Canadian Journal Journal canadien of Physics de physique Published by Publii par THE NATIONAL RESEARCH COUNCIL OF CANADA LE CONSEIL NATIONAL DE RECHERCHES DU CANADA

Volume 51 Number 2 January 15, 1973 Volume 51 numCro 2 15 janvier 1973

Brueckner Self-consistent Study of 1 6 0 and 40Ca Using Phase-Shift Determined Potentials1

R. J. W. HODGSON AND TRAN DUC HOANG Department of Physics, Ut~iversity of Ortawa, Otta,va, Carrada

Received October 10, 1972

A self-consistent Brueckner calculation of the binding energy and single-particle energies of 160 and 40Ca is carried out employing an effective interaction which is determined directly from the two-body scattering data. The interaction is described by its harmonic-oscillator matrix elements. It is found that the results are quite sensitive to the form of the phase shift at high energies.

On effectue un calcul self-consistent de Brueckner pour l'energie de liaison et les energies d'une seule particule pour 1 6 0 et 40Ca en employant une interaction effective qui est determinee directement a partir des resultats de la diffusion a deux particules. On decrit I'interaction en donnant ses elements de matrice du type oscillateur harmonique. On trouve que les rtsultats sont trks sensibles B la forme des dephasages aux hautes energies. [Traduit par le journal]

Can. J . Phys.. 51. 115 (1973)

1. Introduction A considerable amount of work has been con-

ducted during recent years examining techniques for computing total binding energies and single- particle parameters of spherical, doubly magic nuclei using realistic nucleon-nucleon potential elements. The results have been somewhat dis- couraging in that they predict underbinding in the nuclei studied. On the other hand, calculations which have employed effective interactions have yiel-ded'good results: ' -

The purpose of this paper is to present results of a self-consistent binding energy calculation on 1 6 0

and 40Ca which employs an effective potential computed from two-body scattering data. The potential is coinpletely described by its matrix elements between harmonic-oscillator radial wave functions.

A number of techniques have been used to obtain these latter harmonic-oscillator elements from the scattering data. Elliott et al. (1968) em- ployed the Born approximation and an auxiliary potential. Their results, the so-called Sussex matrix elements, have been employed in a number of calculations (Elliott and Jackson 1968; Dey et al. 1969; Seaborn and Cooper 1971 ; Kakkar and

FIG. 1. Bethe-Goldstone diagrams included in pres- 'Work supported in part by the National Research ent calculation. These are first terms of infinite series of

Council of Canada. diagrams.

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116 CAN. J . PHYS. VOL. 51, 1973

Waghmare 1971) yielding reasonably good results. oscillator matrix elements are similar for I > 1 Other techniques have been developed along states, but there are noticeable differences in the S different lines (Koltun 1967; Galonska and and P states. Faessler 1970; Ripa and Maqueda 1971 ; Srivas- In the next section we outline the theory behind tava et al. 1969; Razavy and Hodgson 1970; Ley the present investigation with the results being Koo et al. 1969). The resulting harmonic- presented in Sect. 3.

2. Theory

To calculate the binding energy, we follow the approach of Davies and McCarthy (1971). The total energy of the nucleus is given by

where

12.21

The sum over states A and B includes all occupied single-particle orbitals. Tis the harmonic-oscillator kinetic energy, EA is the single-particle self-consistent energy, and PA is the occupation probability of the state A computed at the self-consistent starting energy o = E, + E,.

The above approach incorporates diagrams as shown in Figs. l a and lb with any number of bubble and single-particle potential insertions in the hole lines. In addition, by incorporating the occupation probabilities (Brandow 1970) an infinite set of diagrams, of which the one in Fig. lc is the first, is included. Each C-matrix element is antisymmetrized and computed on the energy shell with self- consistent starting energy.

The reaction matrix elements have been computed from the harmonic-oscillator matrix elements (Hodgson 1972) which in turn have been obtained from the two-body experimental scattering data. The reaction matrix, defined by

[2.51 Cpa(o> = (ap\ VIy,BG(4)

is computed using the relation

Here bi, is an overlap function

P.81 bia = ( y i I @ u >

between the eigenfunctions of the two-body harmonic-oscillator Hamiltonian H,,

[2.91 @ I j j ; J HoQa = &,@,

and the eigenfunction of the complete two-body Hamiltonian

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HODGSON AND HOANG: BRUECKNER SELF-CONSISTENT STUDY O F 160 AND 40Ca 117

TABLE 1. Parameter associated with high-energy phase shifts*

Model a m n Ez (MeV) E3 (MeV) a' m' n '

#I 0.01 0.5 4 600 1000 0.01 4.0 6 #2 0.01 0.1 4 800 1000 0.01 1 .O 4

*The primed parameters are associated with the ' S , phase shift only.

