Microscopic foundation of the cluster preformation probability in trans-lead cluster radioactivity: absolute magnitude, $A_c\geq28$ scaling, and the origin of the $N_pN_n$/$N_pN_nI$ linearity

Cluster radioactivity — the spontaneous emission of nuclei heavier than an alpha particle but lighter than fission fragments, observed only in trans-lead parents decaying to $^{208}$Pb or its neighbors — is quantified by a decay constant factorized into assault frequency, barrier penetrability, and cluster preformation probability $P_c$. Current determinations of $P_c$ are mutually inconsistent by many orders of magnitude: cluster-formation-model, unified-fission-model, and phenomenological extractions give $\log_{10}P_c$ values differing by 2–12 units for the same emitters. The empirical mapping $P_c=[P_\alpha]^{(A_c-1)/3}$ is applied only for emitted-cluster mass $A_c\leq28$ and is replaced ad hoc by a charge-number formula beyond that, and the observed approximate linearity of $\log_{10}P_c$ on the valence products $N_pN_n$ and $N_pN_nI$ lacks a microscopic explanation. The problem is to construct a microscopic, model-independent theory of $P_c$ that fixes its absolute magnitude, derives its $A_c$ scaling including the high-$A_c$ bend, and explains the valence-nucleon systematics.

Background

Cluster radioactivity is a spontaneous decay mode intermediate between alpha decay and fission, in which a parent nucleus emits a fragment heavier than $^4$He but lighter than typical fission fragments — from $^{14}$C up to $^{34}$Si — with half-lives many orders of magnitude longer than those of alpha decay. It was predicted in 1980 by Šandulescu, Poenaru, and Greiner and first observed by Rose and Jones in 1984 as emission of $^{14}$C from $^{223}$Ra. All experimentally established cases ($^{221}$Fr through $^{242}$Cm) lie in the trans-lead region and decay to the doubly magic nucleus $^{208}$Pb (a nucleus with closed proton and neutron shells, hence especially stable) or its neighbors. In the standard Gamow-type description, the decay constant is factorized as $\lambda=\nu_0PP_c$, where $\nu_0$ is the assault frequency (the rate at which a preformed cluster assaults the Coulomb barrier), $P$ is the WKB (Wentzel–Kramers–Brillouin quasi-classical) tunneling probability through the Coulomb plus nuclear plus centrifugal potential barrier, and $P_c$ is the cluster preformation probability — the probability that the emitted cluster exists as a preformed configuration inside the parent before tunneling. $P_c$ is not directly observable: it must either be extracted from measured half-lives by dividing out a model-computed $\nu_0P$ (a model-dependent extraction) or estimated from nuclear structure models (a microscopic route); the two routes need not agree.

The source study (Qi et al., Phys. Rev. C 108, 014325 (2023)) carried out both routes for the 22 measured trans-lead emitters. In the microscopic route, the alpha preformation probability $P_\alpha$ is computed in the cluster formation model (CFM) as a formation-energy fraction built from experimental binding energies, with four distinct formulae depending on parent parity (even–even, even–odd, odd–even, odd–odd); the cluster preformation probability then follows from the empirical exponential mapping $P_c=[P_\alpha]^{(A_c-1)/3}$ proposed by Blendowske and Walliser (1988) on the basis of microscopic cluster-formation calculations. This mapping is applied only for emitted-cluster mass number $A_c\leq28$, because the log-linear relation between $\log_{10}P_c$ and $A_c$ visibly bends beyond $A_c\approx28$ (as shown by Wei and Zhang (2017) from model-extracted preformation factors); for the three measured decays with $A_c\geq30$, the phenomenological charge-number formula of Ren et al. (2004), $\log_{10}P_c=-(0.01674Z_cZ_d-2.035466)$ with a $-1.175$ offset for odd-$A$ parents, is substituted instead, so the resulting 'microscopic' dataset is in fact a hybrid. In the model-dependent route, $P_c=\ln2/(T_{\rm exp}\nu_0P)$ is extracted using the unified fission model (UFM) of Dong et al. (2009), whose barrier is built from a nuclear proximity potential (a parameterized surface-tension interaction between two gently curved nuclear surfaces) plus the Coulomb and Langer-corrected centrifugal potentials (the standard semiclassical replacement of $l(l+1)$ by $(l+1/2)^2$ in the centrifugal term). The two determinations agree in trend but differ by roughly 2.5–6.5 units in $\log_{10}P_c$, and the model and analytical formula of Wei and Zhang sit another 5–12 units lower; for example, for $^{242}$Cm→$^{208}$Pb+$^{34}$Si the source's Table 1 gives the CFM (microscopic; the Ren-formula entry here, since $A_c=34\geq30$), UFM, Wei–Zhang model, and Wei–Zhang formula values $-11.37$, $-16.51$, $-23.20$, and $-21.64$.

