Quantitative spontaneous-fission half-lives with local-QRPA collective inertia in multidimensional collective space

Spontaneous-fission half-lives computed with nuclear energy density functional (EDF) theory and WKB tunneling are exquisitely sensitive to the collective inertia, the zero-point energy $E_0$, and the assault frequency $n$. A 2024 calculation for even-even fermium isotopes (N=158-164) with the constrained Hartree-Fock-Bogoliubov plus local quasiparticle random-phase approximation (CHFB+LQRPA) found that including dynamical residual effects in the collective inertia makes it much larger than the cranking value, lengthening the half-lives by 4-6 orders of magnitude, yet the results still deviate from experiment by several orders of magnitude, while a constant $E_0=0.5$ MeV alone shifts them by about three orders. The open problem is to make this framework quantitative: extend the CHFB+LQRPA treatment to multidimensional collective spaces including collective variables beyond the axially symmetric quadrupole moment, such as triaxiality, so that both the potential and the inertia are evaluated self-consistently along the least-action fission path, and derive the zero-point-energy corrections and the assault frequency consistently within the same microscopic framework, so as to determine whether dynamical residual effects close the gap with measured half-lives or expose physics beyond the current treatment. As of 2026 the problem is open: the most accurate EDF half-life benchmark still uses cranking inertia, an ad hoc constant assault frequency, and no zero-point subtraction, agreeing with measured half-lives only to within about four orders of magnitude.

Background

Spontaneous fission (SF) is the quantum tunneling of a heavy nucleus through its fission barrier: the nucleus elongates, passes under the barrier, and splits into two fragments. Because the tunneling probability is exponentially small, the half-life is astronomically sensitive to every ingredient of the tunneling action, and a useful theory must control each of them.

The standard microscopic framework is nuclear energy density functional (EDF) theory: the nucleus is described by a generalized Slater determinant (a Hartree-Fock-Bogoliubov, HFB, state), with pairing correlations between like nucleons treated by the Bogoliubov quasiparticle transformation. To map the fission path one solves constrained HFB (CHFB) equations at fixed values of collective shape coordinates, most commonly the axially symmetric mass quadrupole moment $Q_{20}$ (a measure of elongation), obtaining the collective potential $V(Q_{20})$. SF is then treated in the Wenzel-Kramers-Brillouin (WKB) approximation: $T_{1/2}=\ln 2/(nP)$ with penetrability $P=[1+\exp(2S)]^{-1}$ and action $S=\int dQ\,\sqrt{2M(Q)[V(Q)-E_0]}$ between the classical turning points, where $M(Q)$ is the collective inertia, $E_0$ the (zero-point-corrected) ground-state energy, and $n$ the assault frequency, the number of times per second the collective wave packet impinges on the barrier. All of $M$, $E_0$, and $n$ enter exponentially, so ad hoc choices among them move half-lives by orders of magnitude.

The inertia is the central quantity. The workhorse is the cranking formula, the perturbative limit of the adiabatic time-dependent HFB (ATDHFB) theory, which is cheap but ignores dynamical residual effects, the time-odd components of the EDF and the full two-quasiparticle response. The local quasiparticle random-phase approximation (LQRPA) instead builds the inertia from the random-phase-approximation (RPA) normal modes computed locally on top of each CHFB state, so that dynamical residual effects are included; it is far more expensive, and the finite-amplitude method (FAM), which solves RPA-type equations without explicit two-quasiparticle summations, was introduced to reduce the cost.

The source paper (Washiyama, 2024) computed, with the Skyrme EDF SkM*, CHFB potentials and FAM-LQRPA inertias along the axial, mass-symmetric path (only $Q_{20}$; reflection symmetries kept, so left-right asymmetric shapes are excluded) for even-even fermium (Fm, Z=100) isotopes with neutron numbers N=158-164, a region where experiment indicates a mass-symmetric fission mode. The LQRPA inertia substantially exceeds the cranking one and varies strongly along the path, making the SF half-lives 4-6 orders of magnitude longer than with cranking inertia, and still discrepant with the measured half-lives by several orders of magnitude. A constant zero-point energy $E_0=0.5$ MeV alone shifts the half-lives by about three orders of magnitude, exposing the arbitrariness of the current treatment of $E_0$ and of the standard constant assault frequency $n=10^{20.38}\,\mathrm{s}^{-1}$. Allowing triaxiality (the quadrupole coordinate $Q_{22}$) lowers the inner barrier of $^{258}$Fm by about 3 MeV, and the strong coordinate dependence of the LQRPA inertia means the least-action (action-minimizing) path need not coincide with the minimum-potential path.

The broader context: multidimensional calculations with the generator coordinate method (GCM), which expands the collective wave function as a superposition of the constrained HFB states, using cranking inertia and zero-point-energy corrections from the Gaussian overlap approximation (the standard reduction of the GCM to a collective Hamiltonian), have existed since Staszczak et al. and Sadhukhan et al. (2013); least-action SF half-lives of Fm and No isotopes with the Gogny EDF, including pairing fluctuations and zero-point corrections but cranking inertia, were computed by Rodríguez-Guzmán and Robledo (2018, DOI: 10.1103/PhysRevC.98.034308; arXiv:1809.05387); time-dependent generator-coordinate-method yield calculations for Fm isotopes (Regnier et al., 2019) also use cranking-level inertia. The five-dimensional quadrupole collective Hamiltonian with LQRPA inertial functions has been constructed on the $(\beta,\gamma)$ plane (Washiyama, Hinohara, and Nakatsukasa, 2024), but only for low-lying spectroscopy. The most accurate published EDF half-life benchmark (Sánchez-Fernández et al., 2026, BSkG functionals — a family of Skyrme-type EDF parameterizations fitted to nuclear mass data — three collective coordinates, least-action paths, ATDHFB cranking inertia) still agrees with the 136 measured SF half-lives only to within about four orders of magnitude.

Problem Statement

Within nuclear energy density functional theory, compute spontaneous-fission half-lives with a collective inertia that includes dynamical residual effects via the local quasiparticle random-phase approximation (LQRPA) built on constrained Hartree-Fock-Bogoliubov states, with all ingredients of the WKB estimate determined self-consistently rather than chosen ad hoc, and determine whether this brings the predictions into quantitative agreement with measured half-lives (currently discrepant by several orders of magnitude for the Fm isotopes). Specifically:

1. Multidimensional dynamics: the collective potential and the LQRPA collective inertia must be evaluated in a collective space of more than one coordinate that includes at least triaxiality ($Q_{22}$), and the half-life must be computed along the least-action path in that space rather than a one-dimensional axial, mass-symmetric path.

2. Consistent quantum corrections: the vibrational and rotational zero-point-energy corrections and the assault frequency $n$ in $T_{1/2}=\ln 2/(nP)$ must be derived consistently within the same microscopic framework, replacing ad hoc choices such as a constant $E_0$ and the standard constant $n=10^{20.38}\,\mathrm{s}^{-1}$, whose variations move the half-lives by orders of magnitude.

The overarching objective joining these components is the quantitative test of whether dynamical residual effects in the collective inertia, together with consistently treated zero-point energy and assault frequency in the full multidimensional geometry, resolve the orders-of-magnitude discrepancy between EDF-based spontaneous-fission half-lives and experiment, or whether the residual discrepancy points to physics beyond this treatment (e.g., deficiencies of the EDF's time-odd sector or non-adiabatic dynamics).

References

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