Properties of the [15]-DimensionalSU(4)Supermultiplets of Negative-Parity Excitations ofO16Using Realistic Nuclear Forces

Using particle-hole theory in the Tamm-Dancoff approximation and Wigner's $\mathrm{SU}\left(4\right)\mathrm{}$ supermultiplet theory, we determine the negative-parity excited states of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ in terms of [15]-dimensional supermultiplets. Only the supermultiplets with total $L=1\mathrm{}$ are studied in detail, since they are the ones of importance with regard to the giant dipole resonance and to muon capture and electron scattering. Working in terms of nonmixing supermultiplets, we find that each supermultiplet can be theoretically explained in terms of five parameters, while there are eight pieces of experimental data regarding each supermultiplet. Thus, our theory is overdetermined, and we can eliminate the parameters to obtain empirical sum rules, which can then be used to predict unobserved levels or to check the consistency of our calculation, independent of the radial dependence of the nuclear force. Since there are two $L=1\mathrm{}$ supermultiplets, we also determine the mixing between them. Even after mixing, we obtain a sum rule for the $T=0\mathrm{}$ states. We then use the realistic nuclear potentials of Tabakin and Brueckner, Gammel and Thaler to calculate the $L=1\mathrm{}$ spectrum, so that we can compare it with the experimental spectrum and our sum-rule predictions. Theoretically, we find that some of the parameters are small enough to be neglected with respect to the other parameters and that there is very little mixing between the two $L=1\mathrm{}$ supermultiplets. Both of these results allow us to obtain approximate sum rules. We get a total of five sum rules, of which four can be used for making empirical predictions of so-far unobserved experimental quantities. We also calculate the squared matrix elements for muon capture to our theoretically determined $T=1\mathrm{}$ levels and obtain results similar to previous calculations.