Cluster radioactivity - the spontaneous emission of nuclei heavier than the alpha particle from trans-lead parents - is described by preformed-cluster models whose decay constant is the product of a cluster preformation probability PcP_c and a tunneling penetrability. The source study (Zhen Wang and Zhongzhou Ren, Phys. Rev. C 108, 024306 (2023)) generalized the density-dependent cluster model to include deformation-dependent, anisotropic surface diffuseness ("nuclear surface polarization") of the emitted cluster, parametrized by a phenomenological parameter k′k' whose sign selects the polarization mode; it reproduced the measured half-lives within a factor of about 3, but only with an empirically fitted PcP_c and hand-chosen representative modes k′=−1,0,1k'=-1,0,1. The open problem is to derive both quantities - the heavy-cluster preformation probability and the orientation-dependent surface diffuseness (equivalently k′k' or a microscopic diffuseness-anisotropy predictor) - from microscopic nuclear-structure theory (self-consistent energy-density-functional calculations with configuration mixing, such as the generator coordinate method, or ab initio where feasible), and to use them to predict the observed trans-lead cluster-decay half-lives without empirical calibration, and to establish from the microscopic densities themselves which polarization mode, if any, is realized.

Background

Cluster radioactivity is the spontaneous emission from a heavy nucleus of an intermediate-mass fragment - heavier than an α\alpha particle but lighter than typical fission fragments. All confirmed cases occur in the trans-lead region (nuclei heavier than lead) and involve light clusters from 14C^{14}\mathrm{C} to 34Si^{34}\mathrm{Si} emitted by parents between 221Fr^{221}\mathrm{Fr} and 242Cm^{242}\mathrm{Cm}, mostly decaying to the doubly magic 208Pb^{208}\mathrm{Pb} or its neighbors; the phenomenon was first observed in 1984 in 223Ra→14C+208Pb^{223}\mathrm{Ra}\to{}^{14}\mathrm{C}+{}^{208}\mathrm{Pb} (Rose and Jones, Nature 307, 245 (1984)). The partial half-lives are extraordinarily long (roughly 101110^{11} to 102910^{29} years, with branching ratios relative to α\alpha decay of order 10−1010^{-10} to 10−1710^{-17}), so only about two to three dozen cases have been measured. The standard theoretical picture is the preformed cluster model: the decay constant is written as the product of an assault frequency, a cluster preformation probability PcP_c - the probability that the cluster is already formed at the nuclear surface before tunneling - and the penetrability of the Coulomb-plus-nuclear barrier, usually computed in the WKB (Wentzel-Kramers-Brillouin) semiclassical approximation. In the density-dependent cluster model (DDCM) used by the source paper, the cluster-daughter interaction potential is a double-folding integral of the daughter and cluster density distributions with an effective nucleon-nucleon interaction, and the decay width follows from the Schrödinger equation with outgoing Coulomb-wave boundary conditions. The preformation probability is the weak link: it is not computed but fitted to reproduce measured half-lives, or taken from systematics such as the mass-number scaling of Blendowske and Walliser (Phys. Rev. Lett. 61, 1930 (1988)). Nuclear deformation enters through multipole parameters β2\beta_2 (quadrupole), β4\beta_4 (hexadecapole), and β6\beta_6, which describe the shape of the half-density surface; the light emitted clusters are themselves deformed, and their β4\beta_4 and β6\beta_6 values are poorly constrained experimentally. Separately, a microscopic structural effect was identified by Scamps, Lacroix, Adamian, and Antonenko (Phys. Rev. C 88, 064327 (2013)): analyzing density profiles of about 750 nuclei with Skyrme energy-density-functional (EDF) calculations - self-consistent mean-field theory built from an energy functional of the densities - they found that in deformed nuclei the surface diffuseness (the thickness of the density falloff at the surface) becomes anisotropic, decreasing along the main deformation axis while increasing in the transverse direction, an effect they called polarization of the nuclear surface, and showed it modifies dynamical quantities such as fusion barriers. The source paper (Wang and Ren, Phys. Rev. C 108, 024306 (2023)) brought these threads together: it generalized the DDCM so that the cluster density carries a deformation-dependent, orientation-dependent diffuseness controlled by an adjustable parameter k′k', whose sign selects the surface-polarization mode and whose amplitude, together with the deformation parameters, sets the polarization degree. Examining three representative modes (k′=−1,0,1k'=-1,0,1), it reproduced the measured trans-lead cluster-decay half-lives within a factor of about 3, and showed that the choice of polarization mode alters the geometry of the daughter-cluster interaction and hence the computed half-lives. The limitations that define the present problem are these: the sign and amplitude of k′k' are chosen phenomenologically rather than derived from EDF or ab initio densities; the three representative modes are compared only theoretically, and no experiment identifies which polarization mode is realized in the emitted clusters; and PcP_c is determined empirically within the density-dependent cluster-model chain rather than computed from a many-body wave function. Two further contextual gaps compound the task: the higher-multipole deformations (β4\beta_4, β6\beta_6) of the light emitted clusters are poorly known, and no uncertainty budget propagates experimental QQ values, deformation data, and the empirical preformation into the predictions. The only microscopic many-body computation of heavy-cluster preformation to date is the proof-of-principle generator coordinate method (GCM) calculation of Uzawa, Hagino, and Yoshida (Phys. Rev. C 105, 034326 (2022)) - GCM mixes constrained mean-field states along a collective coordinate, here the mass-octupole mode, with Skyrme EDF and BCS (Bardeen-Cooper-Schrieffer) pairing - which reproduced the decay rate of 222Ra→14C+208Pb^{222}\mathrm{Ra}\to{}^{14}\mathrm{C}+{}^{208}\mathrm{Pb} only to within about two orders of magnitude (two to three for 228Th^{228}\mathrm{Th} and 232U^{232}\mathrm{U}), far from the accuracy of calibrated phenomenology.

Problem Statement

Derive, from microscopic nuclear-structure theory - self-consistent energy-density-functional calculations with configuration mixing (for example, a multi-dimensional generator coordinate method including mass-asymmetry, deformation, and orientation coordinates, with pairing), or ab initio methods where feasible - the two empirically calibrated inputs of exotic cluster radioactivity in trans-lead nuclei: (i) the cluster preformation probability PcP_c for the emission of heavy clusters (from 14C^{14}\mathrm{C} to 34Si^{34}\mathrm{Si}) from parents between 221Fr^{221}\mathrm{Fr} and 242Cm^{242}\mathrm{Cm}, and (ii) the surface-polarization degree of freedom of the emitted cluster, i.e., the orientation dependence of its surface diffuseness, expressible as the phenomenological parameter k′k' of Wang and Ren or an equivalent microscopic predictor of diffuseness anisotropy derived from self-consistent densities. Demonstrate the result by predicting the measured cluster-decay half-lives of the observed trans-lead emitters with no empirical calibration of PcP_c or of the polarization parameter. The demonstration must include a defensible statement of which polarization mode, if any, is realized in the emitted clusters, based on the microscopic densities themselves. Completion means the preformation and surface-polarization quantities are no longer free parameters: they are outputs of the microscopic calculation, and the predicted half-lives are compared against the measured trans-lead cluster decays, reproducing the measured values to within a factor of a few.

References

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