Effective-Interaction Theory of Nuclear Spectral Relations. II

A theory of particle-hole relations among spectra, presented in Part I, is extended to include hole-hole (hh) spectra as well as the particle-particle (pp) and particle-hole (ph) cases previously treated. This theory, which incorporates configuration mixing into many-body effective interactions, goes beyond the simple shell-model particle-hole relations of Pandya, Goldstein, and Talmi. A new shell-model relation involving all three partners (the triptych: pp, ph, and hh) is derived. We apply the many-body theory and the triptych relation to the nuclei ${}_{}{}^{42}{}_{}{}^{}\mathrm{Sc}$, ${}_{}{}^{48}{}_{}{}^{}\mathrm{Sc}$, and ${}_{}{}^{54}{}_{}{}^{}\mathrm{Co}$, which are partners in the $f_{\frac{7}{2}}$ shell. We find that we can then relate the spectra of these three nuclei to considerably greater accuracy than do the standard shell-model relations.