations of the correlation function can reproduce all the properties of the deuteron to an ~15 % order of accuracy. It is perhaps interesting to note that for N 2 40 we obtain a faster convergence rate by setting the correlation function to unity, i.e., with a large enough model space there is no need for a correlation function to be put in “by hand.” In heavier nuclei it will be impossible to consider model spaces of the large dimensions which we have used in the deuteron, nor should we expect to require such large model spaces in order to obtain the same level of convergence, since as the nucleons become more bound the importance of the tails of the wavefunctions will be reduced and we have just seen that the higher terms in expansion (49) are almost entirely associated with improving the tail of the wavefunction. 5.2. The Triton

Many workers have studied the triton using the Reid interaction [15-171. The most reliable reports suggest that the calculated binding energy should converge to 7 f 0.2 MeV compared with the experimental value of 8.4819 MeV. The discrepancy of ~1.5 MeV can be attributed to uncertainties in the off-energy shell behavior of the N-N interaction and to three-body forces. These uncertainties are present in all calculations involving more than three nucleons and it is perhaps meaningless to try to reproduce the observed ground state binding energies of nuclei with correlation functions corresponding to X 5 0.15. This view is supported by the nuclear matter calculations of Bethe and his collaborators [18]. First, we consider the triton as three particles in the model space of OS, Op, 1sOd oscillator orbitals. In Fig. 3 we plot the calculated binding energy vs the

‘1 r,-

0.3

$ =50 z - 0.15

O% E

r,-0.3 B P) 5-

8-25 a=o.12

8=25 ii=O.lS I 15

20

I 25

I 30

hw (MN FIG. 3. The calculated binding energy of the triton for various set II parameterization correlation function.

of the

SHELL

MODEL

AND

TWO-BODY

145

CORRELATIONS

oscillator constant for various correlation functions. We see that there are a range of correlation functions corresponding to X ,< 0.2 which will reproduce the experimental binding energy. However, if A? is restricted to be < 0.1, there is no correlation function which will lead to a calculated binding energy within 10 % of the experimental value. Indeed, in set II the minimum value of ol which will yield the correct binding is 0.1. The correlation function corresponding to the smallest value of X which will yield the experimental binding energy has Or= 0.12 and re = 0.25 and ,B = 25F-2 in all states. This corresponds to X0 = 0.17, i.e., our calculation can yield the triton ground state binding energy with an uncertainty of 17%. 5.3. The Alpha Particle For a model of 4He consisting of the oscillator configurations (OS, Op, Is, Odj4 with fro = 20 MeV, the optimum correlation function of set II has ol = 0.1 and rc = 25 F and /3 = 25 F-2 in all states, corresponding to S, = 0.15. In set I the optimum correlation function parameters are re = 0.25 F, rC+ = 0.4 F, rr- = 0.4 F, /3,, = 25 F-2, /3+ = 40 F-2, and p- = 5 F-2. In Fig. 4 we present the IO

-

Set II ~=01,r,=025fm,p=25fm~z

---

Set1 r,=0.25fm.~=25fm~z,r,+=0.4fm rce= 0.4fm.p+ =40fm2, PcO.4

1 o-

fm+

5: Ql -lOI 6

y;y

\-

_

-40 1 I

I

10

20

30

40

hw (MeV)

FIG. 4. The calculated binding energy of the alpha particle. Note that with set 1 there is a definite minimum. With set II there is no clear minimum, however, the uncertainty in the binding energy increases with fiw which increases with X. In both cases the slow variation in the binding energy indicates that we are close to convergence.

calculated binding energy as a function of oscillator constant for both parameterizations of the correlation function. The slow variation of the binding energy with oscillator constant is an indication that we are near convergence.

146

IRVINE

ET

AL.

