Matrix diagonalization algorithm and its applicability to the nuclear shell model

Abstract

An importance-sampling iterative algorithm for diagonalizing shell model Hamiltonian matrices is reviewed and implemented in a spin uncoupled basis. Shell model spaces of dimensions up to N≲109 are considered. The analysis shows that about 10% of the basis states are enough to bring the eigenvalues to convergence. This fraction of states, however, is insufficient to lead to convergence of the transition strengths, thereby limiting the applicability of the method to not too large spaces. In its domain of validity, the method yields a large number of eigensolutions and can be usefully adopted for rather complete studies of low-energy spectroscopy. This is done here for 132,134Xe isotopes. The calculation yields spectra and electromagnetic responses in fairly good agreement with the available experimental data and unveils the properties of the low-energy states of these isotopes, including their proton-neutron symmetry

Progress on the study of nuclear states with mixed proton-neutron symmetry

“…Apparently, the 2 + 1,ms state evolves in different ways as a function of valence particle numbers. The evolution of M1 transitions between mixed-symmetry states and fully symmetric states in the γ-soft nuclei of the xenon isotopic chain have recently been described in a schematic microscopic approach [24] and in the nuclear shell model using a new importance-sampling iterative algorithm for matrix diagonalization [25]. Whether or not the observed differences are related to a critical point of a nuclear shape phase transition near 134 Ba is unknown up to now.…”

Complex nuclear spectra in a large scale shell model approach

“…We have now implemented the algorithm in the m-scheme and proposed a new importance sampling [9,10]. The upgraded method yields a large number of levels for each angular momentum and, therefore, offers a complete description of the low-lying nuclear spectroscopic properties.…”

“…The upgraded method yields a large number of levels for each angular momentum and, therefore, offers a complete description of the low-lying nuclear spectroscopic properties. An illustrative example is provided by the study of the spectra of some heavy Xe isotopes, discussed in [10]. After review briefly the algorithm, we will discuss its convergence properties and present the most meaningful results of the study performed on 130−134 Xe.…”

Level lifetimes and quadrupole moments from Coulomb excitation in the Ba chain and the N = 80 isotones

“…Recent large-scale shell-model calculations [37] predict higher strengths. In the shell model calculations a realistic Hamiltonian (Gmatrix derived from the CD-Bonn potential) is used with the same single particle energies and the same effective charges for all of the isotones as well as for the Xenon isotopes [40].…”