Environmental Engineering Reference
In-Depth Information
Si
nce
Δ
U
U
, the distribution function for the interaction potentials yields
U
D
0 and has the form
U
2
)
1/2
exp
U
2
,
U
2
f
(
U
)
dU
D
(2
π
Δ
(1.17)
2
Δ
where
f
(
U
)
dU
is the probability that the interaction potential lies in the interval
from
U
to
U
C
dU
. The squared deviation of the distribution (1.17) is
Z
1
U
2
)
2
N
0
4
r
2
dr
N
0
e
4
r
D
(
U
)
2
,
Δ
D
U
2
D
2
(
e
'
π
D
4
π
D
UT
0
where the factor 2 takes into account the presence of charged particles of the op-
posite sign, and
N
0
is the mean number density of charged particles of one sign.
Thus, we have for an ideal plasma
U
Δ
U
T
.
1.1.8
Microfields in an Ideal Plasma
The charged plasma particles, electrons and ions, create electric fields in a plasma
that act on atoms and molecules in a plasma. Although the average field at each
point is zero, at this time electric fields are of importance. Guided by the action
of these fields on atoms, for a corresponding time range one can average the fields
from electrons, that is, the action of electrons on atoms as a result of their collision.
Fields from ions at a given point are determined according to their spatial distri-
bution. We will determine below the distribution function
P
(
E
) for electric fields
at a given point for a given time for a random spatial distribution of ions, so by
definition
P
(
E
)
dE
is the probability that the electric field strength at a given point
ranges from
E
to
E
dE
, and we use an isotropic form of the distribution over the
electric field strengths. We take into account the electric field strength from the
i
th
ion at a distance
r
i
from it is
C
e
r
i
r
i
E
i
D
,
and the total electric field of all the ions is
X
E
D
E
i
.
i
Evidently, we include in this sum ions located at a distance
r
i
r
D
from a test
point, that is, we assume that the main contribution to this sum is from ions located
close to a test point. In addition, we take a random distribution of ions in space.
We first find the tail of the distribution function that corresponds to large fields.
The probability of the nearest ion being located in a distance range from
r
to
r
C
dr
is
r
2
drN
i
,
P
(
E
)
dE
D
4
π
