Chemistry Reference
In-Depth Information
3.6.1.1 Generalized Bohm Sheath Criterion
A generalized Bohm sheath criterions was derived by [32] for the limit λ
De
0,
including electrons and negative ions as well as the 1D velocity distribution functions
of positive ions
f
→
(
v
+
)
1
v
2
+
ϕ
=
0
∞
1
m
+
1
v
2
+
1
m
+
·
1
d
(
n
e
+
n
−
)
·
f
(
v
+
)
·
dv
+
=
≤
)
·
.
(3.207)
e
·
n
+
(
0
d
ϕ
0
By use of Dirac δ function for the ion energy distribution function as well as
Maxwellian electrons, only, the Bohm sheath criterion (3.205) is reproduced.
∞
1
m
+
1
v
2
+
1
δ [
v
+
−
v
+
(
0
)
]
·
dv
+
=
m
+
·
v
2
+
(
0
)
−∞
d
ϕ
exp
e
ϕ
=
0
=
1
e
·
d
·
ϕ
1
k
B
·
≤
.
(3.208)
k
B
·
T
e
T
e
Another generalized Bohm sheath criterion is given by [33] taking into account the
electron velocity distribution function
f
(
v
e
)
1
v
2
+
∞
1
m
+
·
1
m
e
1
v
e
·
df
e
dv
e
·
≤−
dv
e
.
(3.209)
−∞
3.6.1.2 Plasma Sheath in Electronegative Gases
Equation 3.207 can be applied for electronegative plasmas with the approximation for
cold positive ions and Maxwellian electrons and negative ions. Taking into account
the quantities α
=
n
−
/
n
e
defined as ratio of negative ion to electron density and
γ
T
−
defined as ratio of the electron to negative ion temperature, it follows for
quasi-neutrality at the sheath edge:
=
T
e
/
n
+
(
0
)
=
(
1
+
α
0
)
·
n
e
(
0
)
,
(3.210)
exp
e
exp
γ
.
·
ϕ
·
e
·
ϕ
n
e
+
n
−
=
n
e
(
0
)
·
+
α
0
·
(3.211)
k
B
·
T
e
k
B
·
T
e
Inserting the electron and negative ion density (3.211) into (3.207) and using
monoenergetic positive ions at the sheath edge it follows
ϕ
=
0
=
1
1
d
(
n
e
+
n
−
)
1
+
α
0
·
γ
1
k
B
·
)
≤
)
·
·
(3.212)
m
+
·
v
2
+
(
0
e
·
n
+
(
0
d
ϕ
1
+
α
0
T
e
and the Bohm criterion for electronegative plasmas is achieved:
k
B
·
T
e
1
+
α
0
·
γ
)
≤
.
(3.213)
m
+
·
v
2
+
(
0
1
+
α
0
