Let us consider the seniority (or pairing quasispin) model
the model for N nucleons moving in a
-fold degenerate shell described by the seniority-pairing
For this model the exact solution can be written in terms of quasispin
quantum numbers [see Eq. (26)].
Alternatively, the ground-state
energy can be expressed in terms of the seniority
quantum number s (see, e.g.,  p. 222):
It is seen that, in the seniority model, filters (6) and (7) give the OES values (41) and single-particle energy spacings (42) which are independent of the particle number N. Values of the OES (41) correctly reproduce the exact pairing gap = , which is defined as a half of the lowest excitation energy in an even system, and which we denote by a prime to distinguish it from the OES. The meaning of is less obvious. It is because the mean-field (Hartree-Fock, HF) treatment of Hamiltonian (14) yields only one 2-fold degenerate single-particle level at energy -G, while our interpretation of assumes that only the Kramers degeneracy is present. Nevertheless, one may compare exact values of with those obtained in an approximated way and see whether the approximate ground-state energies reproduce features of the mass spectrum represented by and .
In the seniority model, the
Hartree-Fock-Bogoliubov (HFB) equations
(which in this case are identical with the BCS equations)
can be solved analytically. Indeed, the BCS occupation
coefficient is given by
For the energy-spacing filters the deviations behave as . While the symmetric filter does not depend on N, both and vary weakly with the particle number. Namely, at the beginning of the shell (N0), Gand , at the middle of the shell = , and at the top of the shell (N) Gand . This behavior follows from a simple identity = reflecting the particle-hole symmetry of the model.
The analysis presented in this section illustrates the advantages of comparing exact and approximate results (similarly as experimental and theoretical results) by looking at appropriate filters. Analytical results available in this model allow the explicit study of pairing and mean-field effects. The high degeneracy of the seniority model does not allow for extracting the energy spacings between the deformed levels; to this end, results for the non-degenerate models are shown in the following sections.