DC sheath
We start with the Poisson equation A<J> = —
d2$
da;2
-—{ni- ne)
n0l [ eg#/fcTe
2q$
d /d$
da; \ da;
E2 - El
d<l> noq
dx eq
£0
3g*/fcTe
^'/kTe
2q$
1
1
2q$'
d$'
0      \ v "H UB
In order to shorten the notation, we will tranform the equation to dimensionless quantities
kfP
A
1 e0kTe
D
q v n0
The eq. (7) can be transformed with the use of (10) to
dip\      / dip
(e"'-7r^)v = 2(^-1 + ^^r-1)
dip
dip
5=0 y
+ 2 (e^ + V1 " 2¥ ~ 2)
An example of the solution calculated by means of eq. (9) is shown bellow:
1
2   Bohm velocity
Because the left-hand part of the eq. (7) is not negative and because d^' < 0, the following inequality must be valid:
^/kTe _        1 < o
J- 9
which leads to
2   ^ 1
mi l _ e-2g*/feTe •
This inequality must be valid also for small velues of the potential <J>, thus
o kT'p
(10)
The ion velocity \JkTejmi is called the Bohm velocity. It is the drift velocity of ions at the plasma-sheath border.
2
3   Floating potential
If there is no net electric current flowing through the sheath, the sheath voltage <J>j/ can be calculated from the equality of the electron and ion flow:
1        8kTe   Sill kTe
7 n0 \ - e feTe    =   no . -
4     V Ttme V mi
_ krTP       ??7 7 , .
«$f/    =--e- In-— (11)
H  11 2       27rme V '
4   Child-Langmuir law for collisionless sheath
The eq. (7) can be modified for assumptions ne pa 0, \rrii v\ <C q<& a Eq <C E and with the help of j = qriQVB to
E     pa--/ =--n0vB /    ^— = — a/-2V-$
o V   ™i "I o
d<J> / 7     /2m, ,     ,:
(-$)
I    =    3  / j_ fm£ 2V e0y 2q
x
9   V mi s2
5    Collisional sheath
(12)
j = qmvi = qnimE (13) We will assume a constant ion mobility /i^:
d2<J> qrii j
^ =   ~^ = ^oftf
d$ d2$ 2j
2__= _—
dx dx2 Eq Hi
— 2j
dx y e0 Mi
2  / 2i a
3 V £oMi
9eoW ^s2
8
=3
(14)
3
6   Sheath with homogeneous ion concentration (matrix sheath)
4