Extending semi-empirical spontaneous-fission half-life systematics to superheavy nuclei with vanishing liquid-drop barriers

Pomorski, Warda and Zdeb (2015) showed that a Świątecki-type semi-empirical formula — the logarithm of the spontaneous-fission (SF) half-life shifted by $k\delta M$ ($k=7.7$ MeV$^{-1}$) against the ground-state shell-plus-pairing mass correction — reproduces SF half-lives of even-even actinide nuclei with an r.m.s. deviation of 1.18 in $\log_{10} T$ (growing to 1.57 when odd-$A$ and odd-odd nuclei are included), but that the deviation rises to 1.96 when superheavy nuclei are included, because for $Z \ge 103$ the liquid-drop fission barrier vanishes and stability is due entirely to ground-state shell effects. The paper leaves open how to extend barrier-based SF half-life systematics into this shell-stabilized regime: the model parameters (the average oscillator frequency entering the action and the assault frequency), the zero-point-energy and collective-inertia conventions, the microscopic origin of the odd-particle hindrance, and predictive power for nuclei without measured masses are all unsettled. The question remains open: the same group's follow-up refits the formula in terms of shell-corrected barrier heights, competing improved empirical formulas keep appearing through 2025 without convergence, microscopic least-action calculations agree with data only to within about three orders of magnitude, and the transition region near $Z=102$–$104$ is still under active investigation.

Background

Spontaneous fission (SF) is the decay mode in which a heavy nucleus in its ground state splits into two comparable fragments by quantum tunneling through the fission barrier. In the semiclassical WKB (Wentzel–Kramers–Brillouin) approximation the half-life is $T_{1/2}^{\mathrm{sf}} \sim \ln 2/(nP)$, where $n$ is the assault frequency — the number of times per unit time the collective vibrational mode attacks the barrier — and $P \approx e^{-2S}$ is the tunneling probability, with the action $S = \frac{1}{\hbar}\int \sqrt{2B(V(s)-E_0)}\,ds$ taken along a path in deformation space; $B$ is the collective inertia (the effective mass of the fission mode) and $V(s)$ the collective potential along the path coordinate $s$. Because the half-life depends exponentially on the barrier height and width and on $B$, an error of 1 MeV in the barrier moves the predicted lifetime by orders of magnitude.

Świątecki (1955) observed that measured SF half-lives, when corrected by the deviation $\delta M$ of the measured ground-state mass from a smooth liquid-drop trend, collapse onto an approximately linear function of the proton number $Z$ (equivalently, of the fissility $Z^2/A$). The liquid-drop model describes only the smooth, macroscopic part of nuclear binding; the residual $\delta M$ is dominated by shell effects — quantum corrections arising from the discrete single-particle level structure, usually computed with the Strutinsky method (a standard smoothing procedure that isolates the oscillatory shell part of the single-particle level sum) together with a pairing correction. In the modern macroscopic-microscopic approach one combines a refined liquid drop such as the Lublin–Strasbourg Drop (LSD) with these microscopic corrections. This regularity is explained by the topographical theorem of Myers and Świątecki: the energy of a nucleus at the saddle point of the fission barrier is almost purely macroscopic (shell effects at the saddle are weak), so the barrier height is approximately $V_B \approx M_{\mathrm{LD}}(\mathrm{saddle}) - M_{\mathrm{exp}}(\mathrm{g.s.})$ — the barrier directly measures the ground-state shell correction, which is the physics behind the $\delta M$ correction.

