Magic numbers for vibrational frequency of charged particles on a sphere

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