December 2013
Intermediate to advanced
392 pages
13h 29m
English
Content preview from Compact Plasma and Focused Ion Beams
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158 Compact Plasma and Focused Ion Beams
On the substitution of Equation 9.7 in Equation 9.6, we can get two inde-
pendent equations for the radial and the axial density. The equation of the
radial part can be written as
∂
∂
∂
∂ab
2
2
11
0
Nr
r
r
D
Nr
r
N
r
RR
r
R
() ()
()
() ,++ =
+/
(9.8)
which is a Bessel’s equation (of order zero) and has a solution given by
Nr NJ
r
D
R
r
()
()
,
/
=
[]
+
+00
12
/ ab
(9.9)
where N
+0
is the plasma density at the center of the tube (r = 0). To satisfy the
boundary condition N
R
(r) = 0 at r = R, where R is the radius of the waveguide
and for a nite N
+0
, we must set a/[D
r
/(
α
+
β
)]
1/2
equal to a constant c, where
c = 2.4 for the rst zero o
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