Calculations of widths in the problem of decay by proton emission

TL;DR

This paper introduces a new quantum method for calculating decay widths in proton emission, accounting for multiple internal reflections and providing more accurate and stable predictions of half-lives compared to semiclassical approaches.

Contribution

The authors develop an exact quantum approach for decay widths that considers multiple internal reflections and the starting point of proton emission, improving accuracy over traditional semiclassical methods.

Findings

Penetrability varies significantly with the starting point R_form.

The new method yields half-life predictions closer to experimental data.

External barrier influence on penetrability can be up to 1.5 times.

Abstract

We develop a new fully quantum method for determination of widths for nuclear decay by proton emission where multiple internal reflections of wave packet describing tunneling process inside proton--nucleus radial barrier are taken into account. Exact solutions for amplitudes of wave function, penetrability $T$ and reflection $R$ are found for $n$-step barrier (at arbitrary $n$) which approximates the realistic barrier. In contrast to semiclassical approach and two-potential approach, we establish by this method essential dependence of the penetrability on the starting point $R_{\rm form}$ in the internal well where proton starts to move outside (for example, for $^{157}_{73}{\rm Ta}$ the penetrability is changed up to 200 times; accuracy is $|T+R-1| < 1.5 \cdot 10^{-15}$). We impose a new condition: in the beginning of the proton decay the proton starts to move outside from minimum of the well. Such a condition provides minimal calculated half-life and gives stable basis for predictions. However, the half-lives calculated by such an approach turn out to be a little closer to experimental data in comparison with the semiclassical half-lives. Estimated influence of the external barrier region is up to 1.5 times for changed penetrability.