Symmetry energy

Figure 4: Top: schematic illustration of the isospin-symmetry-breaking mechanism in MF of odd-odd $N=Z$ nuclei. Bottom: $E_{\mbox{\rm\scriptsize{sym}}}^{\mbox{\rm\scriptsize{(int)}}}$ in odd-odd $N=Z$ nuclei calculated with SLy4, SV, SLy4$_L$, and SkM$^*_L$ EDFs. See text for details.

The spontaneous violation of isospin symmetry in all but isoscalar MF configurations of $N=Z$ nuclei offers a way to study the nuclear symmetry energy. The idea, which is schematically sketched in the upper portion of Fig. 4, invokes the mixed-symmetry antialigned $\vert\bar \nu \otimes \pi \rangle$ (or $\vert \nu \otimes \bar \pi \rangle$) configuration in an odd-odd $N=Z$ nucleus. By applying the isospin projection to the HF state $\vert\bar \nu \otimes \pi \rangle$, one decomposes it into the isoscalar $T=0$ and isovector $T=1$ parts. As argued below, the magnitude of the splitting, $E_{\mbox{\rm\scriptsize {sym}}}^{\mbox{\rm\scriptsize {(int)}}}$, depends on the isovector channel of a given EDF, i.e., its symmetry energy.

For the Skyrme-type EDFs, the symmetry energy in the nuclear matter limit can be decomposed as:[30]

$$a_{\mbox{\rm\scriptsize {sym}}} = \frac{1}{8}\varepsilon_{FG... ...{\mbox{\rm\scriptsize {sym}}}^{\mbox{\rm\scriptsize {(int)}}}.$$ (5)

The first term in Eq. (5) is associated with the isoscalar part of the nucleon-nucleon interaction and primarily depends on the mean single-particle level spacing at the Fermi energy. This term is scaled by the inverse isoscalar effective mass. The second (interaction) term, is related to the isovector part of the Skyrme-EDF: $\delta {\cal H}_{t=1} = C_1^\rho \rho_1^2 + C_1^\tau \rho_1 \tau_1$ (for definitions, see Ref.[16] and references quoted therein).

The value of $E_{\mbox{\rm\scriptsize {sym}}}^{\mbox{\rm\scriptsize {(int)}}}$ appears to be mainly sensitive to the interaction term, which is illustrated in Fig. 4. Indeed, despite the fact that SLy4 and SV EDFs have similar values of $a_{\mbox{\rm\scriptsize {sym}}}$ (equal to 32MeV and 32.8MeV, respectively), the corresponding energy splittings $E_{\mbox{\rm\scriptsize {sym}}}^{\mbox{\rm\scriptsize {(int)}}}$ differ substantially. The reduced values of $\vert E_{\mbox{\rm\scriptsize {sym}}}^{\mbox{\rm\scriptsize {(int)}}} \vert$ in SV are due to its small value of $a_{\mbox{\rm\scriptsize {sym}}}^{\mbox{\rm\scriptsize {(int)}}} = 1.4$MeV,¹ which is an order of magnitude smaller than the corresponding SLy4 value: $a_{\mbox{\rm\scriptsize {sym}}}^{\mbox{\rm\scriptsize {(int)}}} = 14.4$MeV.

An interesting aspect of our analysis of $E_{\mbox{\rm\scriptsize {sym}}}^{\mbox{\rm\scriptsize {(int)}}}$ relates to its dependence on the time-odd terms, which are poorly constrained for Skyrme EDFs. To quantify this dependence, we have performed calculations by using the SLy4$_L$ and SkM$^*_L$ functionals, which have the spin coupling constants adjusted to the Landau parameters.[31,32] These EDFs have different values of $a_{\mbox{\rm\scriptsize {sym}}}$ but the same $a_{\mbox{\rm\scriptsize {sym}}}^{\mbox{\rm\scriptsize {(int)}}} = 14.4$MeV. The similarity of the calculated energy splittings shown in Fig. 4 confirms that this quantity primarily depends on the isovector terms of the functional. Moreover, its significant dependence on the time-odd terms opens up new options for adjusting the corresponding coupling constants to experimental data. This will certainly require the simultaneous restoration of isospin and angular-momentum symmetries, as presented in this study.