Environmental Engineering Reference
In-Depth Information
Figure 2.4
The dependence of the distance of closest approach on the collision impact parame-
ter for the hard-sphere model.
Figure 2.5
The character of scattering for the hard-sphere model:
R
0
is the sphere radius,
θ
is
the scattering angle, and the thick line is the particle trajectory.
For the differential cross section of particles scattering within the framework of the
hard sphere model this gives
R
0
2
d
σ
D
2
π
d
D
d
cos
θ
,
where the azimuthal symmetry of scattering is taken into account. From this we
have for the diffusion
σ
and viscosity
σ
(2)
cross sections of particle scattering on
large angles
Z
(1
Z
(1
2
3
R
0
. (2.11)
σ
D
R
0
,
(2)
cos
2
cos
#
)
d
σ
D
π
σ
D
#
)
d
σ
D
Asisseen,theaveragecrosssections(2.11)areinagreementwiththeestimate(2.8)
for the cross section of scattering into large angles.
The hard-sphere model corresponds to a sharply varied repulsive interaction po-
tential according to the criterion
d
ln
U
d
ln
R
n
D
1 ,
(2.12)
