Environmental Engineering Reference
In-Depth Information
the main contribution to the total scattering cross section is given by values of the
impact parameter that satisfy the relation
Δ
p
(
)
„
/
. This leads to the estimate
t
, w e
t
U
(
t
)
2
σ
1
(2.23)
t
„
v
C
/
R
n
, the total cross
for the total scattering cross section. In particular, if
U
(
R
)
D
section is
C
„
v
2/(
n
1)
σ
.
(2.24)
t
Since the scattering cross section is determined by quantum effects, it approach-
es infinity in the classical limit. This is demonstrated by (2.24), which indeed ap-
proaches infinity in the limit
0.
Particle motion will obey classical laws if the kinetic energy
„!
ε
satisfies the condi-
tion
ε
„
/
τ
,
(2.25)
2
/2, from this it
where
τ
is a typical collision time. Since
τ
/
v
and
ε
D
μ
v
follows that
l
1, where
l
is the collision angular momentum of the
particles. If this criterion is fulfilled, the motion of the particles can be expect-
ed to follow classical trajectories. Furthermore, in this case the total cross sec-
tion
D
μ
v
/
„
for large-angle scattering given
by (2.8). In particular, for a monotonic interaction potential
U
(
R
), (2.8) and (2.23)
give
U
(
σ
t
is greatly in excess of the cross section
σ
0
)/
U
(
t
)
μ
v
/
„
1. It follows from the monotonic nature of
U
(
R
)
t
that
0
,so
t
σ
σ
.
(2.26)
t
Note that if condition (2.25) is not satisfied and scattering has a quantum nature,
then the large-angle scattering cross section and the total scattering cross section
have the same order of magnitude. In particular, this is the situation for elastic
scattering of electrons by atoms and molecules.
2.1.6
Gaseous State Criterion
The condition that defines the gaseous state of a system of particles can be formu-
lated in terms of the collision cross section. A gas is a system of particles with weak
interactions among them. This means that each particle follows a straight trajectory
most of the time. Only occasionally does a particle interacts strongly enough with
another particle to lead to large-angle scattering. This situation can be expressed by
stating that the mean fre
e
path of a particle,
λ
D
1/(
N
σ
), is large compared with
the interaction radius,
p
σ
. Thus, the condition to be satisfied for a system to be in
a gaseous state is
3/2
N
σ
1 .
(2.27)
