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], $d\sim \Delta$, 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 [$\sim$(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.