Valence proton-neutron interactions throughout the mass surface

Empirical average proton-neutron interaction energies, $\delta V_{\mathit{pn}}$, between the last nucleons can be isolated using double differences of masses. We have examined the systematic behavior of $\delta V_{\mathit{pn}}$ throughout the mass surface using the 2003 mass table that includes many new and improved experimental masses. The results are especially revealing for self-conjugate nuclei and in regions of strong shell closures in heavy nuclei. In the former the large $p\text{−}n$ interaction strength can be interpreted as a consequence of the $\mathrm{T}=0$ interaction between protons and neutrons in spatially similar orbitals. In the latter, the bifurcated systematic can be understood in terms of the evolution of proton and neutron orbital overlaps in regions surrounding a shell closure. In regions between shells, anomalies are sometimes encountered that are not fully understood. They might reflect structural effects or could arise from one or more erroneously measured masses. A scheme based on fractional shell filling is presented that may serve as a signature of shell structure in exotic nuclei. A link between empirical $p\text{−}n$ interactions and growth rates of collectivity is pointed out. Finally, our analysis is used to identify candidates for future mass measurements and their needed levels of accuracy, many of which will require new exotic beam facilities.

Figure 1

$\delta V_{\mathit{pn}}$ values in the region of self-conjugate, $Z=N$, nuclei. Empirical $\delta V_{\mathit{pn}}$ values are negative (the empirical $p\text{−}n$ interaction is predominantly attractive). For simplicity, here and in the following figures, we show $\left|\delta V_{\mathit{pn}}\right|$. Only masses based on experimental data from Ref. 8 are used to compute $\delta V_{\mathit{pn}}$ and only values with errors <50 keV are shown.

Figure 2

(Top) $\delta V_{\mathit{pn}}$ values for even-even nuclei near ${}^{208}\mathrm{Pb}$. (Middle and bottom) Similar to the top panel but for even-Z, odd-N nuclei and odd-Z, even-N nuclei, respectively.

Figure 3

(Top) $\delta V_{\mathit{pn}}$ values for even-even nuclei near ${}^{132}\mathrm{Sn}$. (Middle and bottom) Similar to the top panel but for even-Z, odd-N nuclei and odd-Z, even-N nuclei, respectively.

Figure 4

(Top) $\delta V_{\mathit{pn}}$ values for even-even nuclei near N~50. (Middle and bottom) Similar to the top panel but for even-Z, odd-N nuclei and odd-Z, even-N nuclei, respectively.

Figure 5

(Top) $\delta V_{\mathit{pn}}$ values for even-even nuclei in the rare-earth region. (Bottom) Similar to the top panel but for odd-Z, even-N nuclei.

Figure 6

$\delta V_{\mathit{pn}}$ values for Ra isotopes and $Q_{\alpha }$/10 for Rn and Ra isotopes.

Figure 7

(Top) Empirical $\delta V_{\mathit{pn}}$ values as a function of both $(Z,N)$ and f${}_p$ and f${}_n$, the fractional filling for the $\mathrm{Z}=50$–82, $\mathrm{N}=82$–126 shells. All $\delta V_{\mathit{pn}}$ values with errors less than 125 keV are shown. Shading gives the magnitude of $\delta V_{\mathit{pn}}$. (Based on Refs. 9, 11). (Bottom) $R_{4/2}$ systematic against N${}_p{N}_n$ for particle-particle (pp) ($\mathrm{Z}=50$–66, $\mathrm{N}=82$–104) and particle-hole (ph) ($\mathrm{Z}=68$–82, $\mathrm{N}=82$–104) regions. (Based on Ref. 11.)

Figure 8

Calculated $\delta V_{\mathit{pn}}$ values using a δ function interaction [20], for two different assumptions of the order of filling of the neutron shell. Darker shading corresponds to larger $\delta V_{\mathit{pn}}$ values. (Top) Protons and neutrons with normal shell ordering ($\mathrm{Z}=50$–82, $\mathrm{N}=82$–126). The neutron shell is shown at the right. (Bottom) Scenario for neutron-rich nuclei that have 70–112 magic numbers, whereas protons are filling the same orbits as on the top. The ordering of the neutron orbits is shown at the right.