Odd-even staggering filters

The simplest way to quantify the OES of binding energies is to use
the following three-point indicator:

where is the number parity and

By applying filter (1) to experimental nuclear binding energies, , one obtains the experimental neutron and proton OES, . Similarly, by applying this filter to calculated binding energies, one obtains information about theoretical results for the OES. In the following, the same subscripts are used to denote the binding energies and values of indicators (1) obtained for a given model. For example, for the single-particle model described below, the resulting values are and . Note that by using filter (1) we only aim at facilitating the comparison of calculations with data; however, in essence we always compare and analyze experimental and calculated binding energies.

As a simple exercise, let us first calculate the OES for
a system of particles moving independently in a fixed deformed
potential well. In such an extreme
single-particle model, the binding energy is

where

In the above expressions, the single-particle energies

For the ground state of an even-*N* system the
*N*/2 lowest levels are filled. In the neighboring odd-*N*system, the odd particle occupies the lowest available level.
This implies:

Hence, in the absence of the two-body interaction, filter (1) vanishes at odd particle numbers, and it gives half of the single-particle level spacings at even particle numbers. Therefore, following Ref. [4], we assume that any other effect leading to the OES

Consequently, at odd particle numbers *N*=2*n*+1,
we shall use the following filter,

while at even particle numbers

Filters correspond to subtracting values of filter (1) at particle numbers higher or lower by one, while employs the symmetric average of both. The differences between these three filters reflect, therefore, small variations of the OES effects with particle numbers. Up to these small variations, we associate the values of these filters with the spacings of the single-particle levels,

It is clear that filters
use masses of four nuclides
near the given particle number *N*; i.e., they constitute asymmetric
expressions in which either heavier or lighter nuclides dominate. On
the other hand,
uses masses of five nuclides symmetrically
on both sides of *N*. The advantages of using either of these filters
depend, therefore, predominantly on the availability of experimental
data in the isotopic or isotonic chains.
For instance,
in Ref. [4] dealing with light- and medium-mass
nuclei, the filter
was discussed.

It is instructive to relate the asymmetric energy-spacing filters
of Eq. (8) to the particle
separation energies *S*(*N*)=*B*(*N*-1)-*B*(*N*), i.e.,

One sees that the filter depends on separation energies of particles from even systems, while the filter depends on those from odd systems. This does not involve any asymmetry in treating even and odd systems, because, obviously, every particle-separation energy depends on one mass of an even system and on one mass of an odd system.

By using the three-mass filter (6), we hope to avoid mixing the contributions to the OES which originate from single-particle structure with those having other roots, e.g., pairing correlations. Moreover, because it involves three masses only, this filter allows obtaining experimental information on the longest isotopic or isotonic chains. Figures 1 and 2 display experimental values of the neutron and proton OES (6). (It is to be noted that these results slightly differ from those presented in Ref. [4]. Firstly, experimental masses were taken from an updated mass evaluation [23]. Secondly, only the masses having an experimental uncertainty less than 100keV have been considered.)

It is seen that, especially for the light- and medium-mass
nuclei, there exists a
substantial spread of results around the average trend.
This suggests that neutron and
proton OES effects are not only functions of neutron and proton
numbers, respectively, but that a significant cross-talk
between both types of nucleons exists.
Numerous studies of this isotopic dependence of the OES
exist [24,25,26],
usually based on higher-order filters such as the
four-point mass formula [21,22,24]:

Unfortunately, as demonstrated in Ref. [4], the higher-order filters mix the pairing and single-particle contributions to the OES; hence, they are not very useful for the purpose of extracting the pairing component. Since the detailed analysis of the isotopic dependence of the three-mass filter results is not yet available, below we discuss the average trends; i.e., for each

Figure 3 (and solid lines in Figs. 1 and
2) present such average values,
,
for the
neutron and proton OES. Neutron and proton values of the OES
follow a similar pattern. Namely, they systematically decrease
with *A*, and they are reduced around shell and subshell closures,
as expected.
It is also seen that
neutron pairing gaps are
systematically larger than the proton ones.
The shift is mainly due to a mass
dependence of the pairing strength and results from the fact
that at a given value *N*, data are more available for lighter nuclei
than at an identical value of *Z*. Indeed, for *N**Z*, data
exist mostly at *A*(*N*)<*A*(*Z*).
A weak contribution from the Coulomb energy is also expected to
contribute to this shift.