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 $^{24}$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 $^{44-48}$Ti and $^{48}$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 oblique-basis 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. 
V. G. Gueorguiev and J. P. Draayer