Microscopic origin of the first-order shape/phase transitions at N = 40, 60, and 90 and the quantitative link between self-consistent and collective-model potentials

Empirical systematics of energy ratios and B(E2) transition rates establish that even-even nuclei undergo first-order shape/phase transitions from spherical to deformed shapes near neutron numbers N=40 (Zn-Ge-Se), N=60 (Sr-Zr-Mo), and N=90 (Nd-Sm-Gd-Dy). Skyrme-Hartree-Fock+BCS calculations reproduce the changing topology of the potential-energy curves, and fits of the Algebraic Collective Model to experimental spectra yield collective potentials of the shapes used in critical-point descriptions. Three interlocking questions posed by the source work remain open: (i) whether the fitted collective potentials can be placed in quantitative correspondence with the self-consistent potential-energy curves, including zero-point corrections and a deformation-dependent mass tensor; (ii) what fixes the sign of the equilibrium deformation - the prolate tendency at N=90 versus oblate tendencies at N=40 and N=60 in self-consistent calculations - and how it can be pinned down through measured quadrupole-moment signs; and (iii) which mechanism drives the transition in the N=40 region, where the Federman-Pittel mechanism of deformation onset cannot apply because valence protons and neutrons occupy the same major shell.

Background

Nuclear shape/phase transitions. Even-even nuclei (with even numbers of protons and neutrons) can change their equilibrium shape abruptly as neutrons are added along an isotopic chain, passing from spherical to permanently deformed shapes in close analogy with a thermodynamic first-order phase transition. The transition is monitored through structural indicators: the energy ratio $R_{4/2}=E(4_1^+)/E(2_1^+)$, which grows from about 2.0 for a spherical harmonic vibrator to 3.33 for a well-deformed axial rotor; the ratio $R_{2/0}=E(2_1^+)/E(0_2^+)$, which exhibits a maximum in the region of the transition; and the reduced electric-quadrupole transition probability $B(E2;2_1^+\to 0_1^+)$, which is proportional to the square of the quadrupole deformation. Three regions of the nuclear chart are known loci of such first-order transitions at neutron numbers $N=40$ (Zn-Ge-Se chains), $N=60$ (Sr-Zr-Mo), and $N=90$ (Nd-Sm-Gd-Dy). In the $N=90$ isotones the indicators reach the parameter-free predictions of the X(5) critical point symmetry (an exactly solvable reduction of the Bohr Hamiltonian, the standard collective Hamiltonian for the quadrupole surface vibrations and rotations of the nucleus, at the critical point of the spherical-to-deformed transition), while the $N=40$ and $N=60$ regions show the more moderate deformations described by X(3), the variant of X(5) in which the $\gamma$ degree of freedom is frozen at zero (axially rigid shapes).

Two complementary theoretical descriptions are used for these regions. (i) Microscopic: Skyrme-Hartree-Fock plus Bardeen-Cooper-Schrieffer (SHF+BCS) calculations, a self-consistent mean-field method with the Skyrme effective interaction and BCS pairing, constrained on the quadrupole deformation produce potential-energy curves (PECs) as functions of the mass quadrupole deformation $\beta_2$ ($\beta_2>0$ prolate, rugby-ball shaped; $\beta_2<0$ oblate, pancake shaped; the angle $\gamma$ measures departure from axial symmetry). (ii) Macroscopic: the Algebraic Collective Model (ACM), a computationally tractable formulation of the Bohr-Mottelson collective model (the classic geometric model of the nucleus as a vibrating, rotating deformed drop) whose Hamiltonian is a polynomial in $\beta$ and $\cos 3\gamma$ with fourteen coefficients fitted to experimental spectra; its dynamical group structure - the Hamiltonian's building blocks close under the Lie groups $SU(1,1)$ and $SO(5)$ - makes all matrix elements analytic. In the source study, the SHF+BCS PECs display the phase coexistence (competing minima) characteristic of first-order transitions in all three regions, with a tendency towards prolate minima around $N=90$ and towards oblate minima around $N=40$ and $N=60$; for the $^{150}$Nd fit, the part of the ACM potential containing only positive powers of $\beta$ resembles the Davidson potential (the analytic form $a\beta^2+b/\beta^2$), while the total potential resembles the confined beta-soft model (an infinite square well in $\beta$ displaced from the origin), both used in analytic descriptions of shape transitions.

Deformation mechanisms. The classical microscopic explanation of the sudden onset of deformation is the Federman-Pittel mechanism: as valence neutrons enter orbitals of the major shell adjacent to that of the valence protons, the proton-neutron monopole interaction between spatially overlapping orbitals (same orbital angular momentum $l$, angular momenta $j=l\pm 1/2$) grows rapidly and drives quadrupole deformation. This picture presupposes that valence protons and neutrons occupy different, adjacent major shells, as at $N=60$ and $N=90$. At $N=40$ in the Zn-Ge-Se region ($Z=30$-34), however, valence protons and neutrons both fill the same major shell between the magic numbers 28 and 50, so the Federman-Pittel mechanism cannot operate as proposed, and the driver of the transition there is unidentified.

What is missing. Although each description is individually successful, the quantitative bridge between the microscopic PECs and the fitted collective potentials has not been established, and the microscopic origin of the region-dependent shape systematics (including the sign of the deformation) is not understood. These gaps are exactly the open questions formulated in the source work.

Problem Statement

The problem is to give a microscopic account of the first-order spherical-to-deformed shape/phase transitions in the $N=40$, $60$, and $90$ regions that is simultaneously consistent with the empirical indicators, the self-consistent potential-energy curves, and the collective potentials fitted to spectra. Three linked questions, posed as directions for further research by the source work, define its content:

1. Micro-macro correspondence. Establish whether and how the SHF+BCS potential-energy curves map quantitatively onto the ACM collective potentials $V(\beta,\gamma)$ fitted to experimental spectra - the positions and depths of minima, barrier heights, and $\gamma$-dependence - systematically across nuclei of all three regions. The correspondence must incorporate the vibrational and rotational zero-point corrections to the PECs, allow higher-order terms in $\beta^2$ and $\cos 3\gamma$ in the collective potential, and replace the rigid kinetic term of the ACM Hamiltonian by a deformation-dependent inertia derived from the microscopic mass tensor.

2. Origin of the prolate/oblate preference. Determine which single-particle level structure, pairing, and shell effects fix the sign of the equilibrium deformation - the prolate tendency at $N=90$ versus the oblate tendency at $N=40$ and $N=60$ found in the self-consistent calculations - whether these trends are robust under changes of the effective interaction or the pairing treatment, and whether they are consistent with the sign of the quadrupole moments measured for the key transitional nuclei (the observable that distinguishes prolate from oblate shapes, since spectra alone cannot).

3. Mechanism at $N=40$. Identify the microscopic driver of the shape/phase transitional behavior in the $N=40$ region, where the Federman-Pittel mechanism of adjacent-shell proton-neutron overlap is inapplicable because valence protons and neutrons occupy the same major shell: for example cross-shell (particle-hole) excitations, specific proton-neutron interactions, spin-isospin SU(4) supermultiplet correlations (the Wigner SU(4) symmetry, an approximate nuclear symmetry under which protons and neutrons with either spin direction are treated as equivalent), or correlation effects beyond static mean field.

The problem is completely answered when the identified mechanism(s) account, within a controlled microscopic framework, for the empirical systematics of the three transitions, the topology of the self-consistent PECs, and the fitted collective potentials on an equal footing across the three regions. All three questions are mandatory: a resolution must address each of them, and no single question alone resolves the task.

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