Validity of the pseudospin classification of normal-parity quasiparticle states in rotating nuclei at high rotational frequency

In the classic 1982 analysis of Bohr, Hamamoto, and Mottelson, quasiparticle motion in a rotating (cranked) deformed nuclear potential with pairing is organized by pseudospin quantum numbers: for rotational frequencies up to about 400 keV the pseudospin aligns with the rotation axis, quasiparticle states are classified by the pseudospin projection $\tilde{\Sigma}_1=\pm 1/2$, and the signature of the lowest quasiparticle state obeys the rule $r=-i(-1)^{\tilde{N}}$. The analysis rests on the $\Delta v=0$ approximation, which drops pairing-field couplings that change the quasiparticle number by two, and neglects the pseudo-spin-orbit coupling and the admixture of the $j=N+1/2$ intruder state. The open question is whether the pseudospin classification remains useful at higher rotational frequencies, where $\Delta v=2$ interactions significantly mix quasiparticle states and the neglected corrections can no longer be ignored. No prior study is known to address this question: later work pursued superdeformation and pseudospin alignment (weak-pairing regime), the relativistic origin of pseudospin symmetry in static mean fields, and pseudo-SU(3) or Dirac-equation themes, leaving the high-frequency, strong-pairing regime of the original cranking analysis unexplored.

Background

Nuclei set into rapid rotation are described in the cranking framework, in which nucleons move independently in a deformed mean-field potential that rotates about an axis (taken as the 1-axis, perpendicular to the nuclear symmetry 3-axis) with angular frequency $\omega_{\mathrm{rot}}$. The one-body Hamiltonian in the rotating frame is $h_{\mathrm{sp}}-\lambda N-\omega_{\mathrm{rot}}j_1$, where $h_{\mathrm{sp}}$ is the static deformed single-particle Hamiltonian, $\lambda$ the Fermi energy (chemical potential), $N$ the particle-number operator, and $j_1$ the angular-momentum projection on the rotation axis. Pairing correlations (the short-range attractive interaction that couples nucleons into time-reversed pairs) are included through a pairing field, and the elementary excitations are quasiparticles obtained by a Bogoliubov (BCS-like) transformation.

Normal-parity single-particle orbits in a deformed shell — everything except the high-$j$ intruder orbits, the unique-parity orbitals that descend from the next higher major shell (such as $h_{11/2}$ and $i_{13/2}$) and lack a pseudospin partner — are strongly mixed in total angular momentum $j$ and are hard to label individually. Pseudospin symmetry, introduced in 1969 by Hecht and Adler and by Arima, Harvey, and Shimizu, provides a remedy: pairs of orbits $(n,l,j=l+1/2)$ and $(n-1,l+2,j=l+3/2)$ are nearly degenerate and combine into a pseudo-orbital angular momentum $\tilde{l}=l+1$ coupled to a pseudospin $\tilde{s}=1/2$, with pseudo-oscillator quantum number $\tilde{N}=N-1$ (here $N$ denotes the oscillator quantum number of the major shell, not the particle-number operator above). The effective pseudo-spin-orbit coupling — the residual coupling between $\tilde{l}$ and $\tilde{s}$ — is empirically weak, so the pseudospin is approximately conserved in static deformed potentials.

Bohr, Hamamoto, and Mottelson (1982) carried this scheme into the rotating frame. They showed that the Coriolis coupling $-\omega_{\mathrm{rot}}j_1$ contains a term $-\omega_{\mathrm{rot}}\tilde{s}_1$ acting directly on the pseudospin, and that for rotational frequencies $\hbar\omega_{\mathrm{rot}}$ up to about $0.4$ MeV this term dominates the weak pseudo-spin-orbit coupling, so the pseudospin aligns with the rotation axis: the projection of the pseudospin on the rotation axis, $\tilde{\Sigma}_1=\pm 1/2$, becomes an approximate good quantum number, states group into pseudospin doublets with total angular momentum $I=\tilde{R}\pm 1/2$ (where $\tilde{R}$ is the angular momentum carried by the collective rotation together with the pseudo-orbital motion), and the signature of the lowest quasiparticle state follows the rule $r=-i(-1)^{\tilde{N}}$. The signature is the eigenvalue $r=\exp\{-i\pi\alpha\}$ of a rotation by $180^\circ$ about the rotation axis — the quantum number that labels rotational bands in deformed nuclei. The rule and the alignment picture reproduce the patterns of the cranked shell-model spectra (quasiparticle spectra from the rotating mean-field calculation) of Bengtsson and Frauendorf (1979) for neutrons near the Fermi surface at neutron numbers $N_n\approx 91$ and $99$ in the rare-earth region.

