Fundamentals of Plasma Physics

3.7 Non-adiabatic motion in symmetric geometry 105

(a)

(b)

particle

v
vz 0 ẑ

B

adiabatic non-adiabatic adiabatic

region

solenoid coils

flux surface

cusp cusp-trapped particle cusp

Figure 3.16: (a) Cusp field showing trajectory for particle with sufficient initial energy to
penetrate the cusp;(b) two cusps used as magnetic trap to confine particles.

Suppose the particle penetrates the cusp and arrives at some region whereagainψ∼r^2.

Since the particle is now axis-encircling, the relation between canonical momentum and

flux isPθ=−qψ/ 2 π=−q(−Bz 0 πr^2 )/ 2 π=qBz 0 r^2 / 2 from which it is concluded that

r=a.Thus,ifthe particle is able to move across the cusp, it becomes an axis-encircling

particle with thesameradiusr=ait originally had when it was non-axis-encircling. The

minimum energy an axis-encircling particle can have is when it is purely axis encircling,

i.e., hasvr= 0andvz= 0.Thus, for the particle to cross the cusp and reach a location

where it becomes purely axis-encircling, the particle’s initial energy must satisfy

mvz^20

2

≥

m(ωca)^2

2

(3.175)

or simply

vz 0 ≥ωca. (3.176)

Ifvz 0 is too small to satisfy this relation, the particle reflects from the cusp and returns

back to the negativezhalf-plane. Plasma confinement schemes have been designed based

on particles reflecting from cusps as shown in Fig.3.16(b). Here a particle is trapped be-

tween two cusps and so long as its parallel energy is insufficient to violate Eq.(3.176), the