L. Zamick

Nuclear structure calculations with density-dependent effective interactions

Abstract An investigation of density-dependent interactions, such as those of Vautherin and Brink, and Moszkowski and Ehlers, is carried out. The interactions are constrained to give the correct binding energies and mean square radii of closed-shell nuclei such as 16 O and 40 Ca. Such quantities as the single-particle energies, the effective charges for E0 and E2 transitions, the mean energies of electromagnetic collective states, and the low-lying states consisting of either two particles, two holes, or a particle-hole relative to a given closed shell are then considered. Harmonic oscillator wave functions are used, but radial self-consistency is satisfied. The prescription for handling density-dependent or three-body interactions to obtain bulk properties or to do shell model calculations is given. The rearrangement prescription for the interaction of two particles beyond a closed shell is the same as it is for the interaction of a particle and hole of the same closed shell. It is found that the J = 0 breathing mode state and the E0 effective charge are most affected by the density dependence. But this can be simulated by merely adding a monopole-monopole term to a density-independent interaction. The spin dependence of the interactions is also considered. In this respect the Vautherin-Brink interaction (II) has some desirable consequences for particle-hole calculations, i.e. the J = 0, T = 1 “Coulomb mixing state” is very high in energy. However, two identical particles in a J = 0 state have a repulsive interaction energy. This is precisely the opposite of the wellknown pairing property of nuclei. Unfortunately, if one attempts to modify the spin dependence to improve the particle-particle situation one makes the particle-hole situation worse. Evidently, additional components are required if we are to have a universal interaction.

Magnetic moments of nuclei with closed j-j shells plus or minus one nucleon

Abstract The correction to the Schmidt moment of 209 Bi calculated with the Hamada-Johnston potential is shown to be 0.8 n.m. to first order in perturbation theory. This improvement on the zero-order Schmidt moment is compared with previous, smaller first-order corrections calculated with other potentials. A discrepancy with experiment of 0.66 n.m. still remains for this nucleus if one stops at first-order perturbation theory. We also perform complete second-order calculations for 29 Si and 209 Bi which worsen agreement between theory and experiment. The second-order correction for 209 Bi is −0.21 n.m. and for 29 Si −0.06 n.m. (compared with a +0.73 n.m. first-order correction for the latter nucleus). Finally certain Tamm-Dancoff and RPA type diagrams which turn out to be important in the second-order calculations are summed to all orders. Our second-order calculations are performed with the Kallio-Kollveit potential.

IMPORTANCE OF HIGHER ORDER CORRECTIONS TO THE EFFECTIVE CHARGE.

Abstract Rayleigh-Schrodinger perturbation theory through second order is applied to the problem of electromagnetic transitions in closed-shell-plus-one-nucleon nuclei. The concept of effective charge is introduced and its usefulness as a physical parameter is discussed. Effective charges relevant to quadrupole moments and E2 transitions in 17 O and 17 F are calculated in first and second order. The semi-realistic Kallio-Kolltveit interaction is used to evaluate two-body matrix elements. The Tamm-Dancoff and random phase approximations are also discussed, and results of effective charge calculations made in these approximations are presented. In a set of appendices, explicit formulas for the second-order corrections to the reduced matrix element which describes electromagnetic transitions between single-particle states are given (appendices 1 and 2). Also included in these appendices is a discussion of state dependent separation distances in the Kallio-Kolltveit interaction (appendix 3). In appendix 4, results of effective charge calculations carried out with Kuo-Brown two-body matrix elements are presented. Finally, in appendix 5, E2 effective charges relevant to the closed-shell-minus-one-nucleon nuclei 15 O, 15 N, 39 Ca and 39 K are given.

Comparison of various approximations for the ground state of 16O

Abstract The amount and nature of 2 particle-2 hole correlations in the ground state of 16 O was investigated using various approximations. First 2ħω excitations were considered with the particles in the 2s-1d shell and the holes in the 1 p shell. Matrix diagonalization, with spurious states removed, resulted in 22% 2p-2h ground state correlations and a depression of the ground state energy of 7.9 MeV; if spurious states were incorrectly included, then both numbers were nearly doubled. The corresponding numbers for the RPA were 43% and 8.2 MeV using the quasi-boson approximation. This approximation is suggested to be inaccurate. Ordinary perturbation theory gave results remarkably close to matrix diagonalization. The corresponding wave functions had an overlap of .993. The correlations were spread over very many j-j coupling states but could be localized to relatively few states in the L-S representation. Perturbation theory was used in a larger space where the 2 particle-2 hole energy could be greater than 2ħω. This led to 32% 2p-2h correlations and the ground state depression was 24 MeV. The correlations with 2ħω and 4ħω energy were comparable. A dispersion method gave less correlation but qualitatively similar results. In the smaller configuration space the first excited 0 + state was at 24 MeV excitation energy relative to the ground state, but a more realistic estimate is probably about 14 MeV. This state looked slightly simpler in j-j rather than L-S coupling.

