Partial conservation of seniority in semi-magic nuclei

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.

💡 Research Summary

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The paper provides a comprehensive review of the phenomenon of partial seniority conservation in semi‑magic nuclei, focusing on the special case of a single‑j shell with j = 9/2. Seniority (v), originally introduced by Racah in atomic spectroscopy, counts the number of particles that are not coupled into J = 0 pairs. In nuclear physics, for identical nucleons confined to a single‑j orbit, a strong monopole pairing interaction drives the ground state to v = 0, while excited states with v = 2, 4, 6,… appear at higher energies. It is well‑known that for j ≤ 7/2 the seniority quantum number is an exact symmetry of any two‑body interaction; for j ≥ 9/2 the symmetry is generally broken because the two‑body matrix elements can mix states of different seniority.

Surprisingly, however, in the j = 9/2 shell there exist two v = 4 states—one with total angular momentum I = 4⁺ and the other with I = 6⁺—that remain pure seniority eigenstates for arbitrary two‑body interactions. This “partial conservation” means that while the full spectrum does not respect seniority, a specific subspace does. The authors present two complementary proofs. The first is an algebraic demonstration based on coefficients of fractional parentage (CFP) and the quasi‑spin SU(2) algebra: the v = 4 space in the j = 9/2 shell can be decomposed into two orthogonal basis vectors, and one of them is uniquely associated with I = 4⁺ and I = 6⁺. The second proof uses a symbolic shell‑model framework in which the two‑body matrix elements are treated as symbolic parameters. By generating the CFPs and 6‑j symbols symbolically (e.g., with Mathematica or Maple), the authors show that the eigenvectors corresponding to the I = 4⁺ and 6⁺ v = 4 states are independent of the values of the interaction parameters, confirming their invariance.

Experimental evidence for these unmixed states comes from semi‑magic nuclei where only one type of nucleon occupies the g₉/₂ or h₉/₂ orbital. In the N = 50 isotones (⁹²Mo, ⁹²Ru, ⁹²Pd) and the Z = 82 Pb isotopes (²¹⁰–²¹⁶Pb), the observed 4⁺ and 6⁺ levels lie at higher excitation energies than predicted by a simple seniority‑mixing picture, and the measured B(E2) transition strengths are dramatically reduced, often consistent with zero. These features are precisely what is expected when the v = 4 states are pure: electric‑quadrupole operators, which are even‑tensor operators, have vanishing matrix elements between states of the same seniority at mid‑shell, leading to long‑lived seniority isomers. The paper also discusses how these isomers manifest as “seniority isomers” with I = 2j − 1, emphasizing the role of small energy gaps and hindered E2 transitions.

Beyond the j = 9/2 case, the authors speculate that similar partial‑seniority invariants may exist for higher j (e.g., 11/2, 13/2) but the algebra becomes more intricate and the multiplicities increase. They argue that the symbolic shell‑model approach is ideally suited to explore such possibilities because it can handle the combinatorial explosion of CFPs and automatically test invariance against arbitrary interaction parameters. The paper also contrasts seniority coupling with the spin‑aligned neutron‑proton scheme, highlighting that while the latter can produce collective rotational‑like spectra, seniority provides a natural explanation for non‑collective, weak‑transition patterns observed in semi‑magic nuclei.

In the concluding sections, the authors summarize the status of experimental data, outline open questions (such as the need for precise lifetime measurements of the 4⁺ and 6⁺ states in additional isotopic chains, and the exploration of possible partial‑seniority states in mixed‑configuration shells), and propose future directions. These include extending the algebraic proofs to multi‑j spaces, integrating the symbolic shell‑model into large‑scale shell‑model codes, and employing modern ab‑initio interactions to test the robustness of the partial seniority phenomenon. Overall, the paper establishes that partial seniority conservation is a robust, symmetry‑based feature of the nuclear many‑body problem, offering both a diagnostic tool for interpreting experimental spectra and a benchmark for testing nuclear interaction models.