Environmental Engineering Reference
In-Depth Information
by an ion is
α
1/2
e
2
μ
v
σ
D
2
π
.
(2.21)
c
2
The final result shows that particle capture is determined by a long-range part of
the interaction potential. Specifically, relation (2.6) gives, for
r
0
D
r
c
,
U
(
r
c
)/
ε
D
2/(
n
D
. That is, the capture is
determined by the long-range part of the interaction. Since usually we have
n
2), and because of
ε
D
,wehave
j
U
(
r
c
)
j
1,
in the region
r
0
r
c
the interaction potential is small compared with the kinetic
energy of the particles. This means that the main contribution to the capture cross
section is from impact parameters
>
c
that correspond to particle capture.
Capture of particles is associated with a strong interaction. In fact, since
<
)
1/
n
r
c
r
min
(
D
/
ε
r
min
,
small distances between particles occur in the capture process. At these distances
the attractive interaction potential greatly exceeds the kinetic energy of the parti-
cles. A strong interaction of particles in this region is also associated with a strong
scattering of particles. Therefore, one can assume that the capture of particles leads
to their isotropic scattering, and the diffusion scattering cross section (2.8) almost
coincides with the capture cross section of (2.20). For instance, in the case of po-
larization interaction of particles, the diffusion cross section of particle scattering
exceeds the capture cross section by 10%.
2.1.5
Total Cross Section of Scattering
The total cross section for elastic scattering of particles results from integrating
over the differential cross sections for all solid angles, that is,
D
R
d
. In classi-
cal case, the total cross section must be infinite. Classical particles interact and are
scattered at any distances between them, and scattering takes place at any impact
parameters. Therefore, the classical total cross section will tend to infinity if
σ
σ
t
„!
0,
where
is the Planck constant.
We can evaluate the total collision cross section by assuming that the colliding
particles are moving along classical trajectories. The variation of the particle's mo-
mentum is given by the expression
„
Z
1
Δ
p
D
F
dt
,
(2.22)
1
where
F
D@
@
R
is the force with which one particle acts upon the other, and
U
is the interaction potential between the particles. From (2.22) it follows that
U
/
Δ
p
U
(
is the impact parameter. According to the Heisenberg uncertainty
principle, the value
)/
v
,where
Δ
p
can be determined up to an accuracy of
„
/
. Therefore,
