1d32−1f72Energy Splitting atCa40

The low-lying levels of nuclei near closed shells are commonly assumed to be of pure, single-particle model configurations. On this basis the single-particle and single-hole energies used in shell-model calculations are taken from the experimental energy spectra of these nuclei. In recent years, however, both direct-reaction experiments and theoretical calculations have shown that there is considerable configuration mixing in the ${\mathrm{Ca}}^{40}$ ground-state wave function, associated with a depression in the ground-state energy. Since the underlying approximation is not valid, the use of experimental energy differences as the single-particle energies in shell-model calculations is not justified. From calculations of the effects of mixing on the $A=39\mathrm{}$ and $A=41\mathrm{}$ nuclei, as well as on ${\mathrm{Ca}}^{40}$, it is found that the conventional experimental splittings overestimate the important $1d_{\frac{3}{2}}-1f_{\frac{7}{2}}$ energy difference by at least 1 MeV. A smaller error is contained in the "experimental" $2s_{\frac{1}{2}}-1f_{\frac{7}{2}}$ splitting. Similar effects occur in the oxygen region, but the overestimation in the $1p_{\frac{1}{2}}-1d_{\frac{5}{2}}$ splitting is much smaller, since the mixing is considerably less.