Tests of the method

 

As the first test of the method, we considered doubly-magic nuclei. Such nuclei are known to be spherical and thus amenable to reliable calculation using the coordinate-space HFB code [6]. By studying the extent to which our code is able to reproduce the coordinate-space results (referred to in subsequent discussion as exact), we can assess the method. [We should note here that HFB in fact reduces to HF in doubly-magic nuclei, since all pairing correlations vanish].

These calculations were carried out both for nuclei along the beta-stability line and for the very neutron-rich nucleus ²⁸O. Some discussion of this latter nucleus is in order here. ²⁸O is known experimentally to be unbound [44,45], but is predicted to be bound in most mean-field calculations [46]. Due to rapid changes of the single-particle energies with neutron number, shell-model calculations [47] are able to explain the sudden decrease of separation energies that occurs in the chain of oxygen isotopes and renders ²⁶O and ²⁸O unbound. This effect seems to require modifications to the effective interactions currently in use in mean-field studies of light nuclei. Nevertheless, it is common to use ²⁸O as a testing ground of mean-field calculations near the neutron drip line, because according to the standard magic-number sequence it is doubly magic and because it is located (in typical mean-field calculations) just before the two-neutron drip line. This is the philosphy underlying our inclusion of ²⁸O. For comparison, the configurational calculations were performed both in the HO and THO bases. To assess the convergence of the results in the two cases, we varied the number of HO major shells included, considering $N_{{\rm\scriptsize {sh}}}$=8, 12, 16, and 20. For a given number of the major shells, we minimized the total HF energies with respect to the basis parameters, L₀ for the HO basis, and L₀ and cfor the THO basis, so here $N^{{\rm\scriptsize {par}}}_{{\rm\scriptsize {sh}}}$= $N_{{\rm\scriptsize {sh}}}$. We also tested our HO axial-basis results obtained at any given $N_{{\rm\scriptsize {sh}}}$ with those available from Cartesian-basis calculations [15] and the results agreed perfectly. Lastly, for the THO basis, we compared with the calculations of Ref. [11], where spherical symmetry was imposed, and obtained identical results.

As expected, for nuclei within the valley of beta stability the HO and THO results are close to one another and, furthermore, coincide with the exact HFB (HF) results. The situation is quite different for the neutron-rich nucleus ²⁸O, for which the calculations indicate the presence of a significant neutron skin. In Fig. 1, we present the HO and THO results for the total energy and for the proton and neutron rms radii as functions of $N_{{\rm\scriptsize {sh}}}$. For each of the calculated observables, the exact results are shown as a straight line as a function of $N_{{\rm\scriptsize {sh}}}$. Clearly, when we increase the number of major shells, both the HO and THO results for the total energy and for the proton rms radius converge to the exact HFB values. In contrast, the HO neutron rms radius still differs from the exact value, even at $N_{{\rm\scriptsize {sh}}}$=20, while the THO basis gives the correct result.

An explanation of this difference becomes clear when looking at Fig. 2, in which we compare (in logarithmic scale) the HO and THO neutron densities with those from the exact HFB calculations. The HO neutron density fails to reproduce the correct asymptotic behavior at large distances (see also the discussion in Ref. [7]). The THO density, on the other hand, shows perfect agreement with the exact HFB density. There is a difference, of course, near and beyond the box boundary ( $R_{{\rm\scriptsize {box}}}$=20 fm is used in the coordinate HFB calculations). The coordinate-space density rapidly falls to zero at the boundary, while the THO density continues with the correct exponential shape out to infinite distances.

It is clear that the rather small numerical discrepancy between the HO and THO neutron rms radii (Fig. 1) does not reflect the seriousness of the error in neutron densities that arises when using the HO basis. It is also obvious that observables which do not strongly depend on neutron densities at large distances, like the total energy or proton radii, are fairly well reproduced in standard HO calculations. On the other hand, observables that do depend on densities in the outer region, most notably pairing correlations [7], require the correct asymptotic behavior provided by the THO basis.

Encouraged by the excellent results in spherical nuclei, where a comparison with reliable coordinate-space calculations was possible, we next turned to deformed systems. Here, since no coordinate-space HFB results are available, our tests were limited to a study of the convergence of results with increasing number of HO shells. [The exact results would be obtained in either the HO or THO expansion with a complete space, i.e., an infinite number of shells.] Whenever the number of HO shells used in the final HFB calculation was 12 or less, we determined the basis parameters with that same number of shells, $N^{{\rm\scriptsize {par}}}_{{\rm\scriptsize {sh}}}$= $N_{{\rm\scriptsize {sh}}}$. When the number of HO shells of the final calculations exceeded 12, however, we still determined the basis parameters with $N^{{\rm\scriptsize {par}}}_{{\rm\scriptsize {sh}}}=12$.

In Fig. 3, we show convergence results for the ground state of the weakly-bound deformed nucleus ⁴⁰Mg. The top three panels give the results for the total energies, the proton rms radii and the neutron rms radii, respectively. The fourth gives results for the $\beta$ deformation, which is related to the quadrupole moment $\langle Q \rangle$ ( $Q= \sum_{i=1}^A 2z_i^2-x_i^2-y_i^2$) and the rms radius $\langle r^2\rangle$ by

$$\beta =\sqrt{\frac{\pi}{5}}\frac{\langle Q\rangle}{A\langle r^2\rangle}.$$ (61)

The results obtained with $N_{{\rm\scriptsize {sh}}}$=20 are indicated in the figure by horizontal lines. Again, both bases yield very good convergence for the total energy and proton radius. In contrast, noticeable differences between the HO and THO results can be seen for the deformation and neutron rms radius, and they persist to large values of $N_{{\rm\scriptsize {sh}}}$. Although these differences are small in magnitude, they are caused by a very large error in the HO neutron density distribution. This is illustrated in Fig. 4, where we show the neutron densities calculated for the nearby ⁴⁴Mg nucleus. Every point in the figure corresponds to the value of the neutron density at a given Gauss-integration node. Since there are always several nodes near a sphere of the same radius $r=\sqrt{z^{2}+\varrho^{2}}$, there can be some scatter of points, corresponding to different densities in different directions. This is especially true at small distances. At large distances, the scatter is greatly reduced and the densities exhibit to a good approximation spherical asymptotic behavior, exponential in the case of the THO expansion and Gaussian in the case of the HO expansion. Note, however, that some scatter persists in the THO results out to large distances, suggesting that deformation effects are still present there. This is apparently reflecting the importance of deformation of the least-bound orbitals. Clearly, the asymptotic properties of the HO and THO neutron densities are very different from one another, as they were in the spherical calculations (see Fig. 2).