The source paper (Pomorski, Warda and Zdeb, 2015) revisited both decay modes with deliberately minimal models. For alpha decay, a Gamow-like one-parameter model (only the nuclear radius constant $r_0 = 1.23$ fm fitted) reproduces $\log_{10} T_{1/2}^{\alpha}$ of 360 nuclei with $52 \le Z \le 110$ to an r.m.s. deviation of 0.39 for even-even nuclei, with constant hindrance shifts for odd systems; its largest failure is a roughly two-order-of-magnitude underestimation for the $N=127$ isotones of Bi–U, where strong shell structure near the neutron magic number $N=126$ lies outside the model. For SF, the Świątecki formula is refitted as $\log_{10}(T_{1/2}^{\mathrm{sf}}/\mathrm{y}) = -4.1\,\min(Z,103) + 380.2 + h$, with hindrance $h = 0$, $2.5$, $5$ for even-even, odd-$A$ and odd-odd nuclei, applied together with a mass-shift correction $-k\,\delta M$, $k = 7.7\,\mathrm{MeV}^{-1}$, $\delta M = M_{\mathrm{exp}} - M_{\mathrm{LSD}}$. It reproduces the actinide even-even data with an r.m.s. deviation of 1.18 in $\log_{10} T$, rising to 1.96 once the superheavy isotopes are included. The authors justify the Świątecki structure by WKB tunneling through a barrier approximated by two inverted parabolas along the least-action (dynamic) fission path, which gives $\ln T_{1/2}^{\mathrm{sf}} = 2S - \ln n - 0.3665$ with $S = 2V_B/(\hbar\tilde\omega)$, where $\tilde\omega$ is an average oscillator frequency of the two parabolas; neither $\tilde\omega$ nor the assault frequency $n$ is determined within the reduced model, the collective potential requires a zero-point-energy subtraction $V(s) = E_{\mathrm{HFB}}(s) - \varepsilon_0^{\mathrm{fiss}}(s)$, where $E_{\mathrm{HFB}}(s)$ is the Hartree–Fock–Bogoliubov mean-field energy (a mean-field treatment of the nucleus including pairing) at deformation $s$, whose consistent multidimensional treatment is unsettled, and the hindrance shifts for odd systems are empirical stand-ins for the specialization energy — the loss of pairing and shell correlation when an odd nucleon blocks the collective motion — rather than microscopic results.

The crux is the superheavy regime. For $Z \ge 103$ the liquid-drop fission barrier vanishes: superheavy nuclei exist only because large negative ground-state shell corrections create barriers of a few MeV around the predicted closed shells (the island of stability). There the linear-in-$Z$ term of the fitted formula saturates through the ad hoc cap $\min(Z,103)$, the observed spread of $\log T$ at fixed $Z$ is no longer tracked, and predictions fail outright for the many isotopes whose masses are unmeasured. How to extend barrier-based SF half-life systematics into this purely shell-stabilized regime — with parameters fixed, zero-point and inertia conventions settled, the odd-particle hindrance grounded, and predictive power for mass-unknown nuclei — is the open problem.

Problem Statement

Construct a quantitative, predictive description of spontaneous-fission half-lives in the superheavy regime where the macroscopic (liquid-drop) fission barrier vanishes ($Z \ge 103$) and nuclear stability is due entirely to ground-state shell-plus-pairing corrections — that is, extend the Świątecki-type barrier-based systematics through this transition regime rather than patching it with the $\min(Z,103)$ cap, so that the accuracy lost there is restored (the r.m.s. deviation of $\log_{10} T$ grows from 1.18 for even-even actinides to 1.96 once the superheavy isotopes are included) and the evaluation no longer breaks down for nuclei whose masses have not yet been measured.

The source paper and its follow-ups discuss several open aspects that such an extension confronts, listed here as context rather than as completion conditions: the average oscillator frequency $\tilde\omega$ entering the action and the assault frequency $n$ are undetermined within the reduced model, and the mass-shift coefficient $k$ is a fitted constant; the conventions for subtracting zero-point energies from the collective potential and for computing the collective inertia along the least-action fission path in a multidimensional deformation space are unsettled; the odd-$A$ and odd-odd hindrance shifts stand in empirically for the specialization energy, whose microscopic origin and possible deformation dependence are unknown; and predictions fail for the many isotopes whose masses are unmeasured. Any proposed description would, as usual, be expected to be checked against the known spontaneous-fission emitters and to yield testable predictions of SF half-lives for superheavy nuclei whose half-lives and masses have not yet been measured.

References

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