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

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 3α systems are considered in detail. Simple analytical method of constructing the norm kernel for 3α system is suggested. The Pauli-allowed basis functions for the 3α 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= α+α eigenvalues has been given for the first time. Spectrum of the 12 C norm kernel is compared to that of the 5 H system.The question of the role of the Pauli principle in three-cluster systems goes back a long way in history. The 12 C nucleus has been established to exhibit three-alpha-cluster structure forty years ago. Since then many microscopic, macroscopic and semi-microscopic cluster models have been applied to analyze the structure of the ground and excited states of this nucleus. In particular, microscopic 3α calculation was performed within the resonating group method (RGM) by Kamimura [1] and within the generator coordinate method (GCM) by Uegaki [2]. Both calculations give reasonable results for the ground state of the 12 C and some excited states. However, wave functions provided by these models are very complicated and heavy to handle. Furthermore, although RGM ensures correct account of nucleon exchange between different clusters, an antisymmetry requirement on the total wave function can be violated by the improper truncation of model space. The measure of this violation is lacking. For example, in Ref.[1] the RGM calculation was performed with truncation to a space where the angular momentum l of α − α relative wave function was fixed to be equal to zero. At the same time, even the lowest Pauli-allowed state of the 12 C represents the mixture of l = 0, 2 and 4. This raises the question as to whether such truncation is consistent with the requirements of ⋆ This work was partly supported by the Program the Pauli principle. As for the GCM, seven-dimensional numerical integrals are involved in the calculation and the integration over generator coordinate is replaced into summation over some mesh-points. However, the generator coordinate is chosen to be real and its domain is not well-defined, while only complex generator parameters ensure the existence of inverse transition from the generator parameter space to the coordinate space [3].A number of macroscopic models were also applied to studying the 12 C nucleus (see, for example, Refs. [4,5]). Within such models, clusters are considered to be structureless particles interacting via local potentials, which reproduce experimental α − α phase shifts. But the antisymmetrization ...

Influence of the Pauli principle on two-cluster potential energy

“…Now let us consider wave functions of trapped and resonance states in the two-cluster systems. Wave functions of the resonance states in 8,7 Be and 6 Li in oscillator and momentum representation are shown in Fig. 5.…”

“…Wave functions of the trapped state in 5 He, 6 Li and 7 Be in oscillator representation are shown in Fig. 8.…”

“…The influence of the Pauli principle on the kinetic energy of relative motion of two clusters has been investigated in Refs. [3][4][5][6]. In the present paper we will consider only the first part of the cluster-cluster potential.…”

Three-body continuum states in a microscopic cluster model

“…The determination of overlap eigenvalues has also been presented in an algebraic version for three identical clusters in Ref. [38].…”

How the antisymmetrization affects a cluster–cluster interaction: Two-cluster systems

“…The influence of the Pauli principle on the kinetic energy of relative motion of two clusters has been investigated in Refs. [29,34,32,31]. It has been shown that the kinetic energy of a two-cluster system is not equivalent to the kinetic energy of a two-nucleon system due to the antisymmetrization effects.…”