Vibrational modes of thin oblate clouds of charge

A numerical method is presented for finding the eigenfunctions ͑normal modes͒ and mode frequencies of azimuthally symmetric non-neutral plasmas confined in a Penning trap whose axial thickness is much smaller than their radial size. The plasma may be approximated as a charged disk in this limit; the normal modes and frequencies can be found if the surface charge density profile ␴(r) of the disk and the trap bounce frequency profile ␻ z (r) are known. The dependence of the eigenfunctions and equilibrium plasma shapes on nonideal components of the confining Penning trap fields is discussed. The results of the calculation are compared with the experimental data of Weimer et al. ͓Phys. Rev. A 49, 3842 ͑1994͔͒ and it is shown that the plasma in this experiment was probably hollow and had mode displacement functions that were concentrated near the center of the plasma.