Beta-Decay Transition Probabilities and Magnetic Moments in thef72Shell

Abstract

One of the classic weaknesses of the shellmodel analysis of nuclear structure is its general inability to describe the observed magnetic moments and beta-decay transition probabilities. The failure is usually attributed to the sensitivity of the single-particle matrix elements to small configuration admixtures which introduce large changes in the single-particle matrix elements without appreciably affecting the energies of the states involved. It is clear that some explanation of this sort is necessary for s l/2 and p 1/2 particles, since in those cases there is very little uncertainty in the nuclear wave function corresponding to a pure configuration. But in shells of high single-particle spin the trouble may not lie in the assumption of a pure configuration, but rather in the additional assumptions usually made to obtain many-particle wave functions. This note will show in the specific case of the f ll2 shell that proper many-particle wave functions can explain much of the experimental data for magnetic moments and beta-decay transitions without invoking configuration mixing.The calculations of the single-particle operators were made for all nuclei in the f 7/2 shell which have at most two particles (holes) in one of the unfilled shells, and an arbitrary number of par-ticles (holes) in the other. This includes most of the nuclei in the shell. The wave functions used were energy eigenfunctions determined by diagonalizing the energy matrix of the residual interactions between all the nucleons in the shell. The interaction energies were taken to be (in MeV) 0, 1.035, 1.509, 2.248, 2.998, 1.958, 3.400, and 0.617, for angular momenta J = 0-7, respectively. This spectrum is consistent with all available experimental data on the spectrum of Sc 42 . 1 * 2 The interaction energies have been chosen to be charge independent so that the resulting wave functions are states of definite isotopic spin.Following Talmi's procedure, the many-body energy matrix elements were calculated using standard fractional parentage techniques. 3 The details of the calculation and the energy eigenvalues and eigenf unctions calculated by this method will be described in a forthcoming paper now in preparation.The beta-decay transition probabilities were calculated in terms of the single-particle transition probability, log//= 3. 65, taken from the experimental value for the 7-6 transition in the Sc 42 -Ca 42 decay. 4 The results are shown in Table I. In general, the predicted log ft values are raised to around 5.0, which closely matches the Table I. Log/* values.

Odd-Jpairing in nuclei

“…Our previous example was 44 Ti, but the same mathematics holds for 96 Cd. We previously considered various schematic interactions as well and the more realistic MBZ interaction taken from experiment [8], for which detailed wave funtions were subsequently published in the archives [9] (with some modification of the two-body matrix elements). In this work we use the symbol I for the total angular momentum of the state and J otherwise.…”

J=0,T=1pairing-interaction selection rules

“…For a "realistic" interaction in the f 7/2 shell we use the MBZE interaction [3]. This is based on the works of Bayman et al and McCullen et al [4,5] but with improved T=0 two-body matrix elements [4]. From J=0 to J max =7 the matrix elements, which were obtained from experiment are:…”

Degeneracies whenT=0two body matrix elements are set equal to zero and Regge’s6jsymmetry relations

“…In the early 1960's single j shell calculations in the f 7/2 region were performed by McCullen, Bayman, and Zamick (MBZ) [1,2] and Ginocchio and French [3]. In these calculations the two body matrix elements were taken from experiment.…”