Two-body potentials in the collective model

Abstract

The question, “How well can a 1 + 2-body shell-model interaction represent a many-body potential?”, is addressed by optimally expanding the (1 + 2 + 3)-body potential β 3 cos 3γ and the (1 + 2 + 3 + 4)-body potential β 4 of the Bohr-Mottelson collective model in terms of (1 + 2)-body operators. It is found that the correlation of β 4 with its approximation is greater than 97% throughout the sd shell. Although β 3 cos 3γ is also well approximated in the first half of the sd shell where it has more than 80% correlation with its approximation, the correlation drops abruptly at 28Si to 50% and remains low in the second half of the shell. The approximations are primarily sums of the various components of the quadrupole-quadrupole interaction connecting different major oscillator shells. The results suggest that axially-symmetric deformation can be represented by simple (1 + 2)-body operators, whereas asymmetric shapes require non-simple 3-body terms.