Both determinations were found to be approximately linear in the valence product $N_pN_n$ — plotted in the paper in the normalized form $N_pN_n/(N_0+Z_0)$ with $Z_0=82$ and $N_0=126$ — (with $N_p=Z-82$ and $N_n=N-126$, the effective numbers of valence protons and neutrons counted from the $^{208}$Pb core, as used in Casten's $N_pN_n$ scheme, which tracks shell proximity and neutron–proton correlation strength) and in $N_pN_nI$, where $I=(N-Z)/(N+Z)$ is the parent isospin asymmetry; the same linearity was previously established for alpha-decay preformation by Seif et al. (2011) and Deng et al. (2018). The fits to 22 data points give $R^2\approx1$, but the paper provides no uncertainty estimates, no statistical test of the improvement from including $I$, and no mechanism-level explanation. The authors explicitly flag as unresolved: the origin of the multi-unit magnitude discrepancies between methods; the absence of a microscopic mechanism and predictive theory justifying or replacing the exponential mapping for $A_c\geq28$; the untested microscopic foundation and validity domain of the Ren et al. formula; the predictive power of the $N_pN_n$ scheme outside the tested emitters when magic numbers and shell structure drift; and the lack of a microscopic account isolating neutron pairing, deformation, and the neutron–proton interaction behind the linear $N_pN_n$ and $N_pN_nI$ trends.

Problem Statement

Construct a microscopic, model-independent (that is, not resting on fitted phenomenological prescriptions such as the exponential mapping or the Ren et al. charge-number formula) account of the cluster preformation probability $P_c$ in trans-lead cluster radioactivity. The three requested results jointly serve this single objective — replacing the current mutually inconsistent phenomenological prescriptions with a theory whose predictions can be checked against the 22 measured emitters and extended to unmeasured ones. Specifically:

(1) Absolute magnitude: determine $\log_{10}P_c$ for the measured trans-lead emitters and identify which, if any, of the current determinations (CFM-based, UFM-extracted, or the Wei–Zhang model/formula) is quantitatively correct, explaining the origin of the present 2–12 unit discrepancies in terms of identifiable physical effects — approximations in the cluster-formation mapping, missing dynamical correlations, sensitivities of the UFM potential and assault-frequency prescriptions, or limitations of the phenomenological formulas.

(2) Mass-number scaling: derive from microscopic structure the dependence of $P_c$ on the emitted-cluster mass number $A_c$ — justifying or replacing the empirical law $P_c=[P_\alpha]^{(A_c-1)/3}$ and establishing its domain of validity, including the observed bending of $\log_{10}P_c$ versus $A_c$ beyond $A_c\approx28$ that currently forces the ad hoc switch to the Ren et al. charge-number formula, whose microscopic foundation and validity domain should likewise be established.

(3) Valence-nucleon systematics: explain microscopically the observed approximate linearity of $\log_{10}P_c$ in $N_pN_n$ and in $N_pN_nI$ (with $N_p=Z-82$, $N_n=N-126$ counted from the $^{208}$Pb core, and $I=(N-Z)/(N+Z)$), isolating the roles of neutron pairing, deformation, and the neutron–proton interaction, and thereby delimiting the mass regions and shell-structure regimes in which the effective $N_pN_n$ scheme remains predictive — including whether the linearity is a genuine physical trend or partly an artifact of the formulas used to compute $P_c$.

