104 Chapter 3. Motion of a single plasma particle
cross the cusp. Such an analysis is possible because two constants of the motion exist,
namelyPθandH. The energy
H=
Pr^2
2 m+
Pz^2
2 m+
(
Pθ−q
2 πψ(r,z)) 2
2 mr^2
=const. (3.170)can be evaluated using
Pθ=[
mr^2 θ ̇+q
2 πψ]
initial=
q
2 πψ 0 (3.171)since initiallyθ ̇= 0.Here
ψ 0 =ψ(r=a,z=−L) (3.172)is theflux at the particle’s initial position. Inserting initial values of all quantities in
Eq.(3.170) gives
H=
mv^2 z 0
2(3.173)
and so Eq.(3.170) becomes
mv^2 z 0
2=
mv^2 r
2+
mv^2 z
2+
(q
2 π) 2
(ψ 0 −ψ(r,z))^2
2 mr^2
=mv^2 r
2+
mv^2 z
2+
mv^2 θ
2.
(3.174)
The extent to which a particle penetrates the cusp can be easily determined if the particle
starts close enough tor= 0so that theflux may be approximated asψ∼r^2 .Specifically,
theflux will beψ=Bz 0 πr^2 whereBz 0 is the on-axis magnetic field in thez << 0 region.
The canonical momentum is simplyPθ=qψ/ 2 π=qBz 0 a^2 / 2 since the particle started as
non-axis encircling.