I I 0.5 1 .O 1.5 0 5 1.0 1.5

E n e r g y ( G e V l E n e r g y ( G e V l

FIG. 2. High-energy phase shifts for (a) 'So state and (b) 3S1 state.

Q, is the eigenvalue of the Pauli projection operator, which in this representation can be treated exactly. The values of bia and Ei are computed by diagonalizing the matrix

12.1 11 (aaIH0 + Via,,) = .5,6,,, + C Ca[nlSyNL; JT]Cat [iz1l'SyNL; JT] (nlI Vliz'l')

where the Ca are recoupling coefficients defined in Hodgson (1972). The space of available two- particle states used in the determination of G is truncated by placing an upper bound p,:,, on the value of p = 212, + I, + 2n2 + I,. For the present investigation we have used p,,, = 6.

Values of the G-matrix elements were computed for five different values of the starting energy a . Lagrangian interpolation was then used to express each element in terms of a finite polynominal in o. This could then be used to evaluate the matrix elements of aG/ao required in the determination of the occupation probabilities.

The potential itself is defined by the harmonic-oscillator matrix elements (nll Vln'l') which have been computed from the two-body experimental phase-shift data. Two different approaches are examined here. The earliest calculation of these elements was by Elliott and Jackson (1968) and Elliott et al. (1968) who employed an auxiliary potential together with the Born approximation. A second approach, described by Razavy and Hodgson (1970) computed the value of the Born term b,(E) from a knowledge of the phase shifts. The Born term in turn was employed to compute the harmonic-oscillator elements of the two-body potential (Razavy and Hodgson 1970; Hodgson 1971).

One of the variables in this latter procedure is the form which the phase shift takes above the - - exp3rimental limit df ' ~ 5 0 0 MeV. For all but the ,S, states we have assumed that (Chong et al. 1972)

expressed as a function of the momentum. The boundary between the experimentally determined phase shifts and this assumed high energy form occurs at k = kc. The only variable parameters are a, m, and i z .

For the ,s, state we have used the form (2.12) for k > k,, whereas for kc < k < k,

with kc < k2 < k,. Two parameter sets were employed in the present study and are presented in Table 1. The high-energy shape of the S-state phase shifts is shown in Fig. 2.

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CAN. I. PHYS. VOL. 51, 1973

TABLE 2. First-order binding energies for fio = 14.4 MeV

BEJA (MeV)

Model 160 40Ca

'Elliott er al. 1968. ZKakkar and Waghmare 1971. 3Computed for iiw = 10.4 MeV.

TABLE 3. Results for 1 6 0 . Single-particle potential energies (V), single-particle occu- pation probabilities (P), and binding energy per particle. All energies are in MeV

'Elliott et al. 1968. =Barreti, Hewitt, and McCarthy 1972.

3. Results

In Table 2 we present the results of binding energy calculations carried out in first order, with the reaction matrix approximated by the potential matrix. The total potential energy then reduces to the

-- -simple form - - -

PE = (ABI V J A B ) A,B

(2h + ( n l , NL , h ln l l l , n , l , , n1ST.J

The squared quantity is the Moshinsky transfor- consistent equations [2.1] to [2.4], calculations mation bracket. All results were computed with a were performed for 1 6 0 and 40Ca. These results harmonic-oscillator energy ho = 14.4 MeV. are presented in Tables 3 and 4 respectively. The

Employing the reaction matrix and the self- different values of the parameter C correspond to

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HODGSON AND HOANG: BRUECKNER SELF-CONSISTENT STUDY O F 1" AND D°Ca 119

TABLE 4. Results for 40Ca using the The pl12-p312 splitting varies markedly be- Sussex potential. Notation is the same tween the different potential models, but only

a s for Table 3 decreases slowly with increasing C. For the

C = 0.0 C = 20.7 Sussex model this splitting is 4.7 MeV, whereas for parameter set # I it is 13.7 MeV for C = 0.

VO,I,Z -93.5 -93.3 This is to be compared with the result of Barrett V O , ~ I Z -72.9 -73.0 et al. (1972) who find a splitting of 2.9 MeV, and VO,IIZ -66.0 -66.0 the experimentally observed value of 6.2 MeV vo6, 12 -56.0 -56.6 ~ S I I Z -53.3 -54.0 (Cohen et al. 1963).

