On the structure and spectrum of classical two-dimensional clusters with a logarithmic interaction potential

TL;DR

This paper numerically investigates how logarithmic repulsive interactions influence the ground state configurations and vibrational spectra of confined classical two-dimensional particle clusters, revealing unique spectral properties and structural arrangements.

Contribution

It provides new insights into the spectral and structural behavior of 2D particle clusters with logarithmic interactions under various confinement potentials, including a special case with a particle-number-independent breathing mode frequency.

Findings

Particles form a single ring at the boundary in hard wall confinement.

Inner rings can form under general r^n confinement potentials.

The breathing mode frequency is independent of particle number for parabolic confinement.

Abstract

We present a numerical study of the effect of the repulsive logarithmic inter-particle interaction on the ground state configuration and the frequency spectrum of a confined classical two-dimensional cluster containing a finite number of particles. In the case of a hard wall confinement all particles form one ring situated at the boundary of the potential. For a general r^n confinement potential, also inner rings can form and we find that all frequencies lie below the frequency of a particular mode, namely the breathing-like mode. An interesting situation arises for the parabolic confined system(i.e. n=2). In this case the frequency of the breathing mode is independent of the number of particles leading to an upper bound for all frequencies. All results can be understood from Earnshaw's theorem in two dimensions. In order to check the sensitivity of these results, the spectrum of vortices in a type II superconductor which, in the limit of large penetration depths, interact through a logarithmic potential, is investigated.