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 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 ${\omega }_{\max }$ increases with $N^{3/4}$, which is rationalized by studying the lattice dynamics of a two-dimensional triangular lattice. The $N$ dependence of ${\omega }_{\max }$ 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 ${\omega }_{\max }$ 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).

Figure 1

The phonon dispersion curve of 2D triangular lattice. ${\omega }_0=C{a}_0^{-3/2}$ with a unit mass and $C=3.383698$.

Figure 2

The $N$ dependence of (a) $E_{\mathrm{opt}}-E_{\mathrm{fit}}$, (b) ${\omega }_{\max }$, and (c) ${\omega }_{\max }-{\omega }_{\mathrm{eff}}$, where $E_{\mathrm{fit}}$ and ${\omega }_{\mathrm{eff}}$ are expressed by Eqs. (19) and (20), respectively. Dashed lines in (a) and (c) indicate the magic numbers for $E_{\mathrm{opt}}$, i.e., $N=12,32,72,122,132,137,146,182,187$, and 192 identified in Ref. [7].

Figure 3

The distribution of $N=122$ charges on a sphere. Four pentagons are colored red. Two pentagons share a hexagon colored orange. Three hexagons share a vertex colored blue.

Figure 4

The $D$ dependence of $\mathrm{\Delta }{\omega }_{\mathrm{ave}}(N,D)$ for several $N\mathrm{s}$.

Figure 5

The distribution of $|d{v}_i|$ versus ${\varepsilon }_i$ of the charge $i$ for $N=131,132$, and 133. The data is plotted for the top four frequency modes.