Pairing Regularization

The total energy and the average neutron pairing gap in $^{120}$Sn are shown in Fig. 3 after applying the pairing regularization procedure. The pairing strength $V_0$ is kept constant; it reproduces the neutron pairing gap for $^{120}$Sn at the cutoff energy of $\epsilon_{\mbox{\rm\scriptsize {cut}}}=$60MeV.

Figure 3: Total energy (top) and neutron pairing gap (bottom) in $^{120}$Sn for the two values of $N_{osc}$ (left) or using two box sizes (right). Calculations were performed using the mixed pairing interaction.

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' $R\approx \sqrt{2N_{osc}\hbar/m\omega}$ [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 $N_{osc}$.

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 $\rho$ 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.

Figure 4: Total energy of spherical $^{120}$Sn (top) and deformed $^{110}$Zr (bottom) obtained with pairing regularization (black lines) for mixed pairing (left) and volume pairing (right). The results obtained without the rearrangement terms resulting from the variation of $k_{\mbox{\rm\scriptsize{cut}}}$ and $k_F$ in Eq. (12) are also shown using (gray lines).

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 $2j+1$-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 $^{110}$Zr in comparison with spherical $^{120}$Sn. One can see that the fluctuations of the total energy in $^{110}$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.

Figure 5: Two pairing regularization schemes applied to the case of $^{120}$Sn: the Thomas-Fermi approximation [19] (black line) and the free particle Green's function [13] (gray line). Coordinate-space calculations were performed in a 15fm box.