Environmental Engineering Reference
In-Depth Information
One can analyze this problem from another standpoint. We can express the
gaseous state condition as
U
(
N
1/3
)
2
.
μ
v
This criterion signifies a weakly interacting system, since the interaction potential
of a test particle with its neighbors is small compared with its mean kinetic energy.
The notation
U
(
N
1/3
) signifies that the potential is evaluated at the mean distance
between particles,
N
1/3
. On the basis of this expression and relation (2.8), we
have
U
(
N
1/3
)
U
(
0
). For a monotonic interaction potential this is equivalent to
N
1/3
0
, and that inequality returns us to criterion (2.27).
The next step is to apply this criterion to a system of charged particles, that is, to
a plasma. Because of the Coulomb interaction between charged particles,
j
U
(
R
)
jD
e
2
/
R
, criterion (2.8) gives
e
4
/
T
2
σ
(2.28)
for a typical large-angle scattering cross section, where
T
is the average energy of
the particles. Specifically,
T
is their temperature expressed in energy units. The
condition (2.27) that an ensemble of particles be a true gas is transformed in the
case of a plasma to the ideality plasma criterion (1.3):
Ne
6
T
3
1,
where
N
is the number density of charged particles and
T
is their temperature.
2.1.7
Elastic Collisions of Electrons with Atoms
In analyzing elastic scattering of an electron by an atomic particle, we use the
quantum character of this process. Assuming that the effective electron-atom po-
tential is spherically symmetric, one can consider scattering of an electron by a
motionless atom to be independent for different electron momenta
l
.Thisisthe
basis of the partial wave method [29], where the scattering parameters are the
sum of these partial parameters of corresponding
l
. The characteristic of elec-
tron-atom scattering for a given
l
is the scattering phase
l
, which describes the
asymptotic wave function of the electron-atom system at large distances between
the electron and the atom. The connection between the differential cross section
d
δ
2
d
cos
σ
D
2
π
j
f
(
#
)
j
#
of electron-atom scattering, the scattering amplitude
f
(
#
),
σ
is expressed through the scattering phases
and the diffusion cross section
δ
l
,
as [2, 29]
1
X
1
2
iq
1)(
e
2
i
δ
l
f
(
#
)
D
(2
l
C
1)
P
l
(cos
#
),
l
D
0
X
1
4
q
2
σ
D
1) sin
2
(
(
l
C
δ
δ
1
) ,
(2.29)
l
l
C
l
D
0
