Angular Normal Modes of a Circular Coulomb Cluster

TL;DR

This paper analytically derives the eigenfrequencies of angular normal modes in a Coulomb cluster confined to a circle, revealing how these frequencies depend on particle number and symmetry.

Contribution

It provides an exact analytical solution for the eigenfrequencies of angular normal modes in a Coulomb cluster with radial symmetry, a novel result for such systems.

Findings

Eigenfrequencies are derived analytically for the system.

Largest eigenfrequency's dependence on particle number is characterized.

The dynamical matrix is identified as a circulant Laplacian matrix.

Abstract

We investigate the angular normal modes for small oscillations about an equilibrium of a single-component coulomb cluster confined by a radially symmetric external potential to a circle. The dynamical matrix for this system is a Laplacian symmetrically circulant matrix and this result leads to an analytic solution for the eigenfrequencies of the angular normal modes. We also show the limiting dependence of the largest eigenfrequency for large numbers of particles.