Compact Plasma and Focused Ion Beams

24 Compact Plasma and Focused Ion Beams

If the variables are separated,

Hr

z

=Θ

()

()

.

qx

(2.15)

The equation for Θ can be written as

∂

∂q

2

2

2

0

Θ

+=

m ,

(2.16)

and the equation for

ξ

can be written as

∂

∂

∂x

∂

x

x

2

2

2

2

2

1

0

rrr

k

m

r

c

++−













= .

(2.17)

Here m

2

is a separation constant and

k

c

222

=+gwme .

The solution for Θ is

Θ=e

im

q

,

(2.18)

where

θ

can lie between 0 and an integral multiple (m) of 2

π

. Thus, m must

be an integer or zero (= 0, ±1, ±2, ±3 ...). The complex form of Θ indicates that

two solutions are possible—one is Θ = cos m

θ

and the other is Θ = sin m

θ

.

Thus, the modes are degenerate in pairs. The two modes may be interpreted

as two states of polarization ...