Energy and symmetry of self-assembled two-dimensional dipole clusters in magnetic confinement

We report on confined two-dimensional (2D) dipole clusters formed by small ferromagnetic particles floating at the liquid-air interface and confined by nonuniform external magnetic field. The particles self assemble into hexagonally ordered clusters whose lattice constant can be magnetically tuned. We study the area S, the energy E, the chemical potential $\mu ,$ and the lattice constant a, of 2D clusters as functions of particle number N for $N<\mathrm{}130.\mathrm{}$ We develop a continuum approximation which accounts fairly well for the smooth part of $\mu \mathrm{}\left(N\right),S\left(N\right),$ and $a\left(N\right)\mathrm{}$ dependences. In addition to these dependences, we observe quasiperiodic fluctuations with dips at “magic” numbers corresponding to particularly symmetric particle configurations. We demonstrate that these fluctuations are related to the cluster symmetry and to the cluster center of mass position.