5.4. Nuclear Matter

We now go from the limit of extremely light nuclei to the limit of extremely heavy nuclei. Our configuration space is now limited to a single configuration, the fermi sea, described by a single parameter, the fermi momentum, i.e., the density. Extrapolating from the semiempirical mass formula, nuclear matter should have a saturation density corresponding to k, = 1.36 F-l with a binding energy of between 15 and 16 MeV. The most detailed Brueckner-Bethe-Goldstone calculations based on the Reid interaction have yielded saturation at kF = 1.4 F-l and a binding energy (in the two-hole line approximation) of 11 to 12 MeV [18]. For the set II parameterization at kF = 1.5 F-l we find the optimum correlation function corresponds to C?= 0.058 with r, = 0.25 F and ,L3= 25 F-a in all states. In Fig. 5 we plot our calculated binding energy as a function of the fermi momentum to obtain a satisfactory saturation curve comparable with the best BruecknerBethe-Goldstone results.

1 PO

I

1.5

I

2.0

,

2.5

kF(Fm-‘)

FIG.

5. The calculated binding energy of nuclear matter with the set II parameterization.

5.5. 5 < A < 14 With the parameterization of set I it is numerically extremely tedious to obtain the optimum correlation function for each nucleus. However, for the set II parameterization we note that, for the triton, the alpha particle, and nuclear matter, the optimum spatial correlation function corresponded to rC = 0.25 F and fi = 25 F-2. This corresponds to an extremely short-range (< 0.5 F) wound in the two-body wavefunction. For a relatively low density system like the nucleus, such short-range correlations are likely to be more a function of the Reid potential than the specific nucleus under consideration; thus we shall freeze the spatial correlation function at rC = 0.25 F and /z?= 25 F-2 in all states. This corresponds to S, N 0.05

SHELL

at normal nuclear shall consider the that for 5 8 10

MODEL

AND

TWO-BODY

CORRELATIONS

147

densities. For the nuclei with mass numbers 5 < A < 14 we configurations (OS,Op, 1~Od)~ with the additional restrictions < A < 8 : at least two particles in the OSshell < A < 10: four particles in the OSshell < A < 14: four particles in the OSshell and at least four particles in the 0~ shell.

Imposing these restrictions alters the ground state wavefunction by less than 1 %. We now carry out a variational calculation in which the variational parameters are the configuration mixing amplitudes and the oscillator parameter fiw and we fit the single remaining parameter ol to reproduce the observed ground .state binding energy. The results of a typical calculation for 140 are illustrated in Fig. 6. SC

85

c s9c 6 5 5 ‘qk? 95 5 Experimental binding energy 98~732bkV) wOJl7303 a.@o74 hw=19.7

100

105

I

14

16

18

20

22

ho&AeW FIG. 6. The calculated binding energy of I40 as a function of fiw. For various values of C in the set II parameterization of the correlation function with r, = 0.25 F and /f = 25 F-2.

We find that using the average value of E for each mass multiplet we can simultaneously fit the masses of all known members of the multiplet to within a fraction of 1 ‘A. In Fig. 7 we present the results of the average calculated value of C as a function of mass number. We note that the smooth variation of ol with mass number is not inconsistent with the asymptotic value of G = 0.058 that we obtained for nuclear matter.

148

IRVINE ET AL.

A particles in os.op,is.od

o-o’

I 4

,

I 6

,

, 8

,

I ‘0

,

I 12

I

I 14

,

I 16

, A

FIG. 7. The option value of Oras a function of massnumber A.

The optimum value of Ordoes depend on the shell-model configuration restrictions yield a smaller value of (IIwhile more restrictions increase (Y:In the calculations reported here the addition or removal of a single orbital to the configuration space led to changes in Z in the third significant digit.

