The role of the Pauli principle in three-cluster systems composed of identical clusters

TL;DR

This paper investigates how the Pauli principle influences the continuum wave functions of three-cluster nuclear systems, providing analytical methods and classifying eigenfunctions for systems like $^{12}$C.

Contribution

It introduces an analytical approach to construct the norm kernel for three-cluster systems and classifies eigenfunctions, highlighting the Pauli principle's role in nuclear structure.

Findings

Explicit form of basis functions for $3\alpha$ and $3d'$ systems.

Classification of eigenfunctions and eigenvalues of the $^{12}$C norm kernel.

Comparison of the $^{12}$C and $^{5}$H system spectra.

Abstract

Within the microscopic model based on the algebraic version of the resonating group method the role of the Pauli principle in the formation of continuum wave function of nuclear systems composed of three identical $s$-clusters has been investigated. Emphasis is placed upon the study of the exchange effects contained in the genuine three-cluster norm kernel. Three-fermion, three-boson, three-dineutron ($3d'$) and $3\alpha$ systems are considered in detail. Simple analytical method of constructing the norm kernel for $3\alpha$ system is suggested. The Pauli-allowed basis functions for the $3\alpha$ and $3d'$ systems are given in an explicit form and asymptotic behavior of these functions is established. Complete classification of the eigenfunctions and the eigenvalues of the $^{12}$C norm kernel by the $^8$Be$=\alpha+\alpha$ eigenvalues has been given for the first time. Spectrum of the $^{12}$C norm kernel is compared to that of the $^{5}$H system.