Neutron-proton pairing reexamined

We reexamine neutron-proton pairing as a phenomenon that should be explainable in a microscopic theory of nuclear binding energies. Empirically, there is an increased separation energy when both neutron and proton numbers are even or if they are both odd. The enhancement is present at some level in nearly all nuclei: the separation energy difference has the opposite sign in less than 1% of the cases in which sufficient data exist. We discuss the possible origin of the effect in the context of density functional theory (DFT) and its extensions. Neutron-proton pairing from mean-field theory does not seem promising to explain the effect. Gao and Chen have argued that a significant part of the increased binding in odd-odd deformed nuclei might arise as a recoupling energy, and we find a similar result for spherical nuclei. This suggests that the DFT should be extended by angular momentum projection to reach an accuracy capable of treating this effect.

Figure 1

Neutron-proton pairing effect as seen in the neutron separation energy for $N=28$ as a function of proton number $Z$. There is a consistent offset of the separation energies of odd-$Z$ nuclei as compared with the average of the neighboring even-$Z$ nuclei.

Figure 2

Nuclei with measured proton separation energy differences $S_{p2n}$ showing the cases (black squares) with opposite sign from the normal.

Figure 3

$\left|S_{p2n}\right|$ as a function of $A$. The line shows the $A^{-2/3}$ fit Eq. (3) with $K^{\text{'}}=7.3$ MeV.

Figure 4

Chart of nuclides showing those that have $\left|S_{p2n}\right|$ values higher than the average trend $\left|S_{p2n}\right|=7.3/A^{2/3}$ MeV.

Figure 5

Scatter plot of ${\delta }_s$, Eq. (3), compared with $S_{p2n}$ (solid squares) and $S_{n2p}$ (circles). The nuclei plotted are ${}^{40}\mathrm{K}$, ${}^{48}\mathrm{Sc}$, ${}^{50}\mathrm{Sc}$, ${}^{56}\mathrm{Co}$, ${}^{60}\mathrm{Co}$, ${}^{58}\mathrm{Cu}$, ${}^{208}\mathrm{Tl}$, ${}^{208}\mathrm{Bi}$, and ${}^{210}\mathrm{Bi}$.