Properties of Nuclei with Neutrons and Protons in the $1f_{\frac{7}{2}}$ Shell

Elliott's generating procedure is used to derive wave functions for nuclei in which the $1f_{\frac{7}{2}}$ level is filling with neutrons and protons. It is assumed that the low-lying states of these nuclei have well-defined isotopic spin and that the configuration is pure. This representation, which mixes states of different seniority, is used to calculate beta-decay transition probabilities, and magnetic-dipole and electric-quadrupole properties of nuclei. The results are in much better agreement with experiment than are those obtained by use of a seniority classification for the nuclear states. In particular, there is a $K$ selection rule which explains the anomalously long half-life for the ${\mathrm{Ca}}^{47}$ → ${\mathrm{Sc}}^{47}$ and ${\mathrm{Ca}}^{45}$ → ${\mathrm{Sc}}^{45}$ beta decays. The theoretical electric-quadrupole matrix elements are too small by a factor of 3-5, and the theoretical $M1\mathrm{}$ lifetimes too short by approximately a factor of 10. For three particles in the $1f_{\frac{7}{2}}$ level (${\mathrm{Sc}}^{43}$) or three neutrons in the $1g_{\frac{9}{2}}$ state, the eigenfunctions given by the generator formalism are found to be almost identical with those derived from a conventional shell-model calculation.