The rigorous criteria for the phase transitions in Coulomb crystals

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.