Mixed-Mode Calculations in Nuclear Physics

The one-dimensional harmonic oscillator in a box problem is used to introduce the concept of a mixed-mode shell-model scheme. The method combines low-lying “pure mode” states of a system to achieve a better description in situations where complete calculations cannot be done and the dynamics is driven by a combination of modes. The scheme is tested for real nuclei by combining traditional spherical states, which yield a diagonal representation of the single-particle interaction, with collective SU(3) configurations that track deformation. An application to the ds-shell 24 Mg nucleus, using the realistic twobody interaction of Wildenthal, is explored to test the validity of the concept. The results shown that the mixed-mode scheme reproduces the correct binding energy of 24 Mg (within 2% of the exact result) as well as low-energy configurations that have greater than 90% overlap with exact solutions in a space that spans less than 10% of the full space. In the pf-shell, the Kuo-Brown-3 interaction is used to illustrate coherent structures of the low-lying states of 48 Cr. Alternative basis sets are suggested for future mixed-mode shell-model studies. Typically, two competing modes characterize the structure of a nuclear system. One is the single-particle mode that is the underpinning of the meanfield concept; the other is the many-particle collective behavior manifested in the nuclear deformation. The spherical shell model is the theory of choice