Magic numbers for vibrational frequency of charged particles on a sphere

TL;DR

This study investigates the vibrational frequencies of charged particles on a sphere, identifying specific 'magic numbers' where the maximum frequency exhibits notable degeneracy and structural symmetry, extending understanding of the Thomson problem.

Contribution

It introduces the concept of magic numbers based on vibrational frequency degeneracies and links them to geometric structures in the Thomson problem.

Findings

Maximum vibrational frequency scales as N^{3/4}

Identifies specific magic numbers with high degeneracy and icosahedral symmetry

Maximum frequency modes at magic numbers are not defect modes

Abstract

Finding minimum energy distribution of $N$ charges on a sphere is known as the Thomson problem. Here, we study the vibrational properties of the $N$ charges in the lowest energy state within the harmonic approximation for $10\le N\le 200$ and for selected sizes up to $N=372$. The maximum frequency $\omega_{\rm max}$ increases with $N^{3/4}$, which is rationalized by studying the lattice dynamics of a two-dimensional triangular lattice. The $N$-dependence of $\omega_{\rm max}$ identifies magic numbers of $N=12, 32, 72, 132, 192, 212, 272, 282$, and 372, reflecting both a strong degeneracy of one-particle energies and an icosahedral structure that the $N$ charges form. $N=122$ is not identified as a magic number for $\omega_{\rm max}$ because the former condition is not satisfied. The magic number concept can hold even when an average of high frequencies is considered. The maximum frequency mode at the magic numbers has no anomalously large oscillation amplitude (i.e., not a defect mode).