Isospin mixing and Fermi transitions: Self-consistent deformed mean field calculations and beyond

Abstract

We study Fermi transitions and isospin mixing in an isotopic chain $(^{70\text{\ensuremath{-}}78}\mathrm{Kr})$ considering various approximations that use the same Skyrme-Hartree-Fock single-particle basis. We study Coulomb effects as well as the effect of BCS and quasiparticle random phase approximation (QRPA) correlations. A measure of isospin mixing in the approximate ground state is defined by means of the expectation value of the isospin operator squared in $N=Z$ nuclei (which is generalized to $N\ensuremath{\ne}Z$ nuclei). Starting from a strict Hartree-Fock approach without Coulomb interaction, it is shown that the isospin breaking is negligible, on the order of a few per thousand for $(N\ensuremath{-}Z)=6$, increasing to a few percent with Coulomb interaction. Pairing correlations induce rather large isospin mixing and Fermi transitions of the forbidden type $({\ensuremath{\beta}}^{\ensuremath{-}}$ for $N\ensuremath{\le}Z$, and ${\ensuremath{\beta}}^{+}$ for $N\ensuremath{\ge}Z)$. The enhancement produced by BCS correlations is compensated to a large extent by QRPA correlations induced by isospin-conserving residual interactions that tend to restore isospin symmetry.