Vlasov theory of electrostatic modes in a finite length electron column

A Vlasov theory of low-frequency electrostatic modes in a finite length electron column is presented. The column is assumed to have cylindrical symmetry and flat end surfaces at which the electrons undergo specular reflection. The eigenfrequencies and the eigenfunctions are obtained as a series expansion in the small parameter R/L (where R is a characteristic radius and L the half-length). In zeroth order, the modes are simply the modes for an infinitely long column with axial wavenumbers quantized as k=nπ/L. In first order, these modes are weakly coupled. This means that a k=0 diocotron mode can share in the Landau damping of higher k modes. For certain values of the plasma parameters, two zero-order modes are degenerate and the coupling is strong.