Fundamentals of Plasma Physics

3.5 Drift equations 89

distanceLof the mirror-trapped particle slowly decreases. This would typically occur by

reducing the axial separation between the coils producing the magnetic mirror field. Be-

causeJ∼v‖Lis invariant, the particle’s parallel velocity increases on each successive

bounce asLslowly decreases.This steady increase inv‖means that the velocity angle

θdecreases. Eventually,θbecomes smaller thanθtrapwhereupon the particle becomes

detrapped and escapes from one end of the mirror with a large parallel velocity. This mech-

anism provides a slow pumping to very high energy, followed by a sudden and automatic

ejection of the energetic particle.

3.5.8 The third adiabatic invariant

Consider a particle bouncing back and forth in either of the two geometries shown in

Fig.3.10. In Fig.3.10(a), the magnetic field is produced by a single magnetic dipole and

the field lines always have convex curvature, i.e. the radius of curvature is always on the

inside of the field lines. The field decreases in magnitude with increasing distance from the

dipole.

(a) (b)

Figure 3.10: Magnetic field lines relevant to discussion of third adiabatic invariant: (a) field
lines always have same curvature (dipole field), (b) field lines have both concave and con-
vex curvature (mirror field).

In Fig.3.10(b) the field is produced by two coils and has convex curvature near the

mirror minimum and concave curvature in the vicinity of the coils. On defining a cylindrical