Eigenstates of theJ=0,T=1, Charge-Independent Pairing Hamiltonian. II. Seniority-One and -Two States

Abstract

Exact equations for the seniority-one states of $2N+1$ nucleons in an arbitrary external potential well and interacting through a $J=0\mathrm{}$, $T=1\mathrm{}$, charge-independent pairing force are derived and, for a large class of states, solved exactly. The states in this class are characterized as having wave functions that are totally symmetric in the isospins of the paired particles. These states have total isospin $T=\frac{1}{2}, \frac{3}{2}\cdots , \frac{\mathrm{}(2N+1)\mathrm{}}{2}$. The states with the lowest energy consistent with the given values of $N$ and $T$ are contained in this class of states. All the states of $2N+1$ neutrons or protons belong to the charge multiplet with $T=\frac{\mathrm{}(2N+1)\mathrm{}}{2}$. Equations for the seniority-two states of $2N+2$ nucleons are also derived and the solutions to these equations that are totally symmetric in the isospins of the paired particles are considered. These solutions may be classified as charge-symmetric or charge-antisymmetric according to their parity under reflection in isospin space. The equations for the charge-symmetric state are solved exactly. These states have total isospin $T=1, 3, \cdots , N+1\mathrm{}$ for $N$ even and $T=0, 2, \cdots , N+1\mathrm{}$ for $N$ odd with each value of $T$ not equal to zero occurring twice. The equations for the charge-antisymmetric states are solved in an approximation which conserves the isospin of the unpaired particles and which is exact when they occupy degenerate levels. These states have total isospin $T=0, 2, \cdots , N$ for $N$ even and $T=1, 3, \cdots , N$ for $N$ odd with each value of $T$ not equal to zero occurring twice. The calculation of all these states is reduced to the diagonalization of a tridiagonal matrix, whose eigenvalue is given explicitly in terms of $N$ and $T$, and the solution of $N$ coupled, nonlinear, algebraic equations. The wave functions and energies of all the states of these systems having the indicated isospin symmetry and the given value of $T$ are given in terms of the solutions of these equations. An explicit expression for the occupation probabilities of the levels of the single-particle potential (summed over the two charge states) is given. This expression may be evaluated by the solution of an $N\times N$ system of linear algebraic equations.