This analysis rests on two controlled approximations. First, the quasiparticle Hamiltonian was truncated to its $\Delta v=0$ part (Eq. 9b of the source), i.e., pairing-field matrix elements connecting quasiparticle states whose quasiparticle number $v$ differs by two — couplings to the adjoint negative-energy quasiparticle states that admix three-quasiparticle components into the one-quasiparticle states — were dropped. This $\Delta v=2$ mixing is negligible for $\hbar\omega_{\mathrm{rot}}\lesssim 0.4$ MeV but, as the calculated spectra show (Figs. 2 and 4 of the source), becomes significant beyond that frequency, which in normally deformed rare-earth nuclei is precisely the band-crossing (backbending) region, where rotational bands built on different intrinsic configurations cross and the nuclear moment of inertia changes abruptly with spin. Second, the pseudo-spin-orbit coupling and the admixture of the $j=N+1/2$ intruder state, which are responsible for the small computed deviations of the pseudospin projections $\langle\tilde{s}_3\rangle$ and $\langle\tilde{s}_1\rangle$ (the expectation values of the pseudospin components along the symmetry 3-axis and the rotation 1-axis, respectively) from $\pm 1/2$, were neglected in deriving the signature rule and in the schematic model.

The unresolved target is what happens to the pseudospin classification in this higher-frequency region.

Problem Statement

Investigate the consequences of the pseudospin quantum numbers for quasiparticle motion — the motion of the elementary excitations of the paired rotating mean field — in rotating (cranked) deformed nuclear potentials at rotational frequencies beyond the region $\hbar\omega_{\mathrm{rot}}\lesssim 0.4$ MeV analyzed by Bohr, Hamamoto, and Mottelson (1982), in the regime where the $\Delta v=2$ pairing interactions — pairing-field couplings between quasiparticle states differing by two in the quasiparticle number, which admix three-quasiparticle components built on the adjoint negative-energy quasiparticle states — are significant. The question has two linked parts that jointly fix the domain of validity of the pseudospin classification. (1) Determine whether the pseudospin classification of normal-parity quasiparticle states remains useful when $\Delta v=2$ mixing is properly included, assessing each of the following three features of the classification: the alignment of the pseudospin along the rotation axis ($\tilde{\Sigma}_1=\pm 1/2$ as an approximate quantum number, with $\tilde{\Sigma}_1$ the pseudospin projection on the rotation axis), the signature rule $r=-i(-1)^{\tilde{N}}$ for the lowest quasiparticle state (the signature $r$ being the eigenvalue of a $180^\circ$ rotation about the rotation axis, and $\tilde{N}=N-1$ the pseudo-oscillator quantum number, with $N$ the major-shell oscillator quantum number), and the pseudospin-doublet structure of the low-lying quasiparticle spectrum. (2) Quantify how the previously neglected pseudo-spin-orbit coupling (the residual coupling between pseudo-orbital angular momentum and pseudospin) and the admixture of the $j=N+1/2$ intruder state (the unique high-$j$ orbital descending from the next oscillator shell) — the corrections responsible for the deviations of the pseudospin expectation values $\langle\tilde{s}_3\rangle$ and $\langle\tilde{s}_1\rangle$ (the pseudospin components along the nuclear symmetry axis and the rotation axis, respectively) from $\pm 1/2$ — modify the validity of the pseudospin quantum numbers at these higher frequencies. The analysis should treat the pairing field without the $\Delta v=0$ truncation and should cover the normally deformed regime (deformations typical of rare-earth ground states, not the extreme superdeformed shapes) of the source's own examples (neutrons near the Fermi surface at neutron numbers $N_n\approx 91$–99 in the rare-earth region, or an equivalent normal-deformation system), stating the frequency range and physical conditions over which the pseudospin classification holds and where it fails, together with the quantitative criterion used to judge that validity.

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