References

1. L.-J. Qi, D.-M. Zhang, S. Luo, G.-Q. Zhang, P.-C. Chu, X.-J. Wu, X.-H. Li, 'Cluster radioactivity preformation probability of trans-lead nuclei in the $N_pN_n$ scheme', Physical Review C 108, 014325 (2023). DOI: 10.1103/PhysRevC.108.014325, https://doi.org/10.1103/PhysRevC.108.014325 (arXiv:2403.04450)
2. R. Blendowske, H. Walliser, 'Systematics of cluster-radioactivity-decay constants as suggested by microscopic calculations', Physical Review Letters 61, 1930 (1988). DOI: 10.1103/PhysRevLett.61.1930, https://doi.org/10.1103/PhysRevLett.61.1930
3. Z. Ren, C. Xu, Z. Wang, 'New perspective on complex cluster radioactivity of heavy nuclei', Physical Review C 70, 034304 (2004). DOI: 10.1103/PhysRevC.70.034304, https://doi.org/10.1103/PhysRevC.70.034304
4. K. Wei, H. F. Zhang, 'Cluster preformation law for heavy and superheavy nuclei', Physical Review C 96, 021601(R) (2017). DOI: 10.1103/PhysRevC.96.021601, https://doi.org/10.1103/PhysRevC.96.021601
5. K. Wei, H. F. Zhang, 'α decay and cluster radioactivity within the redefined preformed cluster method', Physical Review C 102, 034318 (2020). DOI: 10.1103/PhysRevC.102.034318, https://doi.org/10.1103/PhysRevC.102.034318
6. J. M. Dong, H. F. Zhang, J. Q. Li, W. Scheid, 'Cluster preformation in heavy nuclei and radioactivity half-lives', The European Physical Journal A 41, 197 (2009). DOI: 10.1140/epja/i2009-10819-1, https://doi.org/10.1140/epja/i2009-10819-1
7. L.-J. Qi, D.-M. Zhang, S. Luo, X.-H. Li, X.-J. Wu, C.-T. Liang, 'Systematic calculations of cluster radioactivity half-lives in trans-lead nuclei', Chinese Physics C 47, 014101 (2023). DOI: 10.1088/1674-1137/ac94bd, https://doi.org/10.1088/1674-1137/ac94bd
8. J.-G. Deng, J.-C. Zhao, P.-C. Chu, X.-H. Li, 'Systematic study of α decay of nuclei around the Z = 82, N = 126 shell closures within the cluster-formation model and proximity potential 1977 formalism', Physical Review C 97, 044322 (2018). DOI: 10.1103/PhysRevC.97.044322, https://doi.org/10.1103/PhysRevC.97.044322
9. W. M. Seif, M. Shalaby, M. F. Alrakshy, 'Isospin asymmetry dependence of the α spectroscopic factor for heavy nuclei', Physical Review C 84, 064608 (2011). DOI: 10.1103/PhysRevC.84.064608, https://doi.org/10.1103/PhysRevC.84.064608
10. H. J. Rose, G. A. Jones, 'A new kind of natural radioactivity', Nature 307, 245 (1984). DOI: 10.1038/307245a0, https://doi.org/10.1038/307245a0
11. D. Ni, Z. Ren, 'Half-lives and cluster preformation factors for various cluster emissions in trans-lead nuclei', Physical Review C 82, 024311 (2010). DOI: 10.1103/PhysRevC.82.024311, https://doi.org/10.1103/PhysRevC.82.024311
12. C. Nithya, K. P. Santhosh, 'Theoretical investigation of cluster preformation using empirical formulae and $N_pN_n$ scheme in trans-lead nuclei', Physica Scripta 101, 055303 (2026). DOI: 10.1088/1402-4896/ae3d6e, https://doi.org/10.1088/1402-4896/ae3d6e
13. J. T. Majekodunmi, R. Kumar, M. Bhuyan, 'Quest for a universal cluster preformation formula: A new paradigm for estimating the cluster formation energy', EPL (Europhysics Letters) 143, 24001 (2023). DOI: 10.1209/0295-5075/ace475, https://doi.org/10.1209/0295-5075/ace475
14. K. P. Santhosh, V. K. Anjali, 'Cluster radioactivity using modified generalized liquid model with a statistical cluster preformation probability', Nuclear Physics A 1041, 122787 (2024). DOI: 10.1016/j.nuclphysa.2023.122787, https://doi.org/10.1016/j.nuclphysa.2023.122787
15. M. Lu, N. Wan, 'Effects of different effective nucleon-nucleon interactions on α-decay half-life and extracted α-cluster preformation probability', Physical Review C 110, 044321 (2024). DOI: 10.1103/PhysRevC.110.044321, https://doi.org/10.1103/PhysRevC.110.044321
16. M. Warda, A. Zdeb, R. Rodríguez-Guzmán, 'Microscopic description of cluster radioactivity fission valleys along isotopic and isotonic chains', Physical Review C 113, 034619 (2026). DOI: 10.1103/b7q7-925c, https://doi.org/10.1103/b7q7-925c (arXiv:2603.03519)
17. Y.-Z. Wang, J. Li, W.-H. Zhang, Y.-Q. Li, J.-Z. Gu, 'Effect of the spin–orbit interaction on the structure and two-proton radioactivity of $^{18}$Mg', Nuclear Science and Techniques 37, 217 (2026). DOI: 10.1007/s41365-026-02034-w, https://doi.org/10.1007/s41365-026-02034-w
18. D. Singh, S. Bharmoria, M. Saini, R. Sharma, BirBikram Singh, H. Kaur, 'Investigation of stimulated cluster radioactive decay in nuclei', Nuclear Physics A 1070, 123369 (2026). DOI: 10.1016/j.nuclphysa.2026.123369, https://doi.org/10.1016/j.nuclphysa.2026.123369