6. THE SPECTRA OF LIGHT

NUCLEI

We have now defined an effective, mass dependent interaction which reproduces the binding energies of the light nuclei. There are no free parameters left in this interaction. We shall now examine how the calculated spectra of the nuclei 5 < A < 12, using this Hamiltonian, compare with experiment. In Figs. 8-15 we present the calculated spectra for the mass multiplets A = 5 to 12, respectively. The model space restrictions imposed alter the ground states by less than 1 % compared with the model spaces of Section 5. The theoretical widths indicated in these figures are discussed in Section 3. A dot against a level indicates that there is an unambiguous identification with an observed level which has a definite spin, isospin, and parity assignment. In almost all cases the calculated energy of an identified level is within 0.5 MeV of the absolute experimental energy. In most cases the relative excitation energies are within a few kiloelectron volts. Our calculations show that there is a correspondence between almost 100 eigenstates of the Reid potential and observed nuclear energy levels and that the correspondence is comparable with the best phenomenological shell-model fitting calculations.

SHELL

10

MODEL

AND

TWO-BODY A =5

1

149

CORRELATIONS los,

a=

op, is,od

P2)

0.0954

hw =

;, f-

.

21 MeV

3 1 h’Z’

.

30

5Li

5He

FIG. 8. The calculated spectra of the mass 5 multiplet in a set II parameterization of the correlation function with r, = 0.25 F, /3 = 25 F-a, and G taken from Fig. 7. The widths of the excited levels are a measure of the discrepancy in the center-of-mass kinetic energy compared with that for the ground state. A dot at the side of a level indicates that there is an unambiguous identification with an experimentally known level. A=6

(os,op,is,od,os,a

2)

a = 0.0892

hw

= 20 MeV

./m /:2,1+ . 2,1+ . 0. 1+

2.1+

i

%e

6Li

FIG. 9.

As Fig. 8, for A = 6.

6Be

150

IRVINE ET AL. A= 7 (OS, op, is, od,

C

OS= 4)

a = 0.0924 hw = 19.5 MeV

-1c

2 5 6 $ 5 -2c m 3

-

j,j-

=

j, j13 T’ I-

-

1*1-

=

s,jfsj-

-

j,j-

2 9

-

j,j-

-30

.l

-

S,f-

. .-

f,J-

.-

j,;-

;, ;-

.-

+ 1 f-

j,f-

-40 -

FIG. 10. As Fig. 8, for measurement.

‘He A

‘Li

‘8

= 7. Isotopes which are underlined

-

‘8

do not have a definitive mass

We note that there is a strong correlation between states to which we have assigned a theoretical width on the grounds of center-of-mass motion and levels which have a large experimental particle decay width. While too strong a significance should not be attached to this result we consider it satisfactory for the following reason. A level which has a large particle decay width is unlikely to be well represented in a “small” shell-model space, and we have argued in Section 3 that out theoretical widths are some measure of the convergence of shell-model states. We do not at this stage wish to undertake a detailed analysis of all the levels in Figs. 8-15 but shall content ourselves with the following comments.

SHELL

MODEL

AND

TWO-BODY

A=B(os,op,is,od.

151

CORRELATIONS

OS-~)

a - 0.0891 hw - 20 MeV

s‘ -2o2 6 G E -3o‘u 3 2 9 -4o. $++$

4,0+

-5o. z’;;;,

/1 2 ,o+

. -

o.o+

-6o‘Li

FIG.

A = 5: A = 6:

A = 7: A = 8:

A = 9:

aBe

11.

As

Fig.

8, for

8B

A =

LL

8.

In the model space we have considered there is no evidence of the 3/2f level identified experimentally in 5He and 5Li at ~17 MeV. We find that it is a very bad approximation to treat 6Li as a p-shell nucleus, i.e., only to consider (p)” configurations. Our calculations do support the negative parity assignments which have tentatively been made to levels at 6-10 MeV in 6Li. We would suggest that the T = l+ level at 5.36 MeV has spin 2 and that it mixes with the 2+T = 0 state at 4.57 MeV. Our calculations would confirm the 2+T = 1 assignment of the first excited state in 6He. We would confirm the 712 spin assignment of the 4.63 MeV level in ‘Li. The isotopic mixing calculated for the two 2+ levels at 16.63 and 16.93 MeV in 8Be agrees with experiment but it is extremely sensitive to the value of 0~.It is predicted that the analog 3+ and l+ states also suffer isotopic spin mixing. It is predicted that the 2.691 MeV in gLi has the assignment (1/2)+T = 3/2. The low lying positive parity states at 1.67, 3.03, and 4.7 MeV in sBe and at 2.83 MeV in sB are not reproduced in our calculations. These are are the first low lying states which have definitely been identified experimentally and which we have failed to reproduce.

