Environmental Engineering Reference
In-Depth Information
of this with a simple example. For this goal we take a weakly ionized gas at a low
temperature and find the dependence of the plasma parameter
defined by (1.78)
on the plasma density when there is equilibrium between charged and neutral
plasma particles (ionization equilibrium). From the Saha distribution (1.52) for the
electron number density N e and the atom number density N a in a quasineutral
plasma, we obtain the relation
γ
g m e T e
2
exp
,
3/2
N e
N a
J
T e
D
π
2
where g
g e g i / g a ,where g e , g i ,and g a are the statistical weights of electrons,
ions, and atoms, respectively, T e is the electron temperature, and J is the atomic
ionization potential. We take the total number density of nuclei, N
D
D
N e
C
N a ( N e
D
N i ), as a parameter, and determine the dependence
γ
( N ).
Let us write the Saha distribution in the form
N
e 6
exp
,
γ
T e
C
T 9/2
J
T e
2
γ
D
e
gm 3/2
) 3/2 ]. Concentrating our attention on plasmas with the
maximum departure from ideal plasma conditions, we choose the plasma temper-
ature at a given N such that
e 12 /[
3 (2
where C
D
π
e
γ
is maximal. The condition d
γ
/ dT e
D
0 leads to the
expressions
9 T e /2) 2 m e T e
3/2
exp
,
g 3 T ( J
3 T e /2)
J
T e
N
D
(1.83)
( J
2
π
2
N J
.
9 T e /2
N e
D
(1.84)
J
3 T e /2
The maximum values of
2 J /9, and, in the limit of large N ,the
temperature approaches 2 J /9. In this limit the degree of plasma ionization goes to
zero as N increases, and the plasma parameter
γ
are located at T e
increases with the increase of N .
On the basis of the above equations, we analyze the limiting case where there is ba-
sic violation of the conditions for an ideal plasma. We take into account that (1.83)
and (1.84) are valid for an ideal plasma. The maximum values of
γ
at a given large
N lead to the expressions for the temperature T e and the degree of ionization N e / N :
γ
m e T 0
2
3/2
( T e
T 0 ) 2
4 g
9 N
e 9/2 ,
D
(1.85)
T e
π
2
N e
N D
3
2
( T e
T 0 )
,
(1.86)
T 0
where T 0 is defined as T 0
D
2 J /9. It is evident that the plasma parameter behaves
γ Na 0 1/2
as
in the limit of large N , and the coupling constant of the plasma
Γ Na 0
1/6
is estimated to be
,where a 0 is the Bohr radius. In particular, in the
limit of large N , these expressions for a hydrogen plasma become
N e
N D
3.7 Na 0
5.1
10 3
( Na 0 ) 1/2
1/2
2.5( Na 0 ) 1/6 ,
,
γ D
,
Γ D
(1.87)
 
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