Degenerate shell: Seniority model

Let us consider the seniority (or pairing quasispin) model
[27,28], i.e.,
the model for *N* nucleons moving in a
-fold degenerate shell described by the seniority-pairing
Hamiltonian (14).
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., [28] p. 222):

The corresponding value of is given by:

which implies

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

(43) |

(see e.g. [28] p. 233), and the ground-state energy is:

The resulting three-mass filter (1) can be written as

and hence filters (6) and (7) give

It is seen that in the limit of large (1), the BCS approximation (46) reproduces the leading order of the exact OES result (41). The deviation from the exact result smoothly depends on

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 (*N*0),
*G*and
,
at the middle of the shell
=
,
and at the top
of the shell (*N*)
*G*and
.
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.