Modeling axisymmetric Bernstein modes in a finite-length non-neutral plasma

We have developed a 2-D PIC code to model high-frequency (near the cyclotron frequency) axisymmetric oscillations in a finite-length pure-ion plasma. We previously modeled these modes for infinite-length plasmas, where they are not detectable in the surface charge on the walls because of the axisymmetry and lack of z-dependence. This is not true in a finite-length plasma, however, because the eigenfunction of the oscillation has to have nodes a short distance beyond the ends of the plasma. This gives the modes a $\cos(k_z z)$ dependence, with a $k_z$ such that an integral number of half-wavelengths fit into the plasma. This $z$-dependence makes the mode detectable in the surface charge on the walls. We have modeled the plasma with different $k_z$ values and find that a larger value $k_z$ shifts the frequency downward by a small amount. The damping of the modes also increases as $k_z$ increases. The eigenfunction of the mode with the lowest-order radial dependence is linear in $r$, while higher-order radial modes behave as J$_1(k_r r)$. We will present the results of the properties of these different modes, along with a discussion of their dispersion relation and detectability.