Fundamentals of Plasma Physics

3.5 Drift equations 87

1. trapped particles – these haveW 0 < μBmaxand bounce back and forth between the
   mirrors of the magnetic well,


2. untrapped (or passing) particles – these haveW 0 > μBmaxand are retarded at the
   potential hills but not reflected.
   Sinceμ=mv^2 ⊥ 0 / 2 BminandW 0 =mv 02 / 2 ,the criterion for trapping can be written
   as
   Bmin
   Bmax

<

v^2 ⊥ 0

v^20

. (3.115)

Let us defineθas the angle the velocity vector makes with respect to the magnetic field at

s 0 ,i.e.,sinθ=v⊥ 0 /v 0 and also define

θtrap= sin−^1

√

Bmin

Bmax

. (3.116)

Thus, as shown in Fig.3.9 all particles withθ > θtrapare trapped, while all particles with

θ < θtrapare untrapped. Suppose att= 0the particle velocity distribution ats 0 is

isotropic. After a long time interval long enough for all untrapped particles to have escaped

the trap, there will be no particles in theθ < θtrapregion of velocity space.The velocity

distribution will thus be zero forθ <θtrap;such a distribution function is called aloss-cone

distribution function.

mirror

trapped

loss

cone

v

trap

Figure 3.9: Loss-cone velocity distribution. Particles with velocity angleθ > θtrapare
mirror trapped, others are lost.

3.5.6 J, the Second Adiabatic Invariant

Trapped particles have periodic motion in the magnetic well, and so applying the concept