16 Non-neutral plasmas
16.1 Introduction
Conventional plasmas, the main topic of this book, consist of a quasi-neutral collection
of mutually interacting ions and electrons. This description appliesto the vast majority
of entities called plasma (e.g., fusion, industrial, propulsion, ionospheric,magnetospheric,
interplanetary, solar, and astrophysical plasmas). Non-neutral plasmas are an exception
to this taxonomy and not surprisingly, have certain behaviors which differfrom conven-
tional plasmas (Malmberg 1992, Davidson 2001). Applications of non-neutral plasmas
include microwave oscillators and amplifiers, simulating the vortex dynamics of ideal two-
dimensional hydrodynamics, testing basic non-linear concepts, and confining anti-particles.
16.2 Brillouinflow
Using a cylindrical coordinate system {r,θ,z} we first consider the forces acting on an
infinitely long, azimuthally symmetric cylindrical cloud of cold charged particles all having
thesamepolarity. Because of the mutual electrostatic repulsion of the same-sign charged
particles, the cloud of particles will expand continuously in therdirection and so will
not be in radial equilibrium. The cloud would be even further from equilibrium if, in
addition, it were to rotate with azimuthal velocityu=uθˆθ,because rotation would provide
a centrifugal forcemu^2 θ/rwhich would add to the radially outwards electrostatic force and
so increase the rate of radially outward expansion.
However, if this rotating cloud of same-sign charged particles were immersed in a uni-
form axial magnetic fieldB=Bzˆ, then the magnetic force∼qu×Bwould push the
charged particles radially inwards, counteracting both the electrostatic repulsion and the
centrifugal force. This situation is shown in Fig.16.1 for the situation where the cloud con-
sists of electrons (Malmberg and de Grassie 1975). Negatively biased electrodes are used
at the ends to prevent cloud expansion in thezdirection;if the cloud were made of ions,
then the end electrodes would be biased positively.