Magic numbers for vibrational frequency of charged particles on a sphere
Magic numbers for vibrational frequency of charged particles on a sphere
复制标题
球体上带电粒子振动频率的幻数
作者：
S. Ono
寻找球体上 $N$ 电荷的最小能量分布称为汤姆森问题。在这里，我们研究 $N$ 电荷在 $10le Nle 200$ 和高达 $N=372$ 的选定尺寸的谐波近似内处于最低能态的振动特性。最大频率$omega_{
m max}$ 随着 $N^{3/4}$ 的增加而增加，这是通过研究二维三角晶格的晶格动力学而合理化的。 $omega_{ 的 $N$ 依赖性
m max}$ 标识了幻数 $N=12, 32, 72, 132, 192, 212, 272, 282$ 和 372，反映了单粒子能量的强烈简并性和 $N$ 电荷形成的二十面体结构。 $N=122$ 未被识别为 $omega_{ 的幻数
m max}$ 因为前一个条件不满足。即使考虑高频的平均值，幻数概念仍然成立。幻数处的最大频率模式没有异常大的振荡幅度（即，不是缺陷模式）。
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 $10le Nle 200$ and for selected sizes up to $N=372$. The maximum frequency $omega_{
m max}$ increases with $N^{3/4}$, which is rationalized by studying the lattice dynamics of a two-dimensional triangular lattice. The $N$-dependence of $omega_{
m max}$ 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 $omega_{
m max}$ 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).