Analytical penetration probability including the centrifugal potential: An improved Buck--Merchant--Perez model for alpha-decay half-lives

Technical Summary: An Improved Buck–Merchant–Perez Model for $\alpha$-Decay Half-Lives

Problem Statement
The calculation of $\alpha$-decay half-lives relies on the quantum tunneling of an $\alpha$-particle through a potential barrier formed by nuclear and Coulomb interactions. While the WKB approximation is the standard tool for evaluating penetration probabilities, traditional treatments of the centrifugal potential (arising from non-zero orbital angular momentum $L$ in unfavored transitions) have been limited. Historically, these treatments have been perturbative, assuming a small parameter $\sigma = V_L(R_0)/V_c(R_0) \ll 1$. This approach fixes the inner and outer turning points at their Coulomb values, treating centrifugal hindrance merely as an additive correction to the barrier height while ignoring the modification of the barrier width. Furthermore, the depth of the internal nuclear potential ($V_0$) in the original Buck–Merchant–Perez (BMP) model was often treated as a constant or required nucleus-by-nucleus fitting, failing to provide a unified description across different parity combinations (even-even, odd-A, odd-odd) and angular momentum states.

Methodology
The authors propose a unified framework that addresses these limitations through two primary theoretical advancements:

1. Non-Perturbative Analytical Treatment of the Centrifugal Barrier:
   The authors derive a closed-form, non-perturbative WKB penetration formula for the effective potential $V(r) = V_c(r) + V_L(r)$. Unlike perturbative methods, this approach does not assume small $L$. Instead, it solves the tunneling action integral exactly for arbitrary $L$. Crucially, the inner turning point $R$ is determined self-consistently via the Bohr–Sommerfeld quantization condition, which accounts for the shift of the inner boundary inward as $L$ increases. The outer turning point $R_L$ is also shifted outward. This results in a more accurate calculation of the barrier width and the associated WKB action integral, $J_L$, which is expressed in a closed analytical form dependent on $R$, $Q$, and $L$.

2. Unified Four-Parameter Potential Depth Formula:
   To replace the nucleus-specific fitting of the nuclear potential depth $V_N$, the authors develop a unified formula (Eq. 11) that expresses $V_N$ as a function of fundamental nuclear properties. This formula incorporates:

   • Shell Corrections: Explicitly including the shell correction energy $E_{sh}$ derived from the difference between experimental and macroscopic binding energies.
   • Odd-Even Pairing Effects: A term $-2[(-1)^Z + (-1)^N]$ that captures the staggering between even-even, odd-A, and odd-odd nuclei without requiring separate parameter sets.
   • Orbital Angular Momentum Dependence: A term $-L(L+1)$ that adjusts the potential depth based on the angular momentum carried by the emitted $\alpha$-particle.

   The parameters for this formula were fitted using a least-squares method to 380 favored ($L=0$) ground-state-to-ground-state transitions.

Key Contributions

• Exact Analytical Solution: The derivation of a non-perturbative, closed-form expression for the WKB action integral in the presence of a centrifugal barrier, valid for arbitrary $L$.
• Self-Consistent Radius Determination: The demonstration that the inner turning point shifts inward with increasing $L$ due to the Bohr–Sommerfeld condition, a feature missed by perturbative treatments that fix the radius.
• Unified Parametrization: The development of a single four-parameter formula for the nuclear potential depth that successfully describes both favored and unfavored transitions across the entire nuclear chart ($60 \le Z \le 118$), eliminating the need for distinct parameter sets for different parity or angular momentum configurations.

Results
The model was tested against a dataset of 534 ground-state-to-ground-state $\alpha$ decays:

• Accuracy Improvement: The root-mean-square (rms) deviation of $\log_{10} T_{1/2}$ was reduced to 0.267. This represents a 57% improvement over the original constant-depth BMP model, which yielded an rms deviation of 0.615.
• Performance by Transition Type: The model demonstrated robust performance for both favored transitions (rms = 0.188) and unfavored transitions (rms = 0.398), despite the latter not being included in the fitting process.
• Shell Effects: The inclusion of shell corrections significantly improved the description of half-lives near magic numbers (e.g., $N=126$), where the original BMP model showed pronounced deviations.
• Predictive Capability: The framework was applied to predict the half-lives of unmeasured nuclei in the superheavy region ($Z = 117–120$). The predictions showed excellent agreement with other established approaches (Royer and Deng-Zhang-Royer formulas) and available experimental data for isotopes like $^{293,294}$Ts and $^{294}$Og. The model also identified pronounced shell effects at neutron numbers $N_d = 178, 184,$ and $196$.

Significance
The paper claims that the improved BMP model provides a more physically consistent description of $\alpha$ decay by treating the centrifugal barrier non-perturbatively and linking the nuclear potential depth directly to nuclear structure (shells, pairing, and angular momentum). The authors emphasize that the centrifugal hindrance is not merely an empirical additive correction but is intrinsically connected to the modification of the barrier geometry and the internal potential. By achieving high accuracy with a minimal set of four parameters across a wide range of nuclei, the model offers a reliable quantitative benchmark for future experimental investigations of superheavy nuclei, particularly in the region $Z = 117–120$.