Symmetry restoration in mixed-spin paired heavy nuclei

The nature of the nuclear pairing condensates in heavy nuclei, specifically neutron-proton (spin-triplet), versus identical-particle (spin-singlet) pairing has been an active area of research for quite some time. In this work, we probe three candidates that should display spin-triplet, spin-singlet, and mixed-spin pairing. Using theoretical approaches such as the gradient method and symmetry-restoration techniques, we find the ground state of these nuclei in Hartree–Fock–Bogoliubov theory and compute ground state to ground state pair-transfer amplitudes to neighboring isotopes while simultaneously projecting to specific particle number and nuclear-spin values. We identify specific reactions for future experimental research that could shed light on spin-triplet and mixed-spin pairing.

Figure 1

Two-dimensional probability distributions, Eq. (9), with the projection operator in Eq. (10). Dotted line represents $A=132$ (a) ${}_{66}{}^{132}\mathrm{Dy}$, (b) ${}_{64}{}^{132}\mathrm{Gd}$, (c) ${}_{60}{}^{132}\mathrm{Nd}$.

Figure 2

Two-dimensional probability distribution for ${}_{64}{}^{132}\mathrm{Gd}$ subject to pairing constraints. Dotted line represents $A=132$ (a) No spin-singlet pairing, (b) No spin-triplet pairing.

Figure 3

$I_0$ probability distribution, Eq. (9), with the projection operator in Eq. (11), for ${}_{66}{}^{132}\mathrm{Dy}$ (black circles), ${}_{64}{}^{132}\mathrm{Gd}$ (red stars), and ${}_{60}{}^{132}\mathrm{Nd}$ (blue diamonds).

Figure 4

$I_0$ probability distribution, as in Fig. 3, for ${}_{64}{}^{132}\mathrm{Gd}$ subject to no constraint (red stars), no spin-singlet (black circles), and no spin-triplet (blue diamonds) pairing.

Figure 5

Two-dimensional probability distributions for the three isotopes for $I_0=0$, Eq. (9) with the projection operator in Eq. (12), without any constraint on pairing. Dotted line represents $A=132$ (a) ${}_{66}{}^{132}\mathrm{Dy}$, (b) ${}_{64}{}^{132}\mathrm{Gd}$, (c) ${}_{60}{}^{132}\mathrm{Nd}$.

Figure 6

Two-dimensional probability distributions for the three isotopes for $I_0=1$, Eq. (9) with the projection operator in Eq. (12), without any constraint on pairing. Dotted line represents $A=132$ (a) ${}_{66}{}^{132}\mathrm{Dy}$, (b) ${}_{64}{}^{132}\mathrm{Gd}$, (c) ${}_{60}{}^{132}\mathrm{Nd}$.

Figure 7

$I_0$ probability distribution for ${}_{66}{}^{132}\mathrm{Dy}$ (black circles), ${}_{64}{}^{132}\mathrm{Gd}$ (red stars), and ${}_{60}{}^{132}\mathrm{Nd}$ (blue diamond). Each isotope has been projected to the respective target neutron and proton numbers.

Figure 8

Transfer amplitudes for pair addition processes. Blue circles correspond to $pp$ transfer; red squares correspond to $nn$ transfer, and green triangles correspond to $np$ pair transfer (a) Final isotope is ${}_{66}{}^{132}\mathrm{Dy}$, (b) Final isotope is ${}_{64}{}^{132}\mathrm{Gd}$, (c) Final isotope is ${}_{60}{}^{132}\mathrm{Nd}$.