Electrostatic modes of ion-trap plasmas

The electrostatic modes of a non-neutral plasma confined in a Penning or Paul (rf) trap are discussed in the limit that the Debye length is small compared to the plasma dimensions and the plasma dimensions are small compared to the trap dimensions. In this limit the plasma shape is spheroidal and analytic solutions exist for all of the modes. The solutions for the modes of a Paul-trap plasma are a special case of the modes of a Penning-trap plasma. A simple derivation of some of the low-order quadrupole modes is given. Experimental measurements of these mode frequencies on plasmas of laser-cooled ${\mathrm{Be}}^{+}$ ions in a Penning trap agree well with the calculations. A general discussion of the higher-order modes is given. The modes provide a nondestructive method for obtaining information on the plasma density and shape. In addition, they may provide a practical limit to the density and number of charged particles that can be stored in a Penning trap.