Environmental Engineering Reference
In-Depth Information
2.1.2
Model of Hard Spheres
A typical interaction potential of two atoms as a function of the distance between
them is given in Figure 2.3. If the energy of colliding particles is large enough,
ε
D
,where
D
is the depth of the interaction well, and scattering is determined
by the repulsive part of the interaction potential, one can approximate the interac-
tion potential in this range by the potential with an infinite well that is given in
Figure 2.3 and has the form
U
(
R
)
D
0,
r
>
R
0
I
U
(
R
)
D1
,
r
<
R
0
,
(2.10)
which corresponds to the hard-sphere model, and where
R
0
is a radius of a hard
sphere. The dependence of the collision impact parameter
on the distance of clos-
est approach
r
0
is given in Figure 2.4. The hard-sphere model for atom scattering
is used widely in the study of the kinetics of neutral gases. The hard-sphere model
has a long history and starts from the research of Maxwell [3-5], who on the basis of
the model for gas molecules as colliding billiard balls found both the Maxwell dis-
tribution for an ensemble of free identical particles and the transport parameters
for this ensemble.
The character of particle scattering within the framework of the hard-sphere
model for colliding particles in the center-of-mass frame of reference consists of
two straight lines (Figure 2.5 [6-8]), so their general point is located on the sphere
surface, and the angles
of the trajectory lines with the line passing through this
point and the sphere center are equal. These three lines are located in the same
plane, and according to Figure 2.5 the connection between angles
α
α
and
#
and
between the collision impact parameter
and the scattering angle
#
are given by
# D
π
2
α
,
D
R
0
sin
α
.
Figure 2.3
A typical interaction potential of two atoms (1) and the model potential (2) in the
hard-sphere model.
