Eigenstates of theL=0Charge- and Spin-Independent Pairing Hamiltonian. II. Seniority-1 States

Abstract

Equations for all the seniority-1 states of $2N+1$ nucleons in an arbitrary charge- and spin-independent single-particle potential and interacting through an $L=0$ charge- and spin-independent pairing interaction are derived. These equations are then solved analytically for a large number of physically interesting states. These states are characterized by having wave functions that are totally symmetric functions of the spin-isospin variables of the $N$, $L=0$ pairs of nucleons in the state. The supermultiplet quantum numbers of these states are given by $P=\frac{1}{2}, \frac{3}{2}, \ensuremath{\cdots}$, $N+\frac{1}{2}$, ${P}^{\ensuremath{'}}=\frac{1}{2}$, ${P}^{\ensuremath{'}\ensuremath{'}}=\ifmmode\pm\else\textpm\fi{}\frac{1}{2}$. The wave function of these states is shown to factor into a spin-isospin part times an orbital part. The spin-isospin part of the wave function is shown to satisfy three eigenvalue problems which insure that the state is a supermultiplet, spin, and isospin eigenstate, respectively. The eigenvalues of these three problems are given explicitly in terms of the quantum numbers of the state. The orbital part of the wave function is given analytically in terms of $N$ parameters which we call pair energies. These pair energies are shown to satisfy $N$ coupled nonlinear algebraic equations which depend parametrically upon the interaction strength, single-particle spectrum, and the supermultiplet quantum numbers of the state. Explicit expressions for the energies of the states and the occupation probabilities of the single-particle levels are given in terms of the pair energies. These expressions may be evaluated with ease for values of the parameters that are characteristic of light-weight nuclei.