Vertical modes of two-dimensional dust coulomb clusters in complex   plasmas
  The vertical as well as horizontal oscillation modes in a thermally excited two-dimensional (2D) dust coulomb cluster were investigated for particle numbers between and using both a box_tree simulation and an analytical method. The horizontal mode spectra is shown to agree with published results while the vertical mode spectra obtained from the box_tree simulation and the analytical method are shown to agree with one another. The maximum frequency of the vertical modes is shown to be the vertical oscillation frequency of the whole system acting as a solid plane and the frequency is shown to decrease as the mode number l decreases. The oscillation patterns of the vertical modes are also investigated. For clusters with large numbers of particles, the high frequency modes are shown to have oscillation patterns similar in shape to Bessel-Fourier functions with various index m and n. For specified values of n, modes with higher m (except for) have lower frequencies and for specified m, modes with higher n have lower frequencies. For low frequency modes, the largest amplitude particle motion is concentrated in a few inner rings with the outer rings remaining almost motionless. This is in contrast to the horizontal modes where the strongest motion of the particles is concentrated in the inner rings at high frequency. For clusters with small numbers of particles (), these modes where the strongest vertical motion is concentrated in the inner rings does not exist resulting in oscillation magnitudes for all the modes appearing in the shape of Bessel-Fourier functions. 
K. Qiao and T.W. Hyde