462 Chapter 16. Non-neutral plasmas
is called the and it is seen that real rootsω 0 exist only if the density is sufficiently low for
ω^2 pto be below this limit.
We will consider the situation, typical for non-neutral plasma experiments, where the
density is well below the Brillouin limit so that the two roots are well-separated and given
by
ω 0 − ≃−ωc(
1+ω^2 p/ 2 ω^2 c)
(16.7)
ω0+ ≃ ω^2 p/ 2 ωc; (16.8)the plus and minus signs refer to the choice of signs in Eq.(16.6). The large rootω 0 −is near
the cyclotron frequency and the small rootω0+is much smaller than the plasma frequency
sinceωp<<ωc.The small root is called the diocotron frequency.
The rotation in the magnetic field provides an inward force balancing the outward radial
electrostatic force of the non-neutral plasma. There also exist axial electrostatic forces
due to the mutual electrostatic repulsion between the same-sign charges and these forces
would cause the plasma to expand axially. Since these axial forces cannot be balanced
magnetically, the axial forces are balanced by biased electrodes at the ends of the plasma.
The electrodes have the same polarity as the plasma, thereby providing a potential well in
the axial direction. Any particle which tries to escape axially isrepelled by forces due to
the repulsive bias of the end electrode and reflects before reaching the end electrode.
The fast motion of the charged particles in the axial direction smears out axial structure
so that the non-neutral plasma can be considered axially uniform to first approximation.
A pair of deceptively simple-looking coupled equations relating the electrostatic potential
φand the densityn govern not only the equilibrium but also the surprisingly rich low-
frequency dynamics of a non-neutral plasma. An important feature of these equations is
that theE×Bdrift
u=−∇φ×B
B^2(16.9)
describes an incompressibleflow, so that when this drift is inserted into the continuity
equation
∂n
∂t
+u·∇n+n∇·u=0, (16.10)the∇·uterm vanishes and the density time dependence results from convection only, so
that
∂n
∂t
=
∇φ×B
B^2
·∇n. (16.11)Equation (16.11) and Poisson’s equation
∇^2 φ=−nq
ε 0(16.12)
provide two coupled equations innandφand are the governing equations for the configu-
ration. Although Eqs.(16.11) and (16.12) appear simple, they actually allow quite complex
behavior which will be discussed in the remainder of this chapter. An interesting feature
of these equations is that they have no mass dependence, a consequence of ignoring the
centrifugal force term which affects only the high frequency branch, Eq.(16.7).Thus, the
low frequency dynamics described by Eqs.(16.11) and (16.12) can be thought of as the dy-
namics of a massless, incompressiblefluid governed by a combination ofE×Bdrifts and
Poisson’s equation.