# Self-organization of charged particles in circular geometry

## TL;DR

This paper investigates the self-organization of charged particles confined in circular geometries, proposing a new system of equations to efficiently determine equilibrium configurations and revealing a transition to a hexagonal lattice structure as particle number increases.

## Contribution

It introduces a novel approach to find equilibrium states of charged particles in circular confinement, reducing computational effort and aligning well with molecular dynamics results.

## Key findings

- Equilibrium configurations can be efficiently determined for large particle numbers.
- A transition from circular rings to a hexagonal lattice occurs as particle number exceeds 180.
- The approach shows remarkable agreement with molecular dynamics simulations.

## Abstract

The basic principles of self-organization of one-component charged particles, confined in disk and circular parabolic potentials, are proposed. A system of equations is derived, that allows us to determine equilibrium configurations for an arbitrary, but finite, number of charged particles that are distributed over several rings. Our approach reduces significantly the computational effort in minimizing the energy of equilibrium configurations and demonstrates a remarkable agreement with the values provided by molecular dynamics calculations. With the increase of particle number n>180 we find a steady formation of a centered hexagonal lattice that smoothly transforms to valence circular rings in the ground state configurations for both potentials.