Robustness of microscopically inferred nuclear shape-phase-transition boundaries under complete beyond-mean-field treatment

Nuclear shape phase transitions — rapid changes of the equilibrium quadrupole or octupole shape along isotopic chains — are inferred from order-parameter observables (energy ratios, transition strengths, odd-even staggering) calculated with beyond-mean-field methods based on nuclear energy density functionals (EDFs): collective Hamiltonians with microscopically derived but approximate inertial functions, and interacting-boson Hamiltonians obtained by mapping constrained EDF mean-field energy surfaces. These identifications rest on approximations: axially symmetric octupole constraints, approximate mass parameters and moments of inertia, and the EDF-to-boson mapping itself, while fully microscopic symmetry-restored Generator Coordinate Method (GCM) configuration mixing remains too costly for systematic studies in medium-heavy and heavy nuclei. The problem is to determine the robustness of the inferred shape-phase-transition boundaries — their locations along isotopic chains and their sharpness — under these approximations: re-derive the transitional regions (e.g., the γ-soft/triaxial region around $^{76}$Ge and the octupole transition near $^{226}$Th) with complete microscopic configuration mixing that includes triaxial quadrupole and octupole degrees of freedom simultaneously where both are relevant; and quantify how finite particle number smooths the phase-transition discontinuities as a function of shell structure and correlations. Post-2016 work has built parts of the needed machinery but has not delivered this robustness determination.

Background

Atomic nuclei undergo quantum phase transitions (QPTs): as neutrons or protons are added one by one, the equilibrium collective shape can change rapidly between spherical, axially deformed, triaxial (non-axial), and reflection-asymmetric 'octupole-deformed' (pear-shaped) forms, with nucleon number playing the role of a non-thermal control parameter [Cejnar, Jolie, and Casten, Reviews of Modern Physics 82, 2155 (2010)]. Because a nucleus is a finite system of at most a few hundred fermions, the discontinuities of an idealized thermodynamic-limit transition are smoothed into rapid crossovers; how much of the discontinuity survives at a given particle number, and how this depends on the underlying single-particle shell structure and on correlations, is itself a quantitative question that has so far been studied mainly in algebraic boson models [Dusuel et al., Physical Review C 72, 011301 (2005); Williams et al., Physical Review C 81, 054306 (2010)].

The microscopic framework for these phenomena is nuclear energy density functional (EDF) theory. Self-consistent mean-field (SCMF) calculations with constrained mass multipole moments — the quadrupole deformation $\beta_2$, the triaxiality angle $\gamma$ (measuring deviation from axial symmetry; $\gamma=0$ prolate, $\gamma=60^\circ$ oblate), and the octupole deformation $\beta_3$ (measuring reflection asymmetry) — yield deformation energy surfaces. To compute spectra and transition rates one must go 'beyond mean field' (BMF): restore the symmetries broken by the mean field (angular momentum, particle number, and for octupole shapes parity) and mix the symmetry-breaking intrinsic states, most systematically with the Generator Coordinate Method (GCM), in which the nuclear wave function is a superposition of mean-field configurations labeled by the collective coordinates. Full GCM with angular-momentum and parity projection is computationally demanding, so for medium-heavy and heavy nuclei two economical surrogates are standard. The first is a collective Hamiltonian — the five-dimensional collective Hamiltonian for quadrupole shapes, or a quadrupole-octupole collective Hamiltonian (QOCH) for coupled quadrupole-octupole dynamics — whose potential is the microscopic energy surface and whose mass parameters and moments of inertia are computed approximately (e.g., by adiabatic cranking at the mean-field level). The second is the interacting boson model (IBM), an algebraic model in which valence nucleon pairs are represented by $s$ and $d$ bosons (quadrupole) plus $f$ bosons (octupole); in the EDF-mapped IBM the boson Hamiltonian parameters are fixed by mapping the constrained microscopic energy surface onto the corresponding bosonic surface.

