Drip-line-to-drip-line calculations in Mg

 

The chain of even-Z magnesium isotopes has been the subject of numerous recent theoretical analyses. The extreme interest in this isotopic chain is motivated by recent measurements in ³²Mg [48,49,50], which show a larger-then-expected quadrupole collectivity. Based on the relativistic and non-relativistic mean-field approaches and on shell-model calculations (see Ref. [51] for a review), it is now well documented that shape coexistence and configuration mixing occur in this N=20 nucleus. Moreover, recent advances in radioactive-ion-beam technology allow mass measurements of even heavier isotopes [52], giving hope that the neutron-drip line can be experimentally reached in the Z=12 chain [53].

In this section, we present results of an investigation of the deformation properties of the even-even Mg isotopes from the proton-drip-line to the neutron-drip-line. Our results are complementary to those of recent Skyrme+HFB calculations [8], in which the imaginary-time evolution method of finding eigenstates of the mean-field Hamiltonian h (see Sec. 4) was combined with a diagonalization of the HFB Hamiltonian within a relatively small set of these eigenstates. In that study, a complete set of results was given only for the SIII force and density-dependent pairing was used. Here, we present a complete set of results for the SLy4 force with a density-independent (volume) pairing interaction. These calculations were carried out with $N_{{\rm\scriptsize {sh}}}$=20 and $N^{{\rm\scriptsize {par}}}_{{\rm\scriptsize {sh}}}$=12 HO shells.

In Fig. 5, we plot the total HFB energies per nucleon E/A, the neutron chemical potentials $\lambda_{n}$, the neutron and proton deformation parameters, $\beta_{n}$ and $\beta_{p}$, the neutron, proton, and total quadrupole moments, Q_n, Q_p, and Q_t, the average neutron and proton pairing gaps, $\widetilde{\Delta}_{n}$ and $\widetilde{\Delta}_{p}$, Eq. (53), and the pairing energies $E_{{\rm\scriptsize {pair}}}^{n}$ and $E_{{\rm\scriptsize {pair}}}^{p}$ for the magnesium isotopes as functions of the mass number A. Ground-state values are shown by full symbols connected by lines, while the isolated open symbols correspond to secondary minima of the deformation-energy curves. In the top panel of Fig. 6, we compare the results for the two-neutron separation energies S_2n (open symbols) with those for the related quantity $-2\lambda_{n}$ (full symbols), and in the bottom panel we show the neutron and proton rms radii.

The lightest Mg isotope predicted by these calculations to be bound against two-proton decay is ²⁰Mg. The heaviest bound against two-neutron decay, on the basis of having a positive two-neutron separation energy, is ⁴⁰Mg. On this basis, the position of the two-neutron drip line obtained within the HFB+SLy4 approach is identical to that obtained in the finite-range droplet model [54], relativistic mean field (RMF) [55], and HFB+SIII [8] calculations. The RMF approach with the NL-SH effective interaction [56] predicts the two-neutron drip line at ⁴²Mg, and the relativistic Hartree-Bogoliubov (HB) approach with the NL3 effective interaction [57] predicts it at or beyond ⁴⁴Mg.

On the other hand, from Fig. 6, we see that both ⁴²Mg and ⁴⁴Mg, though having negative values of S_2n, have (small) negative values of the Fermi energy, $\lambda_{n}$. According to the discussion of Sec. 3.2, these nuclei, both of which exhibit oblate shapes, are bound against neutron emission. We will return to this point later.

The most deformed nucleus of the isotope chain is ²⁴Mg with almost the same neutron and proton deformations. At the other end of the chain, due to a large excess of neutrons over protons, significant differences exist between the proton and neutron quadrupole moments. The onset of large deformation in ³⁶Mg causes a decrease of the neutron chemical potential $\lambda_{n}$ with respect to its value in ³⁴Mg. This gives an additional binding of ³⁶Mg, and correspondly to an increase and decrease of the two-neutron separation energies S_2n in ³⁶Mg and ³⁸Mg, respectively (see Fig. 6). In experiment [52], these changes are less pronounced and arrive two mass units earlier, giving rise to a small and large decrease of S_2n in ³⁴Mg and ³⁶Mg, respectively.

Concerning the ground-state deformation properties (full symbols connected by lines in Fig. 5), the proton drip-line nucleus ²⁰Mg displays a well defined spherical minimum (N=8 is a magic number). Then, there is a competition between prolate (^22,24Mg and ^36,38,40Mg) and oblate (^26,30Mg) deformations, while ^28,32Mg are spherical. The last two localized isotopes (with negative Fermi energies), ^42,44Mg, display oblate deformations. Secondary minima of the deformation energy curves (isolated symbols) exist for isotopes ^22,24,26Mg and ^36,38,40Mg.

Non-zero proton pairing correlations are present at all spherical or oblate minima. However, at these shapes, tangible neutron pairing exist only in ^22,24Mg and ^34,36,38Mg. Moreover, for all nuclei with prolate ground-state shapes, i.e., in ^22,24Mg and ^36,38,40Mg, both proton and neutron correlations are small or vanish altogether. These results are at variance with the Gogny-pairing HB calculations of Ref. [57], where non-zero pairing exists in all the heavy Mg isotopes. Also, in Ref. [8], stronger pairing correlations were obtained for the zero-range density-dependent pairing force. However, in that study, the strength parameters were not adjusted to odd-even mass staggering but rather taken from high-spin calculations of superdeformed bands. Our results suggest that the pure HFB-pairing approach is not necessarily the best way to treat pairing correlations in the Mg isotopes, and approximate or exact particle-number projection should probably be employed.

At this point, it is worth expanding a bit on the unusual results for ^42,44Mg. In these two isotopes, the solutions corresponding to prolate shapes are unstable ( $\lambda_{n}$>0), while those corresponding to oblate shapes continue to be bound, i.e., they have $\lambda_{n}$=-0.253 and -0.092MeV for A=42 and 44, respectively. The bound ground states of these two nuclei are thus oblate, whereas in the lighter isotopes the oblate solutions corresponded to secondary minima. This is the origin of the sudden change in two-neutron separation energies, which become negative in ⁴²Mg and ⁴⁴Mg (S_2n=-2.237 and -1.975MeV, respectively. In the case of ⁴²Mg, however, two-neutron emission should be hindered by the fact that the parent and daughter nuclei have dramatically different shapes, and, by this token, ⁴²Mg may still have a substantial half-life even though it is beyond the two-neutron drip line.

The bottom panel of Fig. 6 shows the neutron and proton rms radii, r_n and r_p. At the proton drip-line, the neutron rms radii are smaller than the proton rms radii, and then they increase with increasing neutron number. Around ^24,26Mg, r_n becomes almost equal to r_p, and for nuclei close to the neutron drip-line, r_n takes significantly larger values than r_p. The increase of r_n is fairly linear, similarly as in Refs. [8,56,57], and gives no hint of an existence of unusually larger neutron distributions at the neutron drip line (see also the discussion in Refs. [58,59]).