Odd-even staggering of binding energy for nuclei in the $sd$ shell

In this paper we study odd-even staggering phenomena of binding energy in the framework of the nuclear shell model for nuclei in the $sd$ shell. We decompose the USDB effective interaction into the monopole interaction and multipole (residual) interactions. We extract the empirical proton-neutron interaction, the Wigner energy, and the one-neutron separation energy using calculated binding energies. The monopole interaction, which represents the spherical mean field, provides contributions to the empirical proton-neutron interaction, the symmetry energy, and the Wigner energy. It does not induce odd-even staggering of the empirical proton-neutron interaction or the one-neutron separation energy. Isovector monopole and quadrupole pairing interactions and isoscalar spin-1 pairing interactions play a key role in reproducing an additional binding energy in both even-even and odd-odd nuclei. The Wigner energy coefficients are sensitive to residual two-body interactions. The nuclear shell structure has a strong influence on the evolution of the one-neutron separation energy, but not on empirical proton-neutron interactions. The so-called three-point formula is a good probe of the shell structure.

Figure 1

Differences (in MeV) between the binding energies calculated by the shell model with the USDB interaction and those calculated with $H_m$ and $H_{mQ}$. One sees that the binding energy calculated with $H_m$ is close to that calculated with the USDB interaction for spherical nuclei but not for deformed nuclei.

Figure 2

Integrated proton-neutron interactions, $V_{\mathrm{pn}}$, extracted from experimental data on binding energies and those obtained by the shell model with the USDB interaction and $H_m$. The straight line is plotted according to $V_{\mathrm{pn}}^{\mathrm{SM}}=V_{\mathrm{pn}}^{\mathrm{Expt}.}$. $H_m$ provides an important contribution to $V_{\mathrm{pn}}$.

Figure 3

$\delta V_{1\mathrm{p}-1\mathrm{n}}$ for nuclei in the $sd$ shell.

Figure 4

$V_{\mathrm{pn}}^{\prime }$ versus neutron number $N$ for F, Ne, Na, and Mg isotopes.

Figure 5

$\delta V_{2\mathrm{p}-2\mathrm{n}}$ for nuclei in the $sd$ shell.

Figure 6

Wigner energy coefficients, $W, d$, and $X$, for nuclei in the $sd$ shell. (a) and (b) are the same except that the interactions are different.

Figure 7

One-neutron separation energy, $S_{\mathrm{n}}$, and the three-point formula [Eq. (10)], ${\mathrm{\Delta }}_{\mathrm{n}}^{\left(3\right)}$, for O isotopes in the $sd$ shell.