A computer code for calculations in the algebraic collective model of the atomic nucleus

TL;DR

This paper introduces a Maple software tool for algebraic calculations in the collective model of atomic nuclei, leveraging group theory to analyze a wide class of Hamiltonians involving quadrupole moments.

Contribution

The paper provides a new computational implementation of the algebraic collective model, including derivation and efficient calculation of matrix elements using group-theoretic methods.

Findings

Enables analysis of Hamiltonians quadratic in quadrupole moments

Uses analytical expressions for SO(5) matrix elements

Provides precomputed Clebsch-Gordan coefficients for efficiency

Abstract

A Maple code is presented for algebraic collective model (ACM) calculations. The ACM is an algebraic version of the Bohr model of the atomic nucleus, in which all required matrix elements are derived by exploiting the model's SU(1,1) x SO(5) dynamical group. This paper reviews the mathematical formulation of the ACM, and serves as a manual for the code. The code enables a wide range of model Hamiltonians to be analysed. This range includes essentially all Hamiltonians that are rational functions of the model's quadrupole moments $q_M$ and are at most quadratic in the corresponding conjugate momenta $\pi_N$ ($-2\le M,N\le 2$). The code makes use of expressions for matrix elements derived elsewhere and newly derived matrix elements of the operators $[\pi\otimes q \otimes\pi]_0$ and $[\pi\otimes\pi]_{LM}$. The code is made efficient by use of an analytical expression for the needed SO(5)-reduced matrix elements, and use of SO(5)$\,\supset\,$SO(3) Clebsch-Gordan coefficients obtained from precomputed data files provided with the code.