Odd-even staggering in the neutron-proton interaction and nuclear mass models

In this paper we study odd-even staggering of the empirical neutron-proton interaction between the last neutron and the last proton, denoted as $\delta V_{1n-1p}$, and its consequence in the Garvey-Kelson mass relations (GKs) and nuclear mass models. The root-mean-squared deviations of predicted masses respectively for even-$A$ and odd-$A$ nuclei by using two combinatorial GKs suggest a large odd-even staggering of $\delta V_{1n-1p}$ between even-odd and odd-even nuclei, while the odd-even difference of $\delta V_{1n-1p}$ between even-even and odd-odd nuclei is much smaller. The contribution of the odd-even staggering of $\delta V_{1n-1p}$ between even-$A$ and odd-$A$ nuclei in deviations of theoretical $\delta V_{1n-1p}$ values of the Duflo-Zuker model and the improved $\mathrm{Weizs}\ddot{\mathrm{a}}\mathrm{cker}\text{−}\mathrm{Skyrme}$ model are well represented by an isospin-dependent term. The consideration of this odd-even staggering improves our description of binding energies and one-neutron separation energies in both the Duflo-Zuker model and the improved $\mathrm{Weizs}\ddot{\mathrm{a}}\mathrm{cker}\text{−}\mathrm{Skyrme}$ model.

Figure 1

$\Delta (N,Z)$ for four $(N,Z)$ parities (even-even, even-odd, odd-even, and odd-odd) vs mass number $A$. Here we take the AME2012 database [9] with $N>Z$. Panel (a) presents $\Delta (N,Z)$ ($=\delta V\left(\mathrm{ee}\right)-\overline{\delta V\left(\mathrm{oo}\right)}$) of nuclei with even $N$ and even $Z$, panel (b) presents $\Delta (N,Z)$ ($=\delta V\left(\mathrm{oo}\right)-\overline{\delta V\left(\mathrm{ee}\right)}$) of nuclei with odd $N$ and odd $Z$, panel (c) presents $\Delta (N,Z)$ ($=\delta V\left(\mathrm{eo}\right)-\overline{\delta V\left(\mathrm{oe}\right)}$) of nuclei with even $N$ and odd $Z$, and panel (d) presents $\Delta (N,Z)$ ($=\delta V\left(\mathrm{oe}\right)-\overline{\delta V\left(\mathrm{eo}\right)}$) of nuclei with odd $N$ and even $Z$. One sees in panels (a) and (b) that the distribution of $\Delta (N,Z)$ is nearly equal with respect to zero, while in panels (c) and (d) $\Delta (N,Z)$ deviates from zero systematically. The mean values of $\Delta (N,Z)$ in panels (a)–(d) are 19, $-22$, 60, and -60 keV, respectively.

Figure 2

The RMSDs of predicted masses. Panels (a) and (b) are for even $A$, and panels (c) and (d) are for odd $A$. Results obtained by two combinatorial GKs are labeled by set (i), those by two combinatorial “modified” GKs are labeled by set (ii), and those by considering the pairing energy of Refs. [6, 31] are labeled by set (iii). We consider here nuclei with $N>Z$ and $A\ge 60$ in the AME2012 database [9]. One sees sizable differences of the RMSDs between the first and third cases of set (i) in panels (a) and (d) and much smaller differences in panels (b) and (c). They are reflections of the odd-even difference of $\delta V_{1n-1p}$ between even-odd and odd-even nuclei and that between even-even and odd-odd nuclei, respectively. See the text for details.

Figure 3

Deviations of theoretical $\delta V_{1n-1p}$ values of the DZ and WS models from experimental data [9] without and with improvement of Eq. (5) for $N>Z$. In panels (a) and (b) [(c) and (d)], theoretical nuclear masses are taken from Ref. [3] (Ref. [7] with the radial basis function). Deviations of theoretical values without improvement (denoted here as $\delta V^{\left(\mathrm{th}1\right)}$) are presented in panels (a) and (c), and those of theoretical values with improvement (denoted here as $\delta V^{\left(\mathrm{th}2\right)}$) are presented in panels (b) and (d).