Mass predictions far from stability

Changes of the shell structure, that are expected in nuclei far from stability, are one of the most interesting subjects to be studied in the future exotic-beam facilities [66,67,68]. In light nuclei, such changes are already observed at N=8, 20, and 28 [11]. They are manifested by modifications of nuclear binding energies, that can be interpreted in terms of an appearance of new shells (e.g., N=16 [69]), or as an unexpected collectivity observed in semi-magic nuclei, e.g., in ³²Mg [70]. In light nuclei, such changes can be attributed to shifts of single-particle levels induced by selected channels of the shell-model interactions (see, e.g., Ref. [71]), or to the shape coexistence phenomena related to intruder configurations, as studied either in the shell model [7] or in the mean-field approach [72,73,74]. Both types of effects are strongly connected; indeed, smaller shell gaps in nuclei far from stability may facilitate appearance of intruder configurations at low energies, or polarization induced by the configuration mixing may decrease the shell gaps. Therefore, apart from phenomenological ideas explaining the experimental data, more work is here needed if one wants to discuss the question of whether the shifted levels or intruder states are primary, secondary, or equivalent explanations of the observed changes.

  

Figure 4: Two-neutron separation energies (left) and shell gaps (right, from Ref. [75], reprinted from Nuclear Physics A, M. Samyn et al, ``A Hartree-Fock-Bogoliubov mass formula'', in press, Copyright (2002), with permission from Elsevier Science. ) for nuclei around N=82 as functions of the proton number Z.

In heavier nuclei, predicted shell modifications can be due to the effects of an increasing surface diffuseness [76], changes in the spin-orbit coupling [77,78], or influence of pairing correlations [76,63,79]. Calculated shell quenching effect is illustrated in Fig. 4 for the nuclei along the N=82 isotonic chain. The left panel presents the spherical two-neutron separation energies S_2N which show a conspicuous shell gap between the values corresponding to N=80,82 and N=84,86. When going towards the neutron drip line (i.e., in the direction of decreasing proton number Z), the shell gap decreases, and disappears at the two-neutron drip line (i.e., when the values of two-neutron separation energies go to zero). The shell quenching effect is a generic result obtained in all self-consistent mean-field calculations based on effective two-body interactions. However, the exact magnitude of the effect does strongly depend on what kind of interaction is used. In particular, interactions that give a small effective mass tend to overestimate the shell gaps in stable nuclei, and hence the shell quenching effects are not strong enough to significantly decrease these gaps near the drip lines. Moreover, the type of the pairing interactions used in these calculations crucially influences the degree of the shell quenching [79]. At the moment, our knowledge of the effective interactions is insufficient for reliable predictions of the shell quenching far from stability, and the only method to proceed further seems to be the use of future measurements to fit better interactions.

The best to date effective interactions fitted to nuclear masses were obtained in Refs. [9,75]. Depending on the method to treat the pairing correlations, BCS [9] or HFB [75], the authors obtained the rms deviations between the calculated and measured binding energies equal to 0.738 and 0.766MeV, respectively. The quality of the description is almost identical to that (0.669MeV) obtained within the microscopic-macroscopic FRDM method based on a phenomenological mean-field potential [80]. Needless to say that masses of unknown nuclei, which are obtained by extrapolating either effective interactions or phenomenological mean-fields, are markedly different. This is illustrated in the right panel of Fig. 4, which shows the shell gaps at N=82 defined by S_2N(N=82)-S_2N(N=84). Interactions that were fitted to masses give a weak shell quenching as compared the that obtained for the SkP interaction [50], while the FRDM results show no quenching at all. Obviously, there is still a fair degree of arbitrariness in the values of parameters of the effective interactions, and at the moment it is not clear how well these parameters are fixed by fits to masses or other nuclear properties.

Calculations of Refs. [9,75] were performed by expanding the single-particle wave functions in the harmonic-oscillator (HO) eigenstates, and therefore, they are characterized by unphysical asymptotic properties of the particle and pairing density distributions. It is known [81] that the convergence of results in function of the HO basis size can be extremely slow, and the stability of results tested by increasing the basis by adding one or two HO shells can be misleading. Moreover, the asymptotic properties of densities are essential for a correct description of the pairing channel [51]. It still remains to be investigated to which extend these deficiencies of the method are important for determining masses of weakly bound nuclei. This question can be studied by using methods based on the transformed harmonic oscillator (THO) basis [82,83,84] in which the correct asymptotic behavior is obtained by a proper point-like transformation of the basis wave functions.

  

Figure 5: Two-neutron separation energies in even-even nuclei, obtained within the Skyrme-HFB method for the Skyrme SLy4 interaction and contact pairing force. From Ref. [79].

Preliminary results obtained within such an approach are shown in Fig. 5 [79]. Deformed calculations were performed for a standard Skyrme effective interaction SLy4 [85], and for contact pairing interaction that is intermediate between the volume and surface force [86]. The figure shows the two-neutron separation energies calculated for all even-even nuclei that are bound with respect to two-particle emission, i.e., those that have negative neutron and proton HFB Fermi energies. The results show a very interesting phenomenon occurring at the neutron drip line, namely, negative two-neutron separation energies are sometimes obtained for particle-stable nuclei. This apparently contradictory result illustrates the fact that the HFB particle stability is defined with respect to a fixed configuration, while the two-neutron separation energies can involve binding energies of neighboring nuclei that may have different configurations, e.g., different shapes. Hence, at the neutron drip line we may encounter nuclei for which the two-neutron emission is energetically possible, but can be hindered by different configurations of the parent and daughter nuclei.