α-cluster-model description of nuclei

Abstract

The description of nuclei as a system of $\ensuremath{\alpha}$ particles is considered using a two-variable integrodifferential equation describing $A$-boson systems. The method is based on the assumption that two-body forces are the dominant ones within the system. This allows the expansion of the $A$-body wave function in Faddeev components which in turn can be expanded in potential harmonics that result either in a coupled system of differential equations in the hyper-radius $r$ or, when projected on the ${r}_{ij}$ space, in a single two-variable, integrodifferential equation that includes the two-body correlations exactly. The formalism can be readily applied to systems of up to $A\ensuremath{\sim}20$. Going beyond this number one encounters increasingly difficult numerical problems stemming mainly from the structure of the kernel in the integral. However, these problems can be eliminated by transforming the equation, when $A\ensuremath{\rightarrow}\ensuremath{\infty}$, into a new one having a kernel which has a simple analytical form and is easy to use in calculations. We employed the transformed equation to investigate the possibility of describing nuclei consisting of $A$ $\ensuremath{\alpha}$ particles. It was found that for the Ali-Bodmer potential the $A=5$ system, i.e., the ${}^{20}$Ne, is the most stable while the $A=10$ system, i.e., the ${}^{40}$Ca, the binding energy has a maximum. Various aspects concerning the formation of $A\ensuremath{\alpha}$ nuclei are discussed.