Shell-Model Structure ofCa42−50

The low-lying states of the calcium isotopes ${}_{}{}^{42}{}_{}{}^{}\mathrm{Ca}$ through ${}_{}{}^{50}{}_{}{}^{}\mathrm{Ca}$ are discussed within the framework of the conventional shell model. An inert ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ core is assumed. Calculations are made in several basis-vector spaces involving active $0f_{\frac{7}{2}}$, $1p_{\frac{3}{2}}$, $0f_{\frac{5}{2}}$, $1p_{\frac{1}{2}}$, and $0g_{\frac{9}{2}}$ neutron orbits. In one set of calculations, we use a "realistic" effective interaction derived for this mass region by Kuo and Brown. The shell-model results suggest that, in this effective interaction, the interactions of $f_{\frac{7}{2}}$ neutrons with $p_{\frac{3}{2}}$, $p_{\frac{1}{2}}$, and $f_{\frac{5}{2}}$ neutrons are too strong. In other calculations, with a modified Kuo-Brown interaction, we find the calculated spectra for the low-lying states of the calcium isotopes are in agreement with observed spectra, with several significant exceptions. The exceptions are that the second $0^{+}$ and $2^{+}$ states observed in ${}_{}{}^{42}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{44}{}_{}{}^{}\mathrm{Ca}$, and ${}_{}{}^{46}{}_{}{}^{}\mathrm{Ca}$ are not accounted for in the calculation. Calculated spectroscopic factors for $f_{\frac{7}{2}}$ transfers are in good agreement with experimental spectroscopic factors, and the observed centroids of the $p_{\frac{3}{2}}$ single-particle strengths are reasonably well reproduced. Our results also indicate that "core-excitation" effects are significant above about 2.5 MeV in ${}_{}{}^{42}{}_{}{}^{}\mathrm{Ca}$ through ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$.