Completing the validation and microscopic foundation of the empirical signatures for shape coexistence and shape/phase transitions in even-even nuclei

Bonatsos et al. (J. Phys. G 50, 075105 (2023)) proposed an empirical prescription for locating shape coexistence (SC) between the ground state band and a nearby excited K=0 band in even-even nuclei: candidates should lie within the stripes of proton/neutron numbers 7-8, 17-20, 34-40, 59-70, 96-112 predicted by the proxy-SU(3) dual shell mechanism (avoiding the deformed P≈5 contours at the stripe junctions) and satisfy the thresholds R_4/2 < 3.05, R_0/2 < 5.7, B_02 > 0.1; they further connected the two underlying particle-hole excitation mechanisms to first-order shape/phase transitions. The open problem is to complete the validation and microscopic grounding of this prescription: extend it to lighter masses along the N~Z line, acquire the missing B(E2; 0_2+ -> 2_1+) rates in the Hg, Pb and Po chains, explain the Pt exception, assess whether the thresholds are sufficient rather than merely practical, and quantify the orbital occupancies and configuration mixing that would make the conjectured causal link between intruder-orbital occupation and first-order transition behavior a demonstrated mechanism.

Background

Shape coexistence (SC) is the appearance, in one and the same nucleus, of two low-lying collective bands built on radically different intrinsic shapes - for example a near-spherical ground state band coexisting with a well-deformed excited band. In even-even nuclei the excited band of interest is usually a $K=0$ band (a band whose angular-momentum projection on the nuclear symmetry axis is zero, also called a beta band) headed by the first excited $0^+$ state, $0_2^+$. The phenomenon is reviewed by Heyde and Wood (2011), who mapped the regions ('islands') of the nuclear chart where SC is observed or expected, and from the experimental side by Garrett et al. (2022).

Bonatsos, Martinou, Peroulis, Mertzimekis and Minkov (2023) proposed an empirical prescription for locating SC of the ground state band with a nearby excited $K=0$ band. It rests on the proxy-SU(3) model, an approximation in which the intruder orbitals (orbitals pushed down into a lower shell by the spin-orbit interaction) are replaced by 'proxy' orbitals of similar Nilsson quantum numbers, so that Elliott's SU(3) symmetry (the approximate shell-model symmetry generated by the quadrupole-quadrupole interaction, organizing states into rotational bands) is restored in shells beyond the sd shell. Within this framework the dual shell mechanism - the interplay of the harmonic-oscillator magic numbers (2, 8, 20, 40, 70, 112, 168) and the spin-orbit magic numbers (6, 14, 28, 50, 82, 126) - predicts that SC can occur only within stripes of proton or neutron numbers 7-8, 17-20, 34-40, 59-70, 96-112 (extended to 146-168 in heavier shells), and that the junctions of these stripes, where highly deformed nuclei cluster, should be avoided. This is a necessary condition only. From systematics of all nuclei beyond $Z=N=18$ for which a $K=0$ band and the transition rate $B(E2;0_2^+\to2_1^+)$ are known ($B(E2)$ denoting the reduced electric-quadrupole transition probability), the authors distilled three empirical thresholds: the energy ratio $R_{4/2}=E(4_1^+)/E(2_1^+)$ below 3.05 (well-deformed rotors are excluded), the energy ratio $R_{0/2}=E(0_2^+)/E(2_1^+)$ below 5.7 (the band heads must lie close in energy), and the ratio $B_{02}=B(E2;0_2^+\to2_1^+)/B(E2;2_1^+\to0_1^+)$ above 0.1 (the two band heads must be strongly connected). Well-deformed regions are additionally carved out by the contours of the P factor, $P=N_pN_n/(N_p+N_n)$, where $N_p$ and $N_n$ are the numbers of valence protons and neutrons counted from the nearest closed shell; nuclei with $P>5$ are deformed and do not exhibit this type of SC.

The same paper connected SC to quantum shape/phase transitions (QPTs). In the interacting boson model (IBM), which describes even-even nuclei through $s$ and $d$ bosons (valence nucleon pairs coupled to angular momentum 0 and 2), a first-order phase transition separates the spherical U(5) dynamical symmetry from the axially deformed SU(3) symmetry; the X(5) critical-point symmetry of Iachello (2001) - an analytic solution of the Bohr collective Hamiltonian valid exactly at the phase-transition border - describes nuclei at this border, with the $N=90$ isotones $150$Nd, $152$Sm, $154$Gd as its best experimental realizations. Bonatsos et al. identified two distinct SC mechanisms: (a) neutron-induced SC, where neutrons induce proton particle-hole (p-h) excitations across the major proton shell closures $Z=82$ and $Z=50$ (the classical Hg-Pb and Sn-region coexistence); and (b) proton-induced SC, where protons induce neutron p-h excitations at $N=90$, 60, 40, where the intruder orbitals $1i_{13/2}$, $1h_{11/2}$, $1g_{9/2}$ begin to participate in the onset of deformation; in mechanism (b) a first-order spherical-to-prolate shape/phase transition occurs in parallel, as corroborated by covariant density functional theory calculations of the Bonatsos group (2022).

