# Algebraic cluster models calculations for shape phase transitions of boson-fermion systems

## TL;DR

This paper develops an algebraic cluster model using affine Lie algebra to analyze shape phase transitions in boson-fermion systems, providing a framework for studying multi-body nuclear structures and their energy spectra.

## Contribution

It introduces a solvable extended transitional Hamiltonian within the algebraic cluster model to describe shape transitions in boson-fermion systems, applicable to various multi-body nuclear configurations.

## Key findings

- Energy levels and wave function overlaps calculated
- Behavior of boson number expectation values analyzed
- Coupling effects at shape transition points discussed

## Abstract

The Algebraic Cluster Model(ACM) is an interacting boson model that gives the relative motion of the cluster configurations in which all vibrational and rotational degrees of freedom are present from the outset. We schemed a solvable extended transitional Hamiltonian based on affine $ {{SU(1,1)}} $ Lie algebra within the framework for two-, three- and four- body algebraic cluster models that explains both regions $ O(4)\leftrightarrow U(3) $, $ O(7)\leftrightarrow U(6) $ and $ O(10)\leftrightarrow U(9) $, respectively . We offer that this method can be used to study of $k\alpha + x$ nucleon structures with k = 2, 3,4 and x = 1, 2, . . . , in specific x = 1,2 such as structures $^{9}Be$,$^{9}B$,$^{10}B$ ; $^{13}C$, $^{13}N$, $^{14}N$; $^{17}O$, $^{17}F$. Numerical extraction to the energy levels, the expectation value of boson number operator and behavior of the overlap of the ground-state wave function within the control parameters of this evaluated Hamiltonian are presented. The effect of the coupling of the odd particle to an even-even boson core is discussed along the shape transition and, in particular, at the critical point.