Generalized seniority for the shell model with realistic interactions

The generalized seniority scheme has long been proposed as a means of dramatically reducing the dimensionality of nuclear shell-model calculations, when strong pairing correlations are present. However, systematic benchmark calculations, comparing results obtained in a model space truncated according to generalized seniority with those obtained in the full shell-model space, are required to assess the viability of this scheme. Here, a detailed comparison is carried out, for semimagic nuclei taken in a full major shell and with realistic interactions. The even-mass and odd-mass Ca isotopes are treated in the generalized seniority scheme, for generalized seniority $v⩽3$. Results for level energies, orbital occupations, and electromagnetic observables are compared with those obtained in the full shell-model space.

Figure 1

Pair amplitudes ${\alpha }_a$ for the collective $S$ pair (2), as determined variationally by minimization of $E_{\alpha }$ for the various possible minimal-seniority states: (a) the $v=0$ state ($J=0$), for the even-mass Ca isotopes, or the $v=1$ state based upon a quasiparticle in (b) the $f_{7/2}$ orbital ($J=\frac{7}{2}$), (c) the $p_{3/2}$ orbital ($J=\frac{3}{2}$), or (d) the $f_{5/2}$ orbital ($J=\frac{5}{2}$), for the odd-mass Ca isotopes. Calculations are shown for the FPD6 interaction.

Figure 2

Energy eigenvalue of the $J=0$ ground state, for the even-mass Ca isotopes, calculated in the generalized seniority $v=2$ model space (circles) or full shell-model space (crosses). The energies are considered directly as eigenvalues (top), as two-neutron separation energies $S_{2n}$ (middle), or as the residual difference $\Delta E$ of the generalized seniority result relative to the full shell-model result (bottom). Calculations are shown for the FPD6 (left) and GXPF1 (right) interactions.

Figure 3

Excitation energies $E_x$ of the lowest $J=2$ state (top), lowest $J=4$ state (middle), and first excited $J=0$ state (bottom) of the even-mass Ca isotopes, calculated in the generalized seniority $v=2$ model space (circles) or full shell-model space (crosses). Calculations are shown for the FPD6 (left) and GXPF1 (right) interactions.

Figure 4

(a) and (b) Energy eigenvalue of the lowest $J=\frac{7}{2}$ state, for the odd-mass Ca isotopes, calculated in the generalized seniority $v=1$ model space (open circles), generalized seniority $v=3$ model space (solid circles), or full shell-model space (crosses). (c)–(j) Residual differences $\Delta E$ of the generalized seniority result relative to the full shell-model result, for the energy eigenvalues of the lowest $J=\frac{7}{2}$, $\frac{5}{2}$, $\frac{3}{2}$, and $\frac{1}{2}$ states (top to bottom, respectively). Calculations are shown for the FPD6 (left) and GXPF1 (right) interactions.

Figure 5

Orbital occupations $\langle n_a\rangle$ of the $pf$-shell orbitals, for the lowest $J=0$, 2, and 4 states and first excited $J=0$ state (top to bottom, respectively) of the even-mass Ca isotopes, calculated in the generalized seniority $v=2$ model space (circles) or full shell-model space (crosses). Calculations are shown for the FPD6 (left) and GXPF1 (right) interactions.

Figure 6

Orbital occupations $\langle n_a\rangle$ of the $pf$-shell orbitals, for the lowest $J=\frac{7}{2}$ (top) and $J=\frac{5}{2}$ (bottom) states of the odd-mass Ca isotopes, calculated in the generalized seniority $v=3$ model space (circles) or full shell-model space (crosses). Calculations are shown for the FPD6 (left) and GXPF1 (right) interactions.

Figure 7

Electromagnetic observables for the even-mass Ca isotopes: $Q\left(2_1^{+}\right)$ (top), $B(E2;2_1^{+}\to 0_1^{+})$ (middle), and $\mu \left(2_1^{+}\right)$ (bottom), calculated in the generalized seniority $v=2$ model space (circles) or full shell-model space (crosses). Calculations are shown for the FPD6 (left) and GXPF1 (right) interactions.

Figure 8

Electromagnetic observables for the odd-mass Ca isotopes: $Q\left({\frac{7}{2}}_1^{-}\right)$ (top) and $\mu \left({\frac{7}{2}}_1^{-}\right)$ (bottom), calculated in the generalized seniority $v=1$ model space (open circles), generalized seniority $v=3$ model space (solid circles), or full shell-model space (crosses). Calculations are shown for the FPD6 (left) and GXPF1 (right) interactions.