V O , ~ I Z -47.8 -49.3 Similar results were found for 40Ca, where we PoSl 12 1 . 0 1 . O have only presented values computed using the Popalz 0.99 0.99 Sussex model. The binding energy is high when Pop1,2 0.99 0.99

1 .O 0.97 compared with the expected value of 8.55 MeV

12

Plsllz 0.98 0.97 (Wapstra 1955), yet the splitting of the p and d

P ~ d 3 1 2 1 .0 0.96 single-particle levels is quite good. The p, 12-p312 BEIA 10.3 10.7 difference is now 6.9 MeV and the d312-d512

difference is 8.2 MeV. The latter experimental value is approximately 7 MeV (Cohen et al. 1963).

a change in the energy gap between the occ~~pied Again, the results found using parameter set #1

and unoccupied single-particle orbits. If E , is a are very similar to those of the Sussex model, and the set #2 results give even stronger overbinding. single-particle harmonic-oscillator energy, then

the unoccupied state energies are given by One possible source of error which has not been investigated in the present study is the effect of

e, = E, - C truncaGng the space when cdmputing the G

so that positive C values correspond to decreases in the energy gap.

One of the first observations made was that the two-body reaction matrix elements computed in this study were not as sensitive to the starting energy o in the region of calculation as were the elements obtained by Barrett et al. (1971) for the Hamada-Johnston potential. Hence in our case aG/ao - 1.0 and the P values are very close to unity. In addition, our results are not as sensitive to the energy gap between occupied and unoccu- pied orbits.

The 160 BEIA results are very good in com- parison with the observed values of 7.98 MeV (Mattauch et al. 1965). The Sussex matrix ele- ments and those derived from our phase-shift parajneter set #I, lgad to remarkably similar results. However, by modifying the high-energy phase shifts to the form described by parameter set #2, the binding energy is more than doubled, resulting in an overbound nucleus. This is con- sistent with similar results found in nuclear matter (Hoang and Hodgson, unpublished) where the form of the high-energy phase shift in the S states is found to have a considerable influence on the binding energy. This has also been noted in work performed using rank-two separable poten- tials of short range (Cl~ong et al. 1972).

matrix. Earlier work (Hodgson 1972) indicates that the inclusion of more single-particle orbitals would change the above results, but that the change would not be large.

The technique of performing binding energy calculations on finite nuclei from harmonic- oscillator matrix elements, which in turn are deduced by various techniques from the experi- mental phase shifts, seems to offer a viable alternative to the usual approach. The inter- mediate step of constructing the potential is avoided, and all the required information is contained in the oscillator elements. The difficulty lies in deducing these elements from the phase- shift data. The various approximation procedures used appear quite satisfactory for states having I > 0. Further work needs to be done to see if the result for the S state can be improved.

BARRETT, B. R., HEWITT, R. G. L., and MCCARTHY, R. J . 1971. Phys. Rev. C, 3, 1137. - 1972. N u c ~ . Phys. A, 181, 13. BRANDOW, B. H. 1970. Ann. Phys. (N.Y.), 57, 214. CHONG, K-F., SINGH, Y., SPRUNG, D. W. L., and

SRIVASTAVA, M. K. 1972. P h y s Lett. B, 38, 132. COHEN, B. L., FULMER, R. H., MCCARTHY, A. L., and

MUKHERJEE, P. 1963. Rev. Mod. Phys. 35, 332. DAVIES, K . T. R. and MCCARTHY, R. J . 1971. Phys. Rev.

C, 4, 81.

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120 CAN. I. PHYS. VOL. 51, 1973

DEY, J., ELLIOTT, J. P., JACKSON, A. D., MAUROMATIS, KOLTUN, D. J . 1967. P h y s Rev. Lett. 19, 910. H. A , , SANDERSON, E. A , , and SINGH, B. 1969. Nucl. LEY KOO, E., DELLANO, M., GRILLOT, D. V., and Phys. A, 134, 385. MCMANUS, H. 1969. Nucl. Phys. A, 133, 610.

ELLIOTT, J. P. and JACKSON, A. D. 1968. Nucl. Phys. A, MATTAUCH, J. H. E. , THIEK, W., and WAPSTRA, A. H. 121, 279. 1965. Nucl. Phys. 67, 1 .

ELLIOTT, J. P., JACKSON, A. D., MAUROMATIS, H. A., RAZAVY, M. and HODGSON, R. J . W. 1970. Nucl. Phys. A , SANDERSON, E. A , , and SINGH, B. 1968. Nucl. Phys. 149, 65. A, 121, 241. RIPA, P. and MAQUEDA, E. 1971. Nucl. Phys. A, 166,534.

GALONSKA, J. E. and FAESSLER, A. 1970. Nucl. Phys. A, SEABORN, J. B. and COOPER, B. S. 1971. Nucl. Phys. A, 155, 465. 166, 221.

HODGSON, R. J . W. 1971. Can. J . Phys. 49, 1401. SRIVASTAVA, M. K., JOPKO, A. M., and SPRUNG, D. W. L . - 1972. Can. J. Phys. 50, 940. 1969. Can. J. Phys. 47, 2459. KAKKAR, I. and WAGHMARE, Y. R. 1971. Can. J. Phys.49, WAPSTRA, A. H. 1955. Physica, 21, 385.

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