Nucleon-pair approximations for low-lying states of even-even $N=Z$ nuclei

In this paper we study nucleon-pair correlations of low-lying states for two typical cases of even-even $N=Z$ nuclei. The first case is a single $g_{9/2}$ shell with eight valence nucleons. In this case we study $S$-pair, isoscalar spin-aligned-pair, isoscalar spin-one-pair approximations, with both schematic and realistic interactions. We find that electric quadrupole transition rates and electric quadrupole moments exhibit different patterns depending on the pair-truncated scheme. In the second case we study ground states of ${}^{20}\mathrm{Ne},{}^{24}\mathrm{Mg},{}^{32}\mathrm{S},{}^{36}\mathrm{Ar},{}^{44}\mathrm{Ti},{}^{48}\mathrm{Cr},{}^{60}\mathrm{Zn},{}^{64}\mathrm{Ge},{}^{92}\mathrm{Pd}$, and ${}^{96}\mathrm{Cd}$. For ground states of these nuclei, the $S$-pair approximation is reasonably good (though not as good as for semimagic nuclei); the isoscalar spin-one-pair approximation is not very good. The isoscalar spin-aligned pair approximation is good for ground states of ${}^{44}\mathrm{Ti}$ and ${}^{96}\mathrm{Cd}$, and not very good for ${}^{48}\mathrm{Cr}$ and ${}^{92}\mathrm{Pd}$. The effect of the spin-orbit coupling strength on pairing correlations is studied. Our calculations suggests that the spin-orbit coupling favors the $S$-pair correlation in ground states of $N=Z$ nuclei.

Figure 1

Nucleon-pair approximation for yrast $T=0$ states with spin $I=0,2,4,6,8$, for eight nucleons in the single $j=9/2$ shell with schematic Hamiltonian $H(a,b)=(1-a-b)V_{J=0}+a{V}_{J=2j}+b{V}_{J=1}$, where $a$ and $b$ are adjustable parameters between 0 to 1. (a) Overlaps between pair-truncated wave functions and the shell model wave functions. (b) $E2$ transition rates between yrast $T=0$ states. (c) electric quadrupole moments of yrast $T=0$ states. The abbreviations “LS”, “SA”, “SO1”, and “SO2” represent the lowest seniority scheme [see Eq. (3)], the isoscalar spin-aligned pair approximation [see Eq. (4)], the $P$ pair condensation [see Eq. (5)], and the subspace constructed by three $P$ pairs plus another isoscalar pair [see Eq. (6)], respectively.

Figure 2

Overlaps between wave functions of the “LS+SA+SO2” pair subspace and those of the shell model space. (a) Yrast $T=0$ states, and (b) the second lowest spin $I$ states with $T=0$. Abbreviations “LS”, “SA”, and “SO2” are the same as in Fig. 1.

Figure 3

Overlaps between wave functions of pair-truncated spaces and those of the shell model space for the lowest $I=0,2,...,8$ and $T=0$ states of eight nucleons in a single $j=9/2$ shell, by using effective interactions (a)–(j). The abbreviations “LS”, “SA”, and “SO2” are the same as in Fig. 1.

Figure 4

The yrast states of $T=0$ for eight nucleons in the $j=9/2$ shell by using effective interaction (a). “SM” represents the shell model, and the abbreviations “LS”, “SA”, and “SO2” are the same as in Fig. 1. All the level energies are plotted as relative energies with respect to the $0_1^{+}$ state energy obtained by the SM.

Figure 5

Overlaps between the pair-condensation wave function and the shell-model wave function, and differences between the ground state energy obtained by the pair condensation and that obtained by the shell model, for the ground state of $N=Z$ even-even nuclei by using effective interactions. Solid squares in black represent $S$ pair condensation; solid circles in red represent $P$ pair condensation; solid triangles in blue represent spin-aligned pair condensation. Hallow squares in black represent $S$ pair condensation in semimagic even-even nuclei.

Figure 6

Same as Fig. 5 but for three Hamiltonians, the original effective interaction $H_{\mathrm{eff}}$ [Eq. (7)], and two artificial Hamiltonians, one of which switches off the $T=0$ matrix elements and the other of which switches off the $T=1$ two-body matrix elements, i.e., $H=H_0+V_{\mathbb{T}=1}$ and $H_0+V_{\mathbb{T}=0}$.

Figure 7

Same as Fig. 5 except that the single-particle energies are reparametrized by $x_{\mathrm{SO}}$, a scale of spin-orbit coupling strength parameter.