Effective operators in shell-model calculations

After a brief review of the properties of the nucleon-nucleon interaction and of basic shell-model theory, the microscopic theory for determining effective interactions and operators for use in shell-model calculations is given. Difficulties in applying the usual formalism are discussed. Finally, a new large-basis, no-core shell-model approach is presented, which is able to handle these difficulties. Supported in part by NSF grant No. PHY96-05192. Lectures given at the 10th Indian Summer School of Nuclear Physics: “Theory of Many-Fermion Systems”, Prague, September 8–12, 1997.     

A numerical approach to nuclear shell-model calculations

An application of the Lanczos algorithm for the matrix eigenproblem to nuclear shell-model calculations is described. The method has many advantages over conventional shell-model techniques, one of the most important being a very much slower growth of computation time with increasing dimensionality of the shell-model basis.     

Effective shell-model transition operators for muon-capture calculations

Recently many shell-model calculations have been performed in order to extract the value of the ratio C _P /C _A in light nuclei. Most of these calculations fail to reproduce the value given by the partially conserved axial vector hypothesis, roughly 7. We show that, with the effective transition operators calculated by the perturbative techniques, this discrepancy can be, at least partly, solved. New angular correlation data for ²⁸Si are used for the extraction of C _P /C _A . In the case of ²⁰Ne, the capture rate data are used for the extraction.     

Exact shell-model and projected Hartree-Fock calculations for Mg

The results of a projected Hartree-Fock calculation are compared with those of an exact shell-model calculation for states of J ? 6 in the nucleus ²⁴Mg. The realistic Kuo interaction is used, and gives reasonable agreement with experiment.     

Rearrangement effects in shell-model calculations using density-dependent interactions

A general method is developed to treat density-dependent effective nucleon-nucleon interactions in shell-model calculations. A trial function is introduced, that can generate any excitation mode of the nucleus. The method is based on the evaluation of the expectation value of the full Hamiltonian with respect to this generating Slater determinant wave function. Rearrangement effects, due to the density-dependence of the interaction are systematically included. The proposed formalism is applied in first instance to simplified excitation modes, such as two-quasiparticle and particle-hole configurations. A detailed study on the importance of the various types of rearrngement terms is carried out. A more specific application of the general formalism is presented in carrying out an extended shell-model calculation on ¹¹⁶Sn, including, apart from the neutron 2qp-excitations, proton 1p-1h excitations and coupled excitations of the type (1p-1h) ⊗ (2qp).     

The applicability of nuclear field theory to shell-model calculations

Previous NFT calculations of the four-particle spectrum reached different conclusions concerning the applicability of the lowest-order Rayleigh-Schroedinger perturbation expansion. In the present paper, we show that the inclusion of second-order diagrams and/or diagonalization procedures yields satisfactory results both for the energies and for the transfer matrix elements even for j-shells as small as the $\text{j =}\text{7}\text{2}$ and $\text{j =}\text{11}\text{2}$ shells.     

Event rates for CDM detectors from large-scale shell-model calculations

We investigate the scattering of the CDM candidate LSP (Lightest Supersymmetric Particle) off nuclei. We have computed the associated event rates and annual modulation signals for the ²³Na, ⁷¹Ga, ⁷³Ge and ¹²⁷I CDM detectors by using the nuclear shell model in realistic model spaces and exploiting microscopic effective two-body interactions. Large-scale computations had to be performed in order to achieve convergence of the results. We have tabulated the associated nuclear-structure coefficients for several LSP masses enabling easy interpolation of our results for any other mass. The relevance of the spin-dependent and coherent channels for the event rates is discussed, from both the nuclear-structure and the SUSY-model viewpoints.     

Symplectic shell-model calculations for 20Ne with horizontal configuration mixing

The symplectie shell model, which incorporates vertical $\text{(2n}\text{h}\text{?}\text{ω; n = 1, 2…)}$ major shell configuration mixing as dictated by a quadrupole interaction, is augmented with horizontal (0?ω) mixing induced by realistic single-particle energies and a monopole-pairing interaction. The excitation spectrum and B(E2) rates of the ²⁰Ne ground band are accurately reproduced without the use of an effective charge. The degree of horizontal and vertical mixing is found to be on the order of 20% in the ground state and up to as much as 50% for the 8⁺ level.     

Extended shell-model calculations for the A = 6 nuclei

Extended shell-model calculations for the A = 6 nuclei using the Sussex matrix elements and the Gogny et al. potential are presented. The results are discussed in connection with those obtained in the framework of the lowest order shell-model configurations.     