A=9

(os,op,is,od:os=4) a = 0.0857 hw=

;,;+

19.5

MeV

(He,hw.20;N,hw=17)

;,;-

1,;: f :

-

-50

-60 i -

9He

9Li

FIG.

9B

?Be

12.

As Fig.

-

PC

-

9N

8, for A = 9.

0

z

-10

z T;,:f

f::: 0,3+ g::’ A=

L3

IO (os,op,

is.od;os

‘4)

-

1.3-

-

0,3+

a = 0.0825 hw = 20 MeV -20. -

(He:16,

03+

Li:18,

E -

N:l8,0:14)

;: 9 - -30 6 G 5

*.z+ 4.21. 2. 1 1.2+ $$ i.z+ 0.2+ 2.2+ ,,2+

2 -40 :: 9

-

-50.

-60,

.-

“Li

‘OB

‘%e

FIG.

13.

As Fig.

8, for

0,1+

‘OC

A =

10.

-

‘ON

‘00

A=ll(os,op,ls,od;os=4.

opy

32)

h

a z 0.0798 -20 !

hw = 20 MeV: Be, 6. C __ G

-30

1.2, 2 2 f,$. f. f*

= IS MeV: Li,N

-

= 1.5 MeV. He,O,F

t,;1 I 2

‘,2 1. 2

-80 i

“Ht? -

” Be

“Li mm

FIG. 14. As Fig. 8, for A = 11.

A = 12 (OS, op, IS. od.

OS = 4, sd S2)

a - 0.0771 hw-

,’

=$

-5o-

16,‘*Li,12F

;I = 18, “Be,

‘%

--1,2+ = 20, “6,

-2.2+

‘k,“N

--1,1+

-2,2+ -0,2+ -

-

2,2+

-

0,2+

2,1+

1.1+

.-2,0+

-9o-

.-O/3,

-xx)~

‘*Li

‘%e

“0

12c

‘ZN

FIG. 15. As Fig. 8, for A = 12.

‘20

‘ZF

154

IRVINE ET AL.

A = 10: Our calculations fail to reproduce the low lying negative parity states in loBe and l”B and also the Of level at 6.2 MeV in loBe and the l+T = 0 level at 5.18 MeV in l”B. A = 11: We fail to reproduce the ground state of llBe. The six low lying negative parity states are well reproduced and we have a definite 312 spin assignment for the 5.019 MeV level. A = 12: In laB and 12N the ground state and first excited state, while lying very close to each other, appear inverted in our calculations. Our calculations fail to reproduce the low lying negative parity states and the first excited Of state in 12C.

7. DISCUSSION

We feel that the failure to reproduce certain low lying levels in the nuclei A 2 9 reflects on our choice of model space even though this is much larger than in earlier shell-model calculations. For 12 < A < 15, we consistently fail to reproduce the nonnormal parity states. There are two ways in which our model space can be expanded: Either we can relax the restrictions set out in Section 5.5 (remember that we only checked the effect of these restrictions on the ground state), or we can try to include some (Of, lp) shell configurations. We can report that calculations allowing for up to four particles to be excited from the Op shell to the (Od, 1s) shell do not improve the situation nor do calculations allowing for up to two particles in the (Od, 1s) shell plus up to two particles in the (Of, lp) shell, suggesting that the missing states are indeed highly distorted, In comparing our results with the calculations of Cohen and Kurath [19] we note that none of the states missing from our calculations were included in their fitting procedure. In comparison with experiment the level of agreement on energy levels in our calculations is comparable with theirs for the light Op-shell nuclei, but is not as satisfactory for A > 10. However, if we examine the wavefunctions in detail we find that for A 5 8 we see considerable Os-shell breaking which is of course not allowed for in their model. Confirmation of such details of the wavefunctions require the examination of form factors, electromagnetic moments and transition rates, and beta decay rates. These will be presented in a separate paper. Clearly correlated basis states do not in general form an orthogonal basis. We have numerically checked the orthogonality of all the states corresponding to the levels in Figs. 8-15 and in no case is the overlap greater than one part in 105. This is because our correlation function corresponds to a very small admixture of many configurations far outside our shell-model space.

