Partial conservation of seniority in semi-magic nuclei

TL;DR

This work examines seniority as a unifying quantum number for classifying nuclear states and explains why exact seniority conservation holds for simple single-$j$ shells but can fail for higher-$j$ orbitals. It highlights partial seniority conservation in the $j=9/2$ shell, where two $v=4$ states with $I=4$ and $I=6$ remain unmixed under arbitrary two-body interactions, backed by analytical proofs and symbolic-shell-model results. The review connects the algebraic structure of CFPs and quasi-spin symmetry to solvability, discusses the role of cross-shell mixing and three-body effects in real nuclei, and surveys experimental evidence from Pb, Ni, and $N=50$ isotones. It further discusses the interplay between seniority and spin-aligned np coupling, and outlines future directions, including advanced measurements at FRIB/FAIR and developments in symbolic-shell-model methodologies. Overall, the paper underscores partial dynamical symmetry as a robust lens for understanding pairing and transitions in semi-magic nuclei and guides ongoing experimental and theoretical efforts to map its domain of validity.

Abstract

The concept of seniority plays a central role in nuclear structure physics by classifying many-body states according to the number of unpaired nucleons. While exact seniority conservation holds in single-$j$ systems with $j \leq 7/2$, deviations arise for higher-$j$ orbitals where residual interactions can mix states of different seniority. Surprisingly, certain states in systems with $j \geq 9/2$ exhibit partial conservation of seniority, remaining solvable even when the symmetry is expected to break. This paper reviews the theoretical foundation of the seniority scheme, its connection to pairing interactions and coefficients of fractional parentage, and the conditions under which solvability persists. Particular emphasis is placed on the $j=9/2$ case, where two $v=4$ states with $I=4$ and $I=6$ remain unmixed under arbitrary interactions. We discuss analytical proofs of their existence, numerical studies, and supporting experimental evidence from semi-magic nuclei across five regions of the nuclear chart. Extensions to symbolic shell-model approaches are also presented, highlighting their utility in exploring wave functions and symmetries in many-body systems.

Figures (21)

• Figure 1: Typical spectral patterns for even-even atomic nuclei: (a) Seniority spectrum characterized by relatively high $2^+$ energy and suppressed excitation energies for other yrast states. The $J=2j-1$ state with maximum spin for a pair often appears as an isomeric state due to the small transition energy and nearly-vanishing transition strength sym14122680. These states are also described as non-collective states; (b) Surface vibrations with equally spaced spectrum; (c) Triaxially soft rotor; and (d) Well-defined collective rotation from deformed nuclei. Those three groups are usually associated with collective motions. The ratio of excitation energies of the $2_1^+$ and $4_1^+$ states, $R_{4/2}$, is typically $< 2$ for a seniority spectrum, about $2$ for vibrational nuclei, and close to $3.33$ for rotational nuclei. The ratio of E2 transition strengths, $B_{4/2}$, is mostly $>1$ for collective motions but is near zero for seniority symmetry, as transitions between states of the same seniority are almost forbidden.
• Figure 2: A schematic comparison between the spectrum of a single-$j$ even-$n$ system with no interaction (left), a $J=0$ pairing interaction (middle), and a realistic shell-model effective interaction (right). In the non-interacting case, all states with different spins are degenerate, while with strong pairing, the $I=0$ state with $v=0$ emerges as the ground state. Meanwhile, $v=2$ states with even angular momenta $I=2,4,\dots,2j-1$ can be formed by coupling two unpaired particles. Levels are colored to indicate $v=2$ broken-pair states, with the isomeric state highlighted in a different color. Higher-seniority states appear (mostly at higher energies) when more pairs are broken.
• Figure 3: Illustration of the nuclear shell structure up to $N/Z=82$, characterized by strong spin-orbit splitting and $jj$ coupling. This leads to a spin-zero ($v=0$) ground state for even-even nuclei and a spin-$j$ ($v=1$) ground state for odd-mass nuclei. Orbitals with higher $j$ values, highlighted in green, are of special interest and involve nuclei just above $N/Z=8$ and $20$, as well as those below or above $N/Z=50$ and $82$.
• Figure 4: Systematics of isomers with different angular momenta in even-even nuclei across the nuclear chart. Most $I=8^+$, $10^+$, and $12^+$ states are seniority isomers arising from the coupling of nucleons in the $j=9/2$ ($g/h$), $11/2$ ($h/i$), and $13/2$ ($i/j$) orbitals. Lower-spin seniority isomers with $I=6$ may occur in neutron-deficient Sn isotopes above $N=50$, neutron-rich Sn isotopes above $N=82$, and in $N=28$ isotones. Red circles highlight regions dominated by the coupling of nucleons in various $j=9/2$ orbitals, including Pb and Ni isotopes and the $N=50$ isotones, which have been extensively studied recently. Courtesy of Wenqiang Zhang.
• Figure 5: Illustration of the low-lying states for an $n=4$ system in a $j=9/2$ orbital and the associated E2 cascade. In the seniority scheme, transitions with $\Delta v = 0$ are typically weak, whereas the three $\Delta v = 2$ transitions can be relatively strong. The two $v = 4$ states, labeled as $\alpha$, exhibit partial seniority conservation. These states are expected to be low-lying, though their relative positions depend on the effective interaction and the nuclear region. In addition, there are another set of $v = 4$, $I=4,6$ states which are expected to lie at substantially higher energy.