An algebraic approach for the six nucleon systems

Abstract

The no-core ab-initio shell model approach is successfully employed to study the medium mass nuclei [1]. This many-body technique becomes adaptable as the computational power grows due to technological advances. The theoretical advancements in the no-core frameworks would allow us to explore larger systems more computationally effectively. Using Jacobi coordinates in the harmonic oscillator (HO) basis is a popular approach for the s-shell nuclei [2]. The state vectors of the nuclear system must be antisymmetric and translationally invariant. To ensure the latter, one has to eliminate the center of mass (cm) motion. The Jacobi coordinates allow the explicit removal of the cm coordinate. For the p-shell nuclei, the traditional M-scheme approach is more popular than Jacobi coordinate approach. In said coordinates, the antisymmetrization procedure becomes very complicated due to the large symmetry group algebra. In our research, we employ the coefficients of the fractional parentage to get the antisymmetric state vectors for the six nucleon systems. We do it in binary cluster formalism by using the so-called Λ operators [3]. We antisymmetrize the state vectors in the more compact J-scheme with heavy employment of the angular momenta algebra. The Λ operators are constructed from the two-particle transposition operators of the symmetry group S6. We achieve the simplification of the antisymmetrization procedure by using the symmetry properties of the nucleus and the isospin formalism. The construction of the representations of the Λ operators in the HO basis requires complex representations of the transformations of the Jacobi coordinates. To solve this problem, we introduce an algebraic approach for the six-nucleon antisymmetric state vector construction.