Mixed-symmetry shell-model calculations in nuclear physics

We consider a novel approach to the nuclear shell model. The one-dimensional harmonic oscillator in a box is used to introduce the concept of an oblique-basis shell-model theory. By implementing the Lanczos method for diagonalization of large matrices, and the Cholesky algorithm for solving generalized eigenvalue problems, the method is applied to nuclei. The mixed-symmetry basis combines traditional spherical shell-model states with SU(3) collective configurations. We test the validity of this mixed-symmetry scheme on 24Mg and 44Ti. Results for 24Mg, obtained using the Wilthental USD intersection in a space that spans less than 10% of the full-space, reproduce the binding energy within 2% as well as an accurate reproduction of the low-energy spectrum and the structure of the states - 90% overlap with the exact eigenstates. In contrast, for an m-scheme calculation, one needs about 60% of the full space to obtain compatible results. Calculations for 44Ti support the mixed-mode scheme although the pure SU(3) calculations with few irreps are not as good as the standard m-scheme calculations. The strong breaking of the SU(3) symmetry results in relatively small enhancements within the combined basis. However, an oblique-basis calculation in 50% of the full pf-shell space is as good as a usual m-scheme calculation in 80% of the space. Results for the lower pf-shell nuclei 44-48Ti and 48Cr, using the Kuo-Brown-3 interaction, show that SU(3) symmetry breaking in this region is driven by the single-particle spin-orbit splitting. In our study we observe some interesting coherent structures, such as coherent mixing of basis states, quasi-perturbative behavior in the toy model, and enhanced B(E2) strengths close to the SU(3) limit even though SU(3) appears to be rather badly broken.