Environmental Engineering Reference
In-Depth Information
range of impact parameters between
v
,where
N
is the number density of gas particles and
v
is the relative velocity of the collision.
From this we find that the number of particles scattered per unit time into a given
solid angle is 2
and
C
d
.Theparticlefluxis
N
π
d
N
, so the differential cross section in the classical case is
v
d
σ
D
2
π
d
.
(2.7)
Note that elastic scattering of particles determines transport parameters in gases
and plasmas. If we consider transport processes from the standpoint of the trajec-
tory of a test particle, transport parameters are determined by a significant change
of the particle trajectory that corresponds to particle scattering into a large angle.
Evidently, a typical scattering cross section for large angles is given by the relation
when the interaction potential at the distance of closest approach is comparable to
the kinetic energy of the colliding particles. Therefore, a typical cross section for
scattering into large angles is given by
2
0
, w e
U
(
σ
D
π
0
)
ε
.
(2.8)
In addition, we give one more definition of the differential cross section (2.7) on
the basis of the thought experiment that is represented in Figure 2.2. In this case
scattered particles pass through a gas layer of thickness
l
, where the number density
of scattered particles is
N
and all the beam particles in the scattering angle
# #
0
,
where sin
R
/
L
, are collected by a detector. We take the velocity of incident
particles
v
to be large compared with the thermal velocity of gas atoms, and for
definiteness
R
#
D
0
1. If the intensity of the beam of incident particles
is
I
0
and the intensity of detected particles is
I
, the cross section of scattering into
angles above
L
,or
#
0
#
0
is given by
I
0
I
Nl
σ
(
#
0
)
D
,
I
0
and in this definition we assume
Nl
σ
(
#
0
)
1. Correspondingly, the differential
cross section is given by
1
Nl
dI
I
0
d
σ
D
,
(2.9)
where the intensity variation
dI
is connected with a change of the angle
#
0
for
detected particles, which may be achieved by variation of the distance
L
.
Figure 2.2
A thought experiment for definition of the differential cross section of particle scat-
tering.
