Pairing, quasi-spin and seniority

TL;DR

This paper provides a detailed overview of pairing interactions, quasi-spin formalism, and seniority in nuclear physics, including derivations of matrix elements and eigenvalues relevant for mean-field theories.

Contribution

It offers a comprehensive derivation of the pairing Hamiltonian matrix elements and eigenvalues using quasi-spin formalism, aiding in understanding nuclear pairing phenomena.

Findings

Derivation of matrix elements for the delta interaction

Eigenvalues for the pairing Hamiltonian obtained via quasi-spin formalism

Explicit algebraic techniques for quasi-spin operators

Abstract

We present our concise notes for the lectures and tutorials on pairing, quasi-spin and seniority delivered at SERB school on Role of Symmetries in Nuclear Physics, AMITY University, $2019$. Starting with some generic features of residual nucleon-nucleon interactions, we provide detailed derivation of the matrix elements for the $\delta$-interaction which is the basis for the standard pairing Hamiltonian. The eigen values for standard pairing Hamiltonian are then obtained within the quasi-spin formalism. The algebra involving quasi-spin operators is performed explicitly using the annihilation and creation operators for single nucleon together with the application of Wicks theorem. These techniques are expected to be helpful in deriving the mean-field equations for the $Hartree-Fock$, $Bardeen-Cooper-Schrieffer$ and $Hartree-Fock$ Bogoliubov theories.