Probing mixed-spin pairing in heavy nuclei

The nature of the nuclear pairing condensate is an active topic of investigation, especially as regards its neutron-proton versus identical-particle character, which manifests as the difference between spin-singlet and spin-triplet pairing. In this work, we probe the recently proposed mixed-spin pairing condensates, using a phenomenological Hamiltonian and Hartree-Fock-Bogoliubov theory along with the gradient method. In addition to improving the solution of the many-body problem, we have calculated a series of physical quantities and examined the robustness of the mixed-spin pairing state as the input Hamiltonian is modified. Overall, we find that even though the mixed-spin correlation energy is suppressed in comparison to earlier work, the new pairing behavior persists. We also discuss the possibility of directly probing the mixed-spin pairing phase.

Figure 1

Chart of nuclides with $N\ge Z$ from $N=50$ to $N=75$. Blank squares indicate nuclei with no pairing $(E_{\mathrm{corr}}<0.5$ MeV). Green squares denote a pairing condensate of mostly spin-singlet character, red diamonds those of a spin-triplet character, and blue circles indicate a mixture of the two types of pairing.

Figure 2

Heat maps showing the fluctuations in neutron number (left panel) and proton number (right panel) for heavy nuclei on and off the $N=Z$ line. They range from zero particles to a little over three particles.

Figure 3

Contour plots showing the correlation energy (in MeV) in three different nuclei on the $A=132$ line as a function of spin-singlet and spin-triplet pairing amplitudes. From left to right are ${}_{60}{}^{132}\mathrm{Nd},{}_{64}{}^{132}\mathrm{Gd}$, and ${}_{66}{}^{132}\mathrm{Dy}$. ${}_{60}{}^{132}\mathrm{Nd}$ exhibits mostly spin-singlet pairing, and ${}_{66}{}^{132}\mathrm{Dy}$ exhibits mainly spin-triplet pairing. ${}_{64}{}^{132}\mathrm{Gd}$ exhibits a mixed-state pairing that shows a smooth transition from spin-triplet to spin-singlet as $N-Z$ increases.

Figure 4

Pairing gaps for $55\le N\le 74$ and $N\ge Z$. Most nuclei have intermediate-size pairing gaps (green squares), though two lines exhibit reduced gaps (blue circles).

Figure 5

Correlation energies (in MeV) vs. varying $V_{SO}$ (in MeV ${\mathrm{fm}}^2)$. These are plotted for the same nuclei as in Fig. 3: ${}_{60}{}^{132}\mathrm{Nd}$ in the left panel, ${}_{64}{}^{132}\mathrm{Gd}$ in the middle panel, and ${}_{66}{}^{132}\mathrm{Dy}$ in the right panel. As in Fig. 1, red indicates spin-triplet, green spin-singlet, and blue a mixed state.

Figure 6

Chart of nuclides with $N\ge Z$ from $N=50$ to $N=75$ for $V_{SO}=0$. Notation is as in Fig. 1. A vanishing spin-orbit strength leads to the appearance of many new spin-triplet and mixed-spin nuclei.

Figure 7

Correlation energies (in MeV) vs. varying $v_s$, with $v_t/v_s$ ratios of 1.25, 1.4, 1.5, 1.6, and 1.75. On the left is ${}_{60}{}^{132}\mathrm{Nd}$, on the right is ${}_{66}{}^{132}\mathrm{Dy}$, and in the middle is ${}_{64}{}^{132}\mathrm{Gd}$. All plots indicate a change from spin-singlet (green) towards spin-triplet (red) possibly with an intermediate mixed state (blue) as the $v_t/v_s$ ratio increases, and an increase in correlation energy as $v_s$ increases.