The relevant order parameters are spectroscopic: the ratio $R_{4/2}=E(4_1^+)/E(2_1^+)$, $B(E2)$ transition strengths, the odd-even staggering $S(J)$ of the $\gamma$-band (a discrete second difference of $\gamma$-band energies whose phase distinguishes rigid-triaxial from $\gamma$-soft shapes), and for octupole transitions the band-head energy of the lowest negative-parity band. Two benchmark transitional regions anchor the question. In the germanium chain, empirical $\gamma$-band staggering suggested rigid triaxial deformation at low energy in $^{76}$Ge [Toh et al., Physical Review C 87, 041304 (2013)], while the EDF-based collective Hamiltonian of Nikšić, Marević, and Vretenar [Physical Review C 89, 044325 (2014)] found a $\gamma$-soft mean field whose correlations drive the nucleus toward triaxiality but are too weak to stabilize a $\gamma\approx 30^\circ$ shape, leaving $^{76}$Ge at the transition between triaxial $^{74}$Ge and the $\gamma$-soft heavier isotopes; a systematic EDF analysis [Nomura et al., Physical Review Letters 108, 132501 (2012)] had earlier shown that non-axial medium-heavy nuclei lie between the rigid-triaxial and $\gamma$-unstable geometrical limits. In the thorium chain, QOCH and EDF-mapped $sdf$-IBM studies [Li et al., Physics Letters B 726, 866 (2013); Nomura et al., Physical Review C 89, 024312 (2014)] identified a simultaneous quadrupole and octupole shape-phase transition near $^{226}$Th, where the negative-parity band-head energies form a parabola with its vertex at $^{226}$Th. Vretenar's 2016 review summarized this state of the art together with its limitations: GCM configuration mixing for medium-heavy and heavy nuclei 'presents a more involved and technically difficult problem'; the octupole energy surfaces were computed under the assumption of axial symmetry; and the inertial functions and moments of inertia that govern the collective dynamics are determined by the energy density functional and the effective pairing interaction and are obtained only approximately within these surrogates [Vretenar, EPJ Web of Conferences 117, 04002 (2016)]. The open question is whether the inferred phase-transitional boundaries and their sharpness are genuine predictions of the microscopic theory rather than artifacts of these approximations.

Problem Statement

Determine the robustness of microscopically inferred nuclear shape-phase-transition boundaries — their locations along isotopic chains and their sharpness as seen in order-parameter observables — under the approximations of current beyond-mean-field treatments. The single objective is a reliability statement about the inferred boundaries, supported by two linked deliverables. (1) Complete-correlation calculation: for the transitional nuclei — the γ-soft/triaxial region around $^{76}$Ge, where the relevant non-axial degree of freedom is triaxial quadrupole collectivity, and the quadrupole-octupole transitional region around $^{224}$Ra–$^{228}$Th, where triaxial quadrupole and octupole degrees of freedom are to be included simultaneously — compute low-energy spectra and transition rates (positive-parity for the Ge leg; positive- and negative-parity for the Ra–Th leg) with microscopic configuration mixing that restores the broken symmetries (angular momentum, parity, particle number), via the Generator Coordinate Method or an equivalent symmetry-projected treatment; and quantify, against the same EDF, how the order-parameter systematics (the energy ratio $R_{4/2}=E(4_1^+)/E(2_1^+)$, the reduced electric-quadrupole transition strengths $B(E2)$, the γ-band staggering $S(J)$, and the negative-parity band-head energies) and the resulting phase-boundary locations and sharpness differ from the axially constrained collective-Hamiltonian and EDF-mapped-IBM predictions. (2) Finite-size smoothing: quantify the degree to which the phase-transition discontinuities in these order parameters are washed out at finite particle number, including how the smoothing depends on valence-nucleon number, detailed shell structure, and correlations, distinguishing genuine finite-size smoothing from shell-structure-driven non-uniformities along isotopic chains. An acceptable answer establishes, for the transitional regions studied, whether and how the shape-phase-transition identifications change when the approximations are removed, and how the transition discontinuities are smoothed at finite particle number.

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