SHELL MODEL AND TWO-BODYCORRELATIONS

155

In the present calculations we have used a particular parameterization of the correlation functions. It would be desirable to have a functional variational theory leading to dynamic equations determining the correlation functions. Pandharipande [6] has presented a prescription for calculating the correlation functions in nuclear matter. However, the prescription cannot be justified on strictly variational grounds and requires serious approximations in the treatment of the tensor force. We have developed a constrained variational approach to the calculation of two-body correlations in dense many-body systems [20, 211 and applied it to the study of nuclear matter, treating the tensor force exactly [22]. The application of this method to finite nuclei is in progress.

ACKNOWLEDGMENTS We should like to acknowledge continued discussion with M. Banerjee, R. Bishop, M. Mihailovic, and M. Strayer. One of us (J.M.I.) has benefitted from comments by G. Parish. We are grateful to A. Watt and R. Whitehead for their assistance in establishing the shell-model code at Daresbury.

REFERENCES 1. R. R. WHITEHEAD, 2. E. C. HALBERT,

Nucl. Phys. A 182 (1972), ET AL., “Advances in

290.

Nuclear

Physics,” Vols. 3 and 4, Plenum Press, New York, 1969. 3. R. V. REID, Ann. Physics, 50 (1968), 411. 4. R. JASTROW, Phys. Rev. 98 (1955), 1479. 5. J. W. CLARK AND P. WESTHAUS, Phys. Rev. 141 (1966), 833. 6. V. PANDHARIPANDE, N&cl. Phys. A 166 (1971), 317. 7. J. P. ELLIOTT AND J. H. R. SKYRME, Proc. Roy. Sot. A 232 (19.56), 561. 8. T. T. S. Kuo AND G. E. BROWN, Nucl. Phys. 85 (1966), 40. 9. M. L. RISTIG, W. J. TER Louw, AND J. W. CLARK, Phys. Rev. C 5 (1972), 695. 10. J. M. IRVINE, C. LATORRE, AND V. F. E. PUCKNELL, Adv. Phys. 20 (1971), 661. 11. D. M. GLOECKNER AND R. D. LAWSON, Argonne National Laboratory preprint. 12. S. GARTENHAUS AND C. SCHWARTZ, Phys. Rev. 108 (1957), 482. 13. J. L. FRIAR, Nucf. Phys. A 172 (1971), 257. 14. J. GOLDSTONE, Proc. Roy. Sot. A 239 (1957), 267. 15. M. A. HENNELL AND L. M. DELVES, Phys. Lett. B 40 (1972), 20. 16. A. D. JACKSON, A. LANDE, AND P. U. SAWER, Phys. Left. B 35 (1971), 365. 17. M. R. STRAYER AND P. U. SAUER, Nucl. Phys. A 231 (1974), 1. 18. H. A. BETHE, Ann. Rev. Nucl. Sci. 21 (1971), 93. 19. S. COHEN AND D. KURATH, Nucl. Phys. 73 (1965), 1. 20. J. C. OWEN, R. F. BISHOP, AND J. M. IRVINE, Phys. Left. B 59 (1975), 1. 21. J. C. OWEN, R. F. BISHOP, AND J. M. IRVINE, Daresbury preprint DL/NSF/P17 (Theory) 1975. 22. J. C. OWEN, R. F. BISHOP, AND J. M. IRVINE, Daresbury preprint, DL/NSF/P31 (Theory) 1975.