Linear stability analysis of the Vlasov-Poisson equations in high density plasmas in the presence of crossed fields and density gradients

The equations for the single-particle orbits in a nonneutral high density plasma in the presence of inhomogeneous crossed fields are obtained. Using these orbits, the linearized Vlasov equation is solved as an expansion in the orbital radii in the presence of inhomogeneities and density gradients. A model distribution function is introduced whose cold-fluid limit is exactly the same as that used in many previous studies of the cold-fluid equations. This model function is used to reduce the linearized Vlasov-Poisson equations to a second-order ordinary differential equation for the linearized electrostatic potential whose eigenvalue is the perturbation frequency.