Random interactions explore the nuclear landscape: Predominance of prolate nuclear deformations

We use an ensemble of random but spin- and isospin-invariant interactions in a restricted orbital space as a tool for exploration of the shell-model parameter space to understand the source of collective effects. The mean field with the yrast state sequence 0⁺-2⁺-4⁺, rotational energy ratio E(4⁺)/E(2⁺), and rigid-rotor relation between the prolate quadrupole moment Q(2⁺) and the transition probability B(E2;2⁺→0⁺) appears with a high probability as a result of mixing of single-particle orbitals split by deformation. For a given set of collective observables, we identify the responsible matrix elements of the effective forces. In this framework, we reanalyze an old problem of the predominance of prolate deformations among well-deformed nuclei.