Description of $\alpha$-cluster tail in $^{8}$Be and $^{20}$Ne: Delocalization of $\alpha$ cluster because of quantum penetration

TL;DR

This paper investigates the nature of the outer tail in alpha-cluster wave functions of $^{8}$Be and $^{20}$Ne, revealing that a Yukawa-like tail better describes delocalized alpha clusters due to quantum penetration effects.

Contribution

It introduces and compares various trial wave functions, demonstrating that Yukawa tails effectively capture the delocalized alpha-cluster behavior in weakly bound states.

Findings

Yukawa tail wave function matches the exact wave function better than Gaussian-based functions.

Outer tail characterized by Yukawa-like decay indicates delocalized alpha clusters.

Quantum penetration explains the slow damping of the outer tail in weakly bound states.

Abstract

We analyze the $\alpha$-cluster wave functions in cluster states of $^8$Be and $^{20}$Ne by comparing the exact relative wave function obtained by the generator coordinate method (GCM) with various types of trial functions. For the trial functions, we adopt the fixed range shifted Gaussian of the Brink-Bloch (BB) wave function, the spherical Gaussian with the adjustable range parameter of the spherical Thosaki-Horiuchi-Schuck-R\"opke (sTHSR), the deformed Gaussian of the deformed THSR (dTHSR), and a function with the Yukawa tail (YT). The quality of the description of the exact wave function with a trial function is judged by the squared overlap between the trial function and the GCM wave function. The better result is obtained with the sTHSR wave function than the BB wave function, and further improvement can be done with the dTHSR wave function because these wave functions can describe the outer tail better. The YT wave function gives almost the equal quality to or even better quality than the dTHSR wave function indicating that the outer tail of $\alpha$ cluster states is characterized by the Yukawa-like tail rather than the Gaussian tail. In the weakly bound $\alpha$ cluster states with the small $\alpha$ separation energy and the low centrifugal and Coulomb barriers, the outer tail part is the slowly damping function described well by the quantum penetration through the effective barrier. This outer tail characterizes the almost zero-energy free $\alpha$ gas behavior, i.e., the delocalization of cluster.