Domain of validity of pair-aligned ($J=0,2$-pair) descriptions of nuclear quadrupole deformation: the spherical–deformed phase boundary, the quadrupole pairing strength $G_2$, and the extension beyond the schematic single-$j$ shell

Bohr and Mottelson (Physica Scripta 22, 468, 1980) asked when the quadrupole deformation of an atomic nucleus is generated by the alignment of correlated nucleon pairs with angular momenta $J=0$ and $J=2$ (the fermionic content of the $s$ and $d$ bosons of the interacting boson model, IBM) rather than by the alignment of individual particles in a deformed mean field. For a schematic single-$j$-shell $(j)^n$ model with monopole pairing strength $G_0$, quadrupole pairing strength $G_2$, and quadrupole particle–hole coupling $\kappa$, they derived the instability boundary of the spherical shape and concluded that the pair-aligned regime requires $\kappa\zeta \ll G_0\Omega$ (with $\zeta = 4(N/\Omega)(1-N/\Omega)$), while empirical estimates ($G\Omega\approx 2$ MeV, $\Delta\varepsilon\approx 8$ MeV, $G_2$ only tentatively $\sim G_0/2$ from $^{206}$Pb/$^{210}$Pb) place well-deformed nuclei far outside it. They left open the placement and order of the spherical-to-deformed transition line in realistic single-particle spaces, the empirical determination of $G_2$, and the extension beyond the schematic $(j)^n$ model to realistic multi-$j$ open-shell nuclei. The literature reports major partial progress — $S,D$-pair dominance in realistic deformed intrinsic states, systematic pair-truncation benchmarks, energy-density-functional mappings of the IBM, and critical-point-symmetry analyses of shape transitions — but no empirically anchored quantitative validity map: $G_2$ remains the least constrained ingredient, truncation errors are still benchmarked case by case, and transition-line placement remains model-dependent. The problem of delimiting, microscopically and empirically, where pair-aligned $J=0,2$ truncations capture the low-energy quadrupole collectivity of real nuclei remains open.

Background

Away from closed shells, many nuclei acquire permanently deformed quadrupole shapes and display rotational spectra. Two microscopic mechanisms can generate such deformation. The first, particle alignment, is the standard deformed mean-field picture: individual nucleons occupy the orbitals of a deformed single-particle potential of the Nilsson type (the standard deformed shell model) so that their quadrupole moments add coherently. The second, pair alignment, expresses the same deformed state as a condensate of correlated Cooper-like nucleon pairs, each pair carrying angular momentum $J=0$ or $J=2$; this is the fermionic content of the $s$ boson ($J=0$) and $d$ boson ($J=2$) of the interacting boson model (IBM) of Arima and Iachello, a widely used algebraic description of low-lying even-even nuclear spectra. The interplay with pairing is central: the short-range attractive interaction couples time-reversed orbits into $J=0$ pairs (the seniority/BCS scheme, where BCS is the Bardeen–Cooper–Schrieffer theory of superconductivity adapted to nuclei and seniority counts nucleons not coupled to $J=0$ pairs), and a quadrupole ($J=2$) pairing interaction couples pairs with aligned intrinsic quadrupole shapes. Bohr and Mottelson analyzed this competition for a schematic configuration of $n=2N$ identical particles in a single $j$-shell (a shell of total angular momentum $j$), with degeneracy $\Omega = j+1/2$, a monopole pairing strength $G_0$, a quadrupole pairing strength $G_2$, and a quadrupole particle–hole field of strength $\kappa$ (Physica Scripta 22, 468, 1980). Expanding the energy of a BCS-type condensate to order $\beta^2$ in an axial deformation (where $\beta$ is the dimensionless quadrupole deformation parameter), they found the spherical shape unstable beyond $\kappa_{\rm crit}\zeta/(G_0\Omega) = 1 - (G_2/G_0)(1-\zeta)$ with $\zeta = 4(N/\Omega)(1-N/\Omega)$, and concluded that the pair-aligned ($s,d$-boson) description is self-consistent only where $\kappa\zeta \ll G_0\Omega$, i.e. near the ends of a shell ($\zeta\ll 1$) with the quadrupole pairing nearly as strong as the monopole pairing ($G_2/G_0 \to 1$, i.e. $G_0-G_2 \ll G_0$). Confronting parameters with data — pair binding $G\Omega\approx(25\,\mathrm{MeV})A^{-1/2}\approx 2$ MeV from odd-even mass differences, quadrupole single-particle splitting $\Delta\varepsilon\approx(25\,\mathrm{MeV})\beta\approx 8$ MeV from observed deformations $\beta\approx 0.3$, and a tentative $G_2\sim G_0/2$ from the lowest $2^+$ states of $^{206}$Pb and $^{210}$Pb — they found $\Delta\varepsilon > G_0\Omega$ for strongly deformed nuclei ($A\sim 150$), placing them outside the pair-alignment regime. They further showed that the pair-aligned condensate breaks particle–hole symmetry (maximum band angular momentum $I_{\max}=2N$ versus $2N(\Omega-N)$ for particle alignment) and yields different signatures in one- and two-nucleon transfer, intrinsic quadrupole moments, and band termination (the ending of a rotational band when the valence angular momentum is exhausted). The paper itself flagged what remained unresolved: the phase-transition line between spherical and deformed shapes was located from a continuous-instability analysis, while a first-order transition would shift the line and make its placement sensitive to the detailed single-particle space; the empirical information on $G_2$ was far weaker than for $G_0$, so the boundaries of the pair-alignment region were uncertain; and the whole analysis used a schematic single-$j$ $(j)^n$ configuration, not the realistic open-shell nuclei with several $j$-shells where configuration mixing and higher-multipole interactions enter.

Problem Statement

Determine, for realistic open-shell nuclei with several valence $j$-shells and empirically constrained effective interactions, the region in which the quadrupole-deformed ground state and its low-lying collective bands are described by pair alignment — condensates of $J=0$ and $J=2$ pairs (the $s,d$-boson space of the interacting boson model) — rather than by particle alignment in a deformed mean field. Concretely, following Bohr and Mottelson (1980): (i) map the boundary between spherical and deformed equilibrium shapes in the space of the monopole pairing strength $G_0$, the quadrupole pairing strength $G_2$, and the quadrupole particle–hole coupling $\kappa$, establishing where this transition is continuous (their $\beta^2$ instability analysis) and where it is first order, with the associated sensitivity of the line's placement to the detailed single-particle spectrum; (ii) determine the quadrupole pairing strength $G_2$ from empirical data well enough to fix the boundaries of the parameter region in which pair alignment is valid; and (iii) carry out the corresponding analysis beyond the schematic single-$j$ $(j)^n$ configuration, in realistic multi-shell valence spaces where configuration mixing and interactions of higher multipolarity enter. These three parts jointly serve one objective: a quantitative, empirically grounded statement of where the pair-aligned $J=0,2$ truncation is and is not a faithful description of low-energy quadrupole collectivity, with the location and order of the spherical-to-deformed shape transition as the central landmark, and with observables that distinguish the two alignment schemes — single-nucleon spectroscopic strengths, coherent two-nucleon transfer, intrinsic quadrupole moments, and rotational-band termination — as the confrontation standard.

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