Binding energies

When the tensor terms (3) are added to the EDF, the binding energies are affected through self-consistent changes of all the terms in the EDF. However, qualitatively, the effects of tensor terms on the ground-state energies can be illustrated by integrals of products of the SO densities that appear in Eq. (3). In Fig. 4, values of such integrals are shown for the neutron SO densities squared, $J_n^2(r)$, calculated at magic proton numbers in function of the neutron numbers. Comparison of results obtained without (left panel) and with (right panel) tensor-even interaction included, shows that the effect of the tensor term can, in the first approximation, be treated perturbatively. Due to the fact that the proton SO densities depend weakly on the neutron numbers, for $Z$=28, 50, and 82 the integrals of products $J_n(r)J_p(r)$ show similar a behaviour to those of $J_n^2(r)$, while they are small for $Z$=8 and 20.

From the results shown in Fig. 4, it is clear that, for positive coupling constants, the tensor terms will give characteristic contributions to the ground-state energies of heavy nuclei. These contributions will have a form of inverted arches, spanned between the neutron magic numbers. This feature is conspicuously reminiscent of differences between the theoretical and experimental ground-state energies obtained with tensor terms not included.[24,25,26] It is therefore quite plausible that by including the tensor terms one may be able to remove a major part of the discrepancy between the previously calculated nuclear masses and experiment.

Figure 4: Integrals of the neutron SO densities squared that define contributions of the tensor terms to total binding energies. Left panels correspond to the standard SLy4 parametrization with no tensor terms ($t_e$=0), while the right panels correspond to the tensor-even interaction included ($t_e$=200MeVfm$^5$).