Environmental Engineering Reference
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where the constant a does not depend on the temperature. Thus, the particle kinet-
ic energy per degree of freedom is T /2, and corres po ndingly the average particle
kinetic energy in the three-dimensional space is m
2 /2
D
3 T /2. These relations
v
can be used as the definition of the temperature.
1.2.5
The Saha Distribution
We considered above the distributions of gas particles in bound or free states. Now
we analyze the specific distribution for plasma systems that contain both bound
and free electron states. We must examine the equilibrium between continuous
and discrete electron states. This equilibrium is maintained by the processes
A C C
e
$
A ,
where e is the electron, A C is the ion, and A is the atom. We consider a quasineutral
plasma in which the electron and ion number densities are the same.
Consider an ionized gas in a volume
and denote the average number of elec-
trons, ions, and atoms in this volume by n e , n i ,and n a , respectively (note that
n e
Ω
n i here). Relation (1.43) gives the ratio between the number density of free
and bound states of electrons as
n i
n a
D
) 3 Z exp
d p d r .
p 2 /2 m e
T
g e g i
g a
1
J
C
D
(2
π
Here g e
2, g i ,and g a are the statistical weights of electrons, ions, and atoms
corresponding to their electronic states, J is the atomic ionization potential, and p
isthefreeelectronmomentum,so J
D
p 2 /(2 m e ) is the energy for transition from
the ground state of the atom to a given state of a free electron. It is assumed that
atoms are found only in the ground state.
Integration of this expression over electron momenta yields
C
m e T
2
3/2
exp
Z d r .
n i
n a
g e g i
g a
J
T
D
π
2
Integrating over the volume, we take into account that transposition of the states
of any pair of electrons does not change the state of the electron system. Therefore
R d r D Ω
/ n e . Introducing the number densities of electrons N e
D
n e /
Ω
,ions
N i
D
n i /
Ω
,andatoms N a
D
n a /
Ω
,wededucethat
m e T
2
exp
.
3/2
N e N i
N a
g e g i
g a
J
T
D
(1.52)
π
2
This result is called the Saha distribution [53].
One can write the Saha distribution in the form of the Boltzmann distribu-
tion (1.42) as
g a exp
,
m e T
2
3/2
N i
N a
g c
J
T
g e g i
N e
D
g c
D
,
(1.53)
π
2
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