The rigorous criteria for the phase transitions in Coulomb crystals

TL;DR

This paper establishes rigorous criteria for identifying phase transitions in Coulomb crystals within linear ion traps by analyzing geometric and angular parameters, providing a systematic approach to determine transition points.

Contribution

It introduces a novel analytical and numerical framework based on cross size and polar angle metrics to precisely identify phase transition points in Coulomb crystals.

Findings

Criteria for 1D-2D phase transitions defined

Criteria for 2D-3D phase transitions established

Method applicable to various trap geometries

Abstract

The present work suggests rigorous criteria to determine phase transitions in Coulomb crystals in a linear ion trap. The proposed method is based on the analysis of a cross size $\rho_i$ and relative polar angle between neighboring particles $\Delta\phi_i$ as functions of the system parameters such as number of ions, mass and charge of ions, and trap geometry. The analytical interpretation of the phase transitions relies on the analysis of the cross size $\rho$ and a metric $\Phi$ dependent on the norm of the vector $||\overline{\Delta\phi}||$. We further demonstrate an analysis procedure for the numerical determination of extremes of interpolated functions $\rho$ and $\Phi$. 1D-2D and 2D-3D phase transitions points are defined as points of the greatest growth of functions $\rho$ and $\Phi$, respectively.