Ambiguity of applying the Wildermuth-Tang rule to estimate the quasibound states of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> particles in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> emitters

Abstract

The Wildermuth-Tang (WT) prescription is used to verify the Bohr-Sommerfeld (BS) quantization condition in the $\ensuremath{\alpha}$-decay problem. It gives the global quantum number that relates the number of nodes of the quasibound radial wave function of the $\ensuremath{\alpha}$-daughter system to the shell model and Pauli exclusion principle. Here we examine the applicability of the WT rule in the $\ensuremath{\alpha}$-decay microscopic calculations that start with solving the stationary Schr\"odinger wave equation for different types of the interaction potentials. We found that applying the BS quantization condition along with the WT prescription for the potentials that have no internal pocket yields a large number of nodes in the radial wave function compared to the potentials characterized with an automatic physical internal pocket, which likely produce nodeless or at most a two-node interior wave function. This gives confidence in the latter type of the potentials that efficaciously simulates the Pauli principle by considering the change in the intrinsic kinetic energy. However, it is possible to reproduce the observed half-life data using the potentials that have no automatic internal pocket with applying the BS quantization condition with quantum numbers which are significantly less than that obtained from the WT rule, upon properly normalizing the potential.

Topics & Concepts