Introduction

The odd-even staggering (OES) of binding energies, which
reflects stronger binding of even-particle-number systems
than their odd-particle-number neighbors,
has long been known in atomic nuclei [1]
and recently has been observed also in metal clusters [2] and
in ultrasmall superconducting grains [3].
However, as discussed recently [4],
the apparent similarity between the OES
in all these finite many-fermion systems is deceptive. Although
two basic physical mechanisms are involved:
(*i*)
an effect of spontaneous breaking of the spherical symmetry (Jahn-Teller
effect [5]) and, (*ii*) blocking of pair correlations
by an unpaired fermion, the origin of the OES
is system-specific.

In small metal clusters, OES is believed to be mainly due to the
non-spherical shape of the underlying
mean field [6,7,8] and,
thus far, neither the empirical evidence
nor the theoretical calculations [9] support the presence
of superconductive correlations in metal clusters.
In superconducting grains, on the other hand, OES is believed to be
predominantly due to a blocking effect caused by the presence of
an odd electron
(see Refs. [10,11,12,13,,15,16,17]).
In the smallest grains, where the average spacing
*d* of the electronic energy levels becomes comparable to
the size of the pairing gap [18],
,
the OES must invoke both mean-field and pairing effects.
In the case of metallic grains, it is, however, difficult (if possible at
all) to pin down the detailed structure of
the single-electron level spectrum.
Most of the theoretical models applied to the problem
of superconductivity in grains
use an equidistant level spectrum because of its analytical simplicity
(see Refs. [10,11,14,15]
and references quoted therein). However, more
realistic treatments require
a non-uniform
distribution of single-particle levels as is done, for example,
in Ref. [19] (see also Ref. [20]).

A slightly different situation holds
in atomic nuclei due to the presence of two types of fermions.
The strong and attractive effective proton-neutron
interaction gives rise to the
appreciable symmetry energy contribution [(*N*-*Z*)^{2}] to OES.
Because the symmetry energy smoothly varies with particle number
and is almost insensitive to shell
effects [4], the standard way of extracting
the OES has been by means of the
higher-order binding-energy indicators (filters),
such as the four-point expression of Refs. [21,22]
(see also Ref. [24] and references
therein).
This classical reasoning, which stems from
liquid-drop or Fermi-gas models, has recently been questioned [4].
Namely, it has been
demonstrated, using fully self-consistent Hartree-Fock theory,
that the contribution
to OES due to symmetry energy is, in fact, nearly cancelled out by the
contribution coming from the smoothed single-particle energy.

The aim of this work is to demonstrate the consistency of our new interpretation using the results of simple, exactly solvable models. Section 2 discusses the binding-energy indicators used in this study. It also presents the systematics of experimental pairing gaps obtained using these filters. The many-body models employed in this work (seniority model, equidistant-level model, and pairing-plus-quadrupole model) are described in Sec. 3, together with the results of the calculations. A short summary and conclusions are given in Sec. 4.