Supernova inelastic neutrino-nucleus cross sections from high-resolution electron scattering experiments and shell-model calculations

Highly precise data on the magnetic dipole strength distributions from the Darmstadt electron linear accelerator for the nuclei 50Ti, 52Cr, and 54Fe are dominated by isovector Gamow-Teller-like contributions and can therefore be translated into inelastic total and differential neutral-current neutrino-nucleus cross sections at supernova neutrino energies. The results agree well with large-scale shell-model calculations, validating this model.     

Large-basis no-core shell-model calculations and their application to10C →10B fermi beta decay

We use a 4ħΩ shell-model calculation with a two-body effective interaction derived microscopically from the Reid93 potential to calculate the isospin-mixing correction for the¹⁰C→¹⁰B superallowed Fermi transition. The effective interaction takes into account the Coulomb potential as well as the charge dependence ofT=1 partial waves. Our results suggest the isospin-mixing correctionδ _c ≈ 0.1%, which is compatible with previous calculations obtained by other methods. The obtainedδ _C correction is about a factor of four too small to obtain unitarity of the Cabibbo-Kobayashi-Maskawa matrix with the present experimental data. Presented by B.R. Barrett at the International Conference on “Atomic Nuclei and Metallic Clusters”, Prague, September 1–5, 1997. This work is supported in part by NSF grant No. PHY96-05192.     

Microscopic (p,p′) calculations and polarization charges with large basis shell-model wave functions

Techniques have been developed for performing microscopic model DWBA calculations of inelastic nucleon-nucleus scattering using large basis shell-model wave functions to describe the nuclear states involved. For the case of ¹³⁸Ba at a bombarding energy of 30 MeV, we obtain good fits to the data by including the exchange amplitude in the DWBA and assuming a state and multipole independent polarization charge.     

Continuum shell-model calculation for Pb

Results of a continuum shell-model calculation of the total photoabsorption cross section for ²⁰⁸Pb are presented and compared with the results of experiments and of bound-state calculations.     

Rotational bands from shell-model Hamiltonians

Shell-model Hamiltonians with eigenstates forming rotational bands are considered. Such states have eigenvalues proportional to J(J+ 1) and can be projected from a Slater determinant of deformed orbitals. The latter are linear combinations of single-nucleon wave functions of j-orbits in a major shell. Conditions on matrix elements and single-nucleon energies are obtained in terms of the deformation parameters. An actual effective interaction is constructed yielding exact ground-state rotational bands for ²⁰Ne and ²⁴Mg which gives reasonable agreement with energies of other sd shell nuclei. Unlike the case of SU(3) symmetry, spin-orbit interaction and different single-nucleon energies can be accommodated and the procedure is not confined to oscillator major shells. Other welcome departures of our effective interaction from the SU(3) picture are the absence of rotational spectra in oxygen isotopes and that the ²⁴Mg ground-state band is projected from a Nilsson-type deformed state with axial symmetry.     

Approximation for shell-model level densities

The maximum entropy approach for the calculation of nuclear shell-model level densities developed in a previous paper is extended to the calculation of terms of higher order inN _k ^–1, whereN _K is the dimension of the shell-model subspaces of interest. We present terms of first and second order inN _k ^–1 , i.e. in the loop expansion, and the corresponding diagrams. We investigate the size of these contributions for several examples. We find that even for subspace dimensions as small as ten, the saddle-point approximation is quite reliable, the leading terms of the loop expansion are small, and the terms of next order are negligible.On leave from Department of Nuclear Physics, Charles University, S-18000 Prague 8, Czechoslovakia     

Hidden configurations and effective interactions: A comparison of three different ways to construct renormalized hamiltonians for truncated shell-model calculations

We discuss in general some criteria and methods for constructing effective Hamiltonians. Then three different methods are compared for constructing an effective Hamiltonian to be used in nuclear shell-model calculations for A = 17–20, allowing (A-16) active nucleons in the d_5/2, s_1/2 vector space. For all three methods, the aim is to obtain a d_5/2, s_1/2 model which will simulate the results of a given full d_5/2, s_1/2, d_3/2 model. The three methods for finding the effective Hamiltonian are.

1. (a) conventional low-order perturbation theory;

2. (b) a projection technique, in which we construct a Hamiltonian whose eigenvalues excactly match a selected subset of d_5/2, s_1/2, d_3/2 eigenvalues, and whose eigenvectors excatly match the projections of d_5/2, s_1/2, d_3/2 eigenvectors on the d_5/2, s_1/2 space; and

3. (c) least-square fit to selected d_5/2, s_1/2, d_3/2 energies.

For all three methods, we first restrict the effective Hamiltonian to a linear combination of 1-body and 2-body operators. Then for the perturbation and projection techniques, we also calculate the 3-body-operator terms in the effective Hamiltonian.When the effective Hamiltonians are limited to 1-body and 2-body terms, the leastsquare method yields the best overall fit to the low-lying spectrum of d_5/2, s_1/2, d_3/2