[Paper Review] Systematic study on proton radioactivity of spherical proton emitters within two--potential approach

Jiu-Long Chen, Xiao‐Hua Li+3 more|arXiv (Cornell University)|Oct 28, 2021

Nuclear physics research studies54 references9 citations

TL;DR

This study systematically calculates proton radioactivity half-lives for spherical proton emitters with Z ≥ 69 using the two-potential approach (TPA) with a parameterized cosh-type nuclear potential. By fitting 32 experimental data points, the authors determine optimal potential parameters and propose a simple analytic expression for the formation probability as a function of the daughter nucleus mass number A_d^{1/3}, which significantly improves agreement with experimental half-lives, reducing standard deviation by 45.1% compared to methods ignoring formation probability.

ABSTRACT

In the present work we systematically study the half--lives of proton radioactivity for spherical proton emitters with ${Z\ge 69}$ based on two--potential approach. While the nuclear potential of the emitted proton--daughter nucleus is adopted by a parameterized cosh type, the parameters of the depth and diffuseness for nuclear potential are determined by fitting experimental data of 32 spherical proton emitters. In order to reduce the deviations between experimental half-lives and calculated ones, we propose a simple analytic expression for formation probability of proton radioactivity with the same orbital angular momentum $l$. The results indicate that the formation probability can be simply described by a formula of $A_d^{1/3}$. Moreover, the linear relationship between the formation probability and the fragmentation potential also exists. The calculated half-lives can well reproduce the experimental data.

Motivation & Objective

• To systematically calculate proton radioactivity half-lives for spherical proton emitters with Z ≥ 69 using the two-potential approach (TPA).
• To determine the optimal parameters (depth V₀ and diffuseness a) of the cosh-type nuclear potential by fitting experimental half-lives of 32 spherical proton emitters.
• To develop a simple analytic expression for the proton formation probability that improves agreement with experimental data.
• To investigate the relationship between formation probability and nuclear structure properties, particularly the mass number A_d of the daughter nucleus.

Proposed method

• The total potential V(r) is constructed as the sum of a cosh-type nuclear potential V_N(r), Coulomb potential V_C(r), and centrifugal potential V_l(r), with a renormalized factor λ.
• The two-potential approach (TPA) splits the potential into internal bound and external scattering states, enabling calculation of the quasibound state width Γ via the WKB approximation.
• The proton decay width Γ is calculated using the formula Γ = S_p ℏ²F_P / (4μ), where S_p is the formation probability, μ is the reduced mass, and F_P is the penetration probability.
• The formation probability S_p is extracted from experimental half-lives and fitted as a function of A_d^{1/3}, yielding a simple analytic expression S_p ∝ A_d^{1/3}.
• The fragmentation potential V_frag = (Z_d e²)/R − Q_p is used to correlate formation probability with nuclear structure, revealing a linear relationship.
• Theoretical half-lives are compared with experimental data using logarithmic deviations and standard deviation (σ) to assess accuracy.

Experimental results

Research questions

• RQ1 Can the two-potential approach with a cosh-type nuclear potential accurately reproduce experimental proton radioactivity half-lives for spherical Z ≥ 69 emitters?
• RQ2 What is the functional dependence of the proton formation probability on the mass number A_d of the daughter nucleus?
• RQ3 How does the inclusion of a mass-number-dependent formation probability improve the accuracy of half-life predictions compared to models ignoring it?
• RQ4 Is there a linear correlation between the formation probability and the fragmentation potential V_frag?
• RQ5 Can a simple analytic expression S_p ∝ A_d^{1/3} effectively describe the formation probability across different orbital angular momenta l?

Key findings

• The optimal parameters for the cosh-type nuclear potential are determined as V₀ = 58.405 MeV and a = 0.537 fm by fitting 32 experimental half-lives.
• The formation probability S_p is well described by a simple analytic expression proportional to A_d^{1/3}, with a strong linear correlation observed across different l values.
• A linear relationship is found between the calculated formation probability S_p^calc and the fragmentation potential V_frag, confirming its link to nuclear structure.
• The standard deviation σ of the calculated half-lives is reduced to 0.163 when formation probability is included, representing a 45.1% improvement over the 0.297 σ obtained without it.
• The method achieves a ratio of T_calc / T_expt within a factor of 2.01–2.43 (σ = 0.163), significantly outperforming models that neglect formation probability.
• The improvement in accuracy is confirmed by a 34.3% reduction in σ for the UDLP model when formation probability is included, validating the importance of S_p in half-life calculations.