E0 properties in the lead region

Abstract The theory of finite Fermi systems is applied to monopole properties in the lead region-isotope shifts, isomer shifts, and E0 transition rates. A density-dependent delta interaction is used. It is noted that the L = 0 properties are much more sensitive to the density dependence than are the other properties such as M1 and E2 transitions.

Effective charges for electric multipole transitions: I. First order perturbation theory

Abstract The effective charge for electromagnetic transitions in nuclei is shown to be dependent on the multipolarity of the transitions, being smaller for E3 transitions than for E2 transitions. The effective charge is also dependent; it usually increases as the orbital angular momentum of the polarizing nucleon increases.

Collective models of giant states with density-dependent interactions

Abstract Using a simple density-dependent interaction −αδ(r 1 −r 2 )+γρ σ ( 1 2 (r 1 +r 2 ))δ(r 1 −r 2 ) , with the parameters α and γ adjusted to give the correct binding energy and the correct radius, we show that the energy of the giant quadrupole state is √2. This is the same result obtained by Mottelson, Hamamoto and Suzuki, using, however, different methods. We then consider finite-range terms in the interaction, such as those considered by Vautherin and Brink, and obtain a fairly simple modification of the above result. In contrast to the above quadrupole result the energy of the giant monopole state is not unique but depends linearly on σ, the power of the density. The Inglis cranking model is used to get the relevant mass parameters. This model appears to work very well for high frequency vibrations.

Theory of core excitation components in the ground states of the calcium isotopes

Wave functions of the ground states of the calcium isotopes are calculated. The configurations are restricted to f72(n)JT and [d32−2 IH THf72(n+2) IP TP] JT where IH, TH, IP, TP are the angular momenta and isobaric spins of the two holes and (n + 2) particles respectively. Using the abreviated notation [THTP] T for the core excitation components we show that the matrix elements between these components and the shell model states f72(n) assume a very simple form for all possible TH, TP and n. A relationship between the neutron and proton strengths in d, p reactions to opposite parity states and the average number of neutron holes and proton holes in the ground states of the calcium isotopes is worked out and is discussed in simple physical terms. It is found that the [TH = 1 TP = T + 1] T states are mainly responsible for the neutron strengths in (d, p) reactions, whereas the [TH = 1 TP = T − 1]T states contribute most to the proton strengths in (3He, d) reactions to opposite parity states. In order to reproduce the experimental trend in which the neutron strength decreases as one goes through the even calcium isotopes one must choose the energies of the [TH = 1 TP = T + 1]T states to be an increasing function of n. This is contrary to a previous prediction by this author. The work of Gerace and Green on the calcium isotopes is discussed and compared with the present work. We obtain much smaller neutron strengths for the (d, p) reactions in 40Ca and 42Ca to J = 32+ states than are obtained by some of the experimentalists. For example, the reaction 40Ca (d, p) 41Ca J = 32+ state has 0.78 as the value of the neutron strength according to Belote et al1. In our model this would imply that the percentage of core-excitation component in the ground state of 40Ca is at least 78%.

PARTICLE-HOLE INTERACTION AND CORE POLARIZATION.

Abstract The discrepancy between particle-hole matrix elements of the nuclear interaction as obtained using a realistic two nucleon potential and as derived from experiment is partly but not completely removed by considering the effects of core polarization. Simple formulae for the corrections to the monopole interaction between a particle and a hole are given.

Microscopic and macroscopic calculations of the breathing mode state

Abstract The energy of the breathing mode in nuclei can be calculated both from a macroscopic point of view via the nuclear compressibility and from a microscopic point of view in the TDA and RPA formalisms. We compare results of both types of calculation for 16 O using several self-consistent effective interactions and a couple of more traditional interactions motivated by nuclear structure. We comment on the degree of consistency between the results of both types of calculation, some of the technical considerations in carrying out the calculations, and some of the extremes to which the effective interaction can be pushed without changing the breathing mode energy.