Several pieces of the construction are unfinished. First, the Pt isotopes form an acknowledged exception: they show clear shape evolution yet fall outside the proxy-SU(3) stripes, and their evolution is describable in the IBM without any p-h excitations (McCutchan, Casten and Zamfir 2005). Second, for the Hg, Pb and Po isotopes - among the best SC candidates in the chart - the key observable $B(E2;0_2^+\to2_1^+)$ is missing: as compiled in Tables III and IV of the source paper, no direct experimental value exists for $180,182,184,186,188,190$Hg, $184,186,188,190,192,194$Pb and $196,200$Po (for $198$Po only one value with very large error bars is available, quoted in the IBM study of Garcia-Ramos and Heyde 2015), so the $B_{02}>0.1$ filter cannot be tested there. Third, the prescription was derived only for SC of the ground state band and 'does not exclude the possibility of SC exhibited by two excited bands within the same nucleus', and the authors stress that the dual-shell criterion is 'only a necessary condition and not a sufficient one'. Fourth, the prescription has not been tested at lower masses, where Heyde and Wood's map shows an elongated SC island (island A) along the $N\approx Z$ line.

Problem Statement

The common objective is to complete the validation and the microscopic foundation of the empirical prescription of Bonatsos et al. (2023) for shape coexistence (SC) of the ground state band with a nearby excited $K=0$ band in even-even nuclei - the prescription being: search within the proxy-SU(3) dual-shell stripes of proton/neutron numbers 7-8, 17-20, 34-40, 59-70, 96-112 (and 146-168), avoiding the $P\approx5$ deformed contours at the stripe junctions, and require the thresholds $R_{4/2}<3.05$, $R_{0/2}<5.7$, $B_{02}>0.1$ - together with its claimed connection to first-order shape/phase transitions. Within this single objective, the following linked results are requested.

(1) Empirical extension to lighter masses: determine, from evaluated data (ENSDF, the Evaluated Nuclear Structure Data File of the National Nuclear Data Center) and/or new measurements, whether the same thresholds and the stripe localization apply in the lighter-mass region, taken as the even-even nuclei bounded below by the source paper's $Z=N=18$ calibration limit and covering in particular the elongated SC island A on the $N\approx Z$ line of the Heyde-Wood (2011) map; state the nucleus set surveyed and the SC criterion against which each verdict is reached, and for each nucleus in that set whether the prescription succeeds or fails.

(2) The missing interband rates: determine, experimentally (e.g. Coulomb excitation, fast-timing or conversion-electron spectroscopy) or by systematics validated against nuclei with directly measured $B(E2;0_2^+\to2_1^+)$ rates independently of the $B_{02}>0.1$ filter itself, the values (or bounds) of $B(E2;0_2^+\to2_1^+)$ for the SC candidate isotopes $180,182,184,186,188,190$Hg, $184,186,188,190,192,194$Pb and $196,198,200$Po, and test whether they satisfy $B_{02}>0.1$.

(3) The Pt exception: identify the microscopic origin of the fact that the Pt isotopes evolve shapes (and show coexistence behavior) while lying outside the dual-shell stripes and being describable without particle-hole excitations, i.e. either explain the exception within (or as a delimited failure of) the dual-shell mechanism or demonstrate that the prescription must be amended.

(4) Sufficiency and scope of the thresholds: assess whether the three thresholds are sufficient - not merely a useful practical filter - for the occurrence of ground-state/$K=0$ SC, including the scope question of whether SC involving two excited bands rather than the ground-state band falls within the reach of the thresholds; make the operational criterion explicit - independent of the three thresholds themselves and anchored in the physical definition of SC as two low-lying bands built on radically different intrinsic shapes - and either prove sufficiency within that explicitly stated criterion, exhibit counterexamples, or show the question is not well posed as stated, in the last case identifying the specific condition that the question violates.

(5) Microscopic linkage: quantify, through microscopic calculations (mean-field, beyond-mean-field, shell-model or configuration-mixing IBM with stated interactions and parameters) and comparison with data, the orbital occupancies and configuration mixing that produce the two SC mechanisms (proton p-h excitations across the $Z=82$, 50 closures; neutron p-h excitations at $N=90$, 60, 40), and establish or refute the conjectured causal linkage between intruder-orbital occupation and the IBM-like first-order spherical-to-prolate transition behavior - going beyond the parallel occurrence of the two phenomena to a demonstrated mechanism.

An answer may address any subset of (1)-(5) completely, provided it states which components it closes; the problem as a whole is completed when every component is settled by evidence of the indicated kind.

References

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