Geometry of N-particle clusters in two-dimensions: Some exact results

TL;DR

This paper analytically studies two-dimensional classical particle clusters under magnetic fields, revealing exact ground state configurations and transition points, independent of interaction specifics.

Contribution

It introduces a simple analytical method to determine exact ground states and transition points for 2D particle clusters in magnetic fields, independent of interaction details.

Findings

Minimum energy configurations are interaction-independent.

First shell transition occurs between 5 and 6 particles.

Results are exact and analytically proven.

Abstract

We report on a study of a finite system of classical confined particles in two-dimensions in the presence of a uniform magnetic field and interacting via a two-body repulsive potential. We develop a simple analytical method of analysis to obtain ground state energies and configurations. We prove analytically the minimum energy configurations are independent of the nature of two body interaction and the magnetic field. In particular we prove that the first transition from a single shell occurs when the number of particles changes from five to six. These results are exact.