The total energy and the average neutron pairing
gap in Sn are shown in
Fig. 3 after applying the pairing
regularization procedure. The pairing strength
is kept constant; it reproduces the neutron pairing gap for
Sn at the cutoff energy
of
60MeV.
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In the left panels of Fig. 3, we show results obtained
in the HO basis, while the results
from the solution of the HFB equations in coordinate space are displayed in the right panels.
One can correlate the coordinate-space
and HO representations by introducing an
`effective box size'
[8]. Using this formula, the basis of 20 HO shells
corresponds to a box radius of about 14.5fm.
Figure 3 demonstrates that the
regularization procedure is stable with respect to
the cutoff energy. Moreover, one obtains reasonable results already for
fairly low cutoff energies of about 40MeV.
The variations in the total energy in coordinate-space
calculations do not exceed 40keV, while they are about 150keV
in the HO expansion. The latter number
does not decrease significantly with
.
The differences in applying the pairing regularization procedure in
the coordinate-space and HO calculations can be explained by the
different way the quasiparticle space is expanded in both
approaches. The particle density is defined by the lower
components of the quasiparticle wave functions, which are localized
within the nuclear interior. On the other hand, the
abnormal density is defined by the products of the upper
and lower components of the quasiparticle wave function. For
the quasiparticle energies that are greater than the modulus of the chemical potential, the upper
components of the quasiparticle wave function are not localized. Therefore, contrary to
the normal density, the abnormal density strongly depends on the
completeness of the s.p. basis outside the nuclear interior.
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In the coordinate-space calculations, the box boundary conditions provide discretization of the spectrum for the quasiparticle continuum states that are not localized. On the other hand, all the HO basis states are localized. Results of stability with respect to the cutoff energy for the coordinate-space and HO calculations are, therefore, different. As far as the description of nonlocalized states is concerned, the coordinate-space method is superior over the HO expansion method.
Fluctuations in the total energy shown in
Fig. 3 coincide with -folded degenerate
angular-momentum multiplets of states in spherical nuclei that enter
the pairing window with increasing cutoff energy. This can be
confirmed by performing a similar analysis for a deformed nucleus where
the magnetic degeneracy is lifted. Such results
are shown in Fig. 4 for deformed
Zr
in comparison with spherical
Sn.
One can see that the fluctuations of the total energy in
Zr are down to about
40keV.
The steep increase of the total energy at the cutoff energies below 30MeV results from neglecting quasiparticle states with significant occupation probability. This effect is more severe for the mixed-pairing than for volume-pairing calculations due to the surface-peaked character of mixed pairing fields. On the other hand, the stability with respect to the cutoff energy is similar in both cases.
We have also tested the importance of the rearrangement terms arising as a result of the regularization procedure. The gray lines in Fig. 4 show results obtained without taking into account the second and third term of Eq. (12). These terms lead to changes in the total energy of a few keV and can be safely neglected.
Finally, we have tested the Thomas-Fermi approximation used in the pairing regularization procedure. Instead of adopting the Thomas-Fermi ansatz, one can perform regularization using the free particle Green's function [13]. As illustrated in Fig. 5, the convergence of the latter method is very slow; the Thomas-Fermi method is clearly superior.
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