Tensor interaction and the nuclear binding energies

Figure 3: Tensor contributions to the total binding energy calculated by using the spherical Hartree-Fock-Bogoliubov model with the SLy4$_T$ functional of Ref.[28]. Vertical and horizontal lines indicate the tensorial magic numbers. From Ref.[43].

Figure 4: Differences between theoretical and experimental binding energies (positive) in selected double-magic nuclei. Black dots represent the results obtained using conventional SLy4 force. Black and white diamonds label the results calculated using the SLy4$_T$ and the SkO$_{T^\prime}$ functionals, respectively.

As discussed in Ref.[43], the tensor contribution to the nuclear binding energy shows interesting generic topological patterns closely resembling those of the shell-correction, see Fig. 3. The single-particle tensorial magic numbers at $N(Z)$=14, 32, 56, or 90, corresponding to the maximum spin-asymmetry in the $1d_{5/2}$, $1f_{7/2}\oplus 2p_{3/2}$, $1g_{9/2}\oplus 2d_{5/2}$ and $1h_{11/2}\oplus 2f_{7/2}$ spherical s.p. configurations, respectively, are clearly seen in the figure. Note, that the calculated tensorial magic numbers are shifted due to configuration mixing toward the classic magic numbers of $N(Z)$=8, 20, 28, 50, and 82. The topological features shown in Fig. 3 are fairly independent of a specific parameterization of the force. Indeed, they simply reflect the order of s.p. levels, which is rather unambiguously established and relatively well reproduced by the state-of-the art nuclear MF models, at least in light and medium-mass nuclei.

Values of the tensor and SO strengths deduced from the s.p. properties are at variance with those obtained from mass fits[37,40]. A large reduction of the SO strength, which is particularly strong for the low-$m^*$ forces like SLy4, has a particularly destructive impact on theoretical binding energies, see Fig. 4.

A multidimensional fit to masses shows that the mass performance of the SLy4$_T$ force can be improved by a tiny refinements of the remaining coupling constants[28], however, the accuracy of the original SLy4 cannot be regained. This indicates that a spectroscopic quality parameterization that would perform reasonably well on binding energies must have large effective mass, $m^*$ $\geq 0.9$. One of the candidates is the SkO$_{T^\prime}$ functional of Ref.[40]. This functional, at least for the classical set of double-magic nuclei shown in Fig. 4, is of a similar accuracy as SLy4, and it outperforms both the SLy4$_T$ and SLy4 $_{T\mathrm{min}}$ of Ref.[28].

Figure 5: Two-neutron separation energy $S_{2n}$ (upper part) in oxygen isotopes. Empirical data [44] are labeled by black dots. Theoretical values obtained using the SkO and SkO$_{T^\prime}$ functionals are marked by gray and white dots, respectively. Lower part shows contribution of the tensor term to the $S_{2n}$ as a function of shell filling.

This allows for reasonable quantitative estimates of the tensor influence, for example, on two-neutron separation energies, potential energy surfaces (PES's), and onset of deformation. An example of calculation of the two-neutron separation energies for oxygen nuclei is shown in Fig. 5. One clearly sees here the way the tensor interaction induces in oxygen isotopes a breaking of stability against the two-neutron emission around $^{26}$O. Indeed, as shown in the lower panel of the figure, in $^{26}$O a decrease of $S_{2n}$ is directly related to the $d_{3/2}$ sub-shell occupation that reduces the spin-asymmetry and tensor contribution to the binding energy.

By deforming the nucleus one can easily change the spin asymmetry and, in turn, tensor effects. Fig. 6 shows the PES's versus quadrupole deformation in $^{80}$Zr (left) and $^{120}$Sn (right), calculated by using the quadrupole-constrained HFB method. At the spherical shape, nucleus $^{80}$Zr is spin-saturated. By deforming the system, one increases the spin-asymmetry by enforcing the occupation of the $g_{9/2}$ sub-shell. By adding to SkO a strong attractive tensor field (SkO$_{TX}$), one pulls the deformed minimum down. The consecutive reduction of the SO strength (SkO$_{T^\prime}$) provides for a compensation mechanism, and it shifts the $g_{9/2}$ sub-shell and the deformed minimum up in energy, back to its original position obtained for the SkO functional. In $^{56}$Ni, a similar compensation mechanism is found in the yrast super-deformed bands, see Ref.[40]. In case of $^{120}$Sn, the PES's calculated by using the SkO and SkO$_{T^\prime}$ parameterizations are again very close to each other. Note however, that both these curves differ substantially from the PES calculated using the SLy4 parameterization.

Figure 6: Potential energies versus axial quadrupole deformation parameter $\beta_2$ in $^{80}$Zr (left) and $^{120}$Sn (right), calculated by using the SkO, SkO$_{TX}$, SkO$_{T^\prime}$, and SLy4 functionals. See text for details.