Fundamentals of Plasma Physics

106 Chapter 3. Motion of a single plasma particle

particle is confined between the two cusps.
Cusps have also been used to trap relativistic electron beams in mirror fields (Hudgings,
Meger, Striffler, Destler, Kim, Reiser and Rhee 1978, Kribel, Shinksky, Phelps and Fleischmann
1974). In this scheme an additional opposing solenoid is added to one end of a magnetic
mirror so as to form a cusp outside the mirror region. A relativistic electron beam is in-
jected through the cusp into the mirror. The beam changes from non-axis-encircling into
axis-encircling on passing through the cusp as in Fig.3.16(a). If energy is conserved, the
beam is not trapped because the beam will reverse its trajectory and bounceback out of the
mirror. However, if axial energy is removed from the beam once it is in the mirror, then the
motion will not be reversible and the beam will be trapped. Removal of beam axial energy
has been achieved by having the beam collide with neutral particles or by having the beam
induce currents in a resistive wall.

3.7.3 Stochastic motion in large amplitude, low frequency waves

The particle drifts (E×B, polarization, etc.) were derived using an iteration scheme which
was based on the assumption that spatial changes in the electric and magnetic fields are suf-
ficiently gradual to allow Taylor expansions of the fields about their values at the gyrocen-
ter.
We now examine a situation where the fields change gradually in space relative to the
initial gyro-orbit dimensions, but the fields also pump energy into the particle motion so
that eventually the size of the gyro-orbit increases to the point that the smallness assumption
fails. To see how this might occur consider motion of a particle in an electrostatic wave

E=ˆykφsin(ky−ωt) (3.177)

which propagates in a plasma immersed in a uniform magnetic fieldB=Bz.ˆ The wave
frequency is much lower than the cyclotron frequency of the particle in question. This
ω << ωccondition indicates that the drift equations in principle can be used and so ac-
cording to these equations, the charged particle will have both anE×Bdrift

vE=

E×B

B^2

=ˆx

kφ

B^2

sin(ky−ωt). (3.178)

and a polarization drift

vp=

m

qB^2

dE⊥

dt

=ˆy

mkφ

qB^2

d

dt

sin(ky−ωt). (3.179)

If the wave amplitude is infinitesimal, the spatial displacements associated withvEand
vpare negligible and so the guiding center value ofymay be used in the right hand side of
Eq.(3.179) to obtain

vp=−ˆy

ωmkφ

qB^2

cos(ky−ωt). (3.180)

Equations (3.178) and (3.180) show that the combinedvEandvpparticle drift motion
results in an elliptical trajectory.
Now suppose that the wave amplitude becomes so large that the particle is displaced
significantly from its initial position. Since the polarization drift is in theydirection, there
will be a substantial displacement in theydirection. Thus, the right side of Eq.(3.179)