Proton pairing vibrational states around the doubly magic nucleus ${}^{208}\mathrm{Pb}$

Proton pairing vibrational states in ${}^{208}\mathrm{Pb}$ isotones with $76\le Z\le 88$ protons are described like neutron pairing vibrational states in lead isotopes with $120\le N\le 132$ neutrons within the model of pair addition and pair removal phonons. The anharmonicity calculated with three parameters reproduces the known energies within less than 5%. The interaction between the pairing phonons for protons and neutrons is similar in a striking manner.

Figure 1

Binding energies $E^{\mathrm{rel}}$ [Eq. (3)] of neutrons (solid lines with diamonds) and protons (dotted lines with crosses, reduced by a factor of 4) relative to ${}^{208}\mathrm{Pb}$.

Figure 2

Pairing vibration states determined by experiment (solid lines with diamonds) and calculated in the harmonic approximation [Eq. (6) with vanishing anharmonic terms $V_{+-},V_{++}$, and $V_{--}$, dotted lines] (a) for ${}^{208}\mathrm{Pb}$ isotones and (b) for lead isotopes (adapted from Fig. 6-62 in [14]). The dashed lines show the prediction in the anharmonic approximation [Eq. (6)]. The parameter $V_{+-}$ [Eqs. (6, 7, 8)] derives from the energies of the pairing state $\left[1_{-}{1}_{+}\right]$ shown at top.