Symmetry energies for $A=24$ and 48 and the USD and KB3 shell model Hamiltonians

Calculations in the $\mathit{sd}$ and $\mathit{pf}$ shells reported some time ago by Satuła et al. [Phys. Lett. B 407, 103 (1997)] are redone for an extended analysis of the results. As in the original work, we do calculations for one mass number in each shell and consider in each case the sequence of lowest energies for isospins 0, 2, and 4, briefly the symmetry spectrum. Following further the original work, we study how this spectrum changes when parts of the two-nucleon interaction are turned off. The variation of its width is explored in detail. A differential combination ${\epsilon }_{\text{W}}$ of the three energies was taken in the original work as a measure of the so-called Wigner term in semiempirical mass formulas, and it was found to decrease drastically when the two-nucleon interaction in the channel of zero isospin is turned off. Our analysis shows that the width of the symmetry spectrum experiences an equally drastic decrease, which can be explained qualitatively in terms of schematic approximations. We therefore suggest that the decrease of ${\epsilon }_{\text{W}}$ be seen mainly as a side effect of a narrowing of the symmetry spectrum rather than an independent manifestation of the two-nucleon interaction in the channel of zero isospin.

Figure 1

The energy $E\left(T\right)$ for $A=24$ and $T=0$, 2, and 4 calculated from the Hamiltonian USD as a function of the angular momenta $J_{\text{cut}}$ and $J_{\text{cut}}^{\prime }$. The increasing sequence of abscissa in the first part of the axis is $J_{\text{cut}}$, and the subsequent decreasing sequence is $J_{\text{cut}}^{\prime }$.

Figure 2

As Fig. 1 for $A=48$ and the Hamiltonian KB3.

Figure 3

Parameter ${\epsilon }_{\text{W}}$ and ratio $r_{\text{W}}$ extracted from the energies in Fig. 1.

Figure 4

Parameter ${\epsilon }_{\text{W}}$ and ratio $r_{\text{W}}$ extracted from the energies in Fig. 2.

Figure 5

Parameter ${\epsilon }_{\text{W}}$ and ratio $r_{\text{W}}$ extracted from the results in Table I of Ref. [5].