Effective shell-model interaction through second order for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>sd</mml:mi></mml:math>shell

Abstract

We evaluate the effective shell-model interaction ${\mathcal{V}}_{\mathrm{eff}}$ for the $\mathrm{sd}$ shell, through second order as a function of an energy shift between the valence and unoccupied oscillator states. The Brueckner reaction matrix $G$ for the Reid soft-core potential is obtained in a single-particle basis sothe Pauli operator is treated exactly. In terms of second order in $G$, intermediate-state sums and the dependence of energy denominators on passive particles and holes are both treated with care. Significant dependence on the intermediate-state spectrum is found for ${\mathcal{V}}_{\mathrm{eff}}$ in both the $T=0$ and $T=1$ components. An energy shift yielding a total gap of about 49 MeV is found to yield $A=18$ two-particle spectra in reasonable agreement with experiment in view of the expected role of 4p-2h intruder states. On the other hand, a comparison of the matrix elements of ${\mathcal{V}}_{\mathrm{eff}}$ with those determined by Chung and Wildenthal from spectra of $A=18 \mathrm{to} 22$ nuclei indicates a preference for a total gap between 35 and 40 MeV. These empirically determined gaps may be compared with a total gap of about 30 MeV, motivated by a study of the self-consistent particle spectrum by Gambhir and McCarthy.NUCLEAR STRUCTURE Effective interaction, theory; convergence of intermediate-state sums; $A=18$ spectra; shell-model matrix elements in the $\mathrm{sd}$ shell; realistic microscopic shell-model interaction through second order in the reaction matrix.