Spectra and symmetry in nuclear pairing

We apply the algebraic Bethe ansatz technique to the nuclear pairing problem with orbit dependent coupling constants and degenerate single particle energy levels. We find the exact energies and eigenstates. We show that for a given shell, there are degeneracies between the states corresponding to less than and more than half full shell. We also provide a technique to solve the equations of Bethe ansatz.

Figure 1

Energy eigenvalues for $N=1\text{−}6$ pairs of nucleons in the first nuclear $\mathit{sd}$ shell. The coefficients $c_j$ and the pairing strength $\left|G\right|$ are those from Ref. [4]. Note the spectral symmetry discussed in the text and also the absence of the zero energy states for the nuclei corresponding to more than half full shell.

Figure 2

Excitation spectra for Ni isotopes and comparison with experimental results (in parentheses) [10].