Mixed-Symmetry Shell-Model Calculations

The one-dimensional harmonic oscillator in a box problem is used to introduce the concept of an oblique-basis shell-model theory. The method is applied to nuclei by combining traditional spherical shell-model states with SU(3) collective configurations. An application to Mg, using the realistic two-body interaction of Wildenthal, is used to explore the validity of this oblique-basis, mixed-symmetry shell-model concept. The applicability of the theory to the lower pf -shell nuclei Ti and Cr using the Kuo-Brown-3 interaction is also discussed. While these nuclei show strong SU(3) symmetry breaking due mainly to the single-particle spin-orbit splitting, they continue to yield enhanced B(E2) values not unlike those expected if the symmetry were not broken. Other alternative basis sets are considered for future obliquebasis shell-model calculations. The results suggest that an oblique-basis, mixed-symmetry shell-model theory may prove to be useful in situations where competing degrees of freedom dominate the dynamics. Two dominate but often competing modes characterize the structure of atomic nuclei. One is the single-particle shell structure underpinned by the validity of the mean-field concept; the other is the many-particle collective behavior manifested through nuclear deformation. The spherical shell model is the theory of choice when single-particle behavior dominates [1]. When deformation dominates, the Elliott SU(3) model can be used successfully [2]. This manifests itself in two dominant elements in the nuclear Hamiltonian: the single-particle term, H0 = ∑ i εini, and a collective quadrupole-quadrupole interaction, HQQ = Q · Q. It follows that a simplified Hamiltonian H = ∑ i εini − χQ · Q has two solvable limits associated with these modes. To probe the nature of such a system, we consider a simpler problem: the onedimensional harmonic oscillator in a box of size 2L [3]. As for real nuclei, this system