Interactions between Configurations of Three Nucleons in the1d,2sShell

Abstract

Calculations for the nuclei with mass 19 were based on a shell model with harmonic oscillator wave functions, central Gaussian interactions between the three outer nucleons, and a single-particle operator which leads to the energy differences between $1d_{\frac{3}{2}}$, $2s_{\frac{1}{2}}$, and $1d_{\frac{5}{2}}$ levels observed in ${\mathrm{O}}^{17}$. Energy matrices for all configurations of the $1d$, $2s$ shell were calculated. Wave functions of four nuclear states were obtained. They lead to the observed $\mathrm{ft}$ value for the allowed unfavored ${\mathrm{O}}^{19}$ ${\beta }^{-}$ transition, as well as for the favored ${\mathrm{Ne}}^{19}$ ${\beta }^{+}$ transition. Energy differences between $\frac{5}{2}+$ and ½+ states of ${\mathrm{F}}^{19}$ and ground states of ${\mathrm{O}}^{19}$ and ${\mathrm{F}}^{19}$ are correctly given. Disagreement with the measured energy of the ½+ state of ${\mathrm{O}}^{19}$ and the magnetic moment of ${\mathrm{F}}^{19}$, however, indicates that at least modifications of this model are necessary to obtain general agreement.