Environmental Engineering Reference
In-Depth Information
where the interaction potential U ( R ) is a function of distance and is of the order of
the energy of colliding particles. In particular, if the interaction potential is approx-
imated by the dependence
AR n ,
U ( R )
D
(2.13)
criterion (2.12) has the form
n
1.
As a demonstration of this, Table 2.1 contains the parameters of the interaction
potential for two inert gas atoms at distance R 0 , where the interaction potential
U ( R 0 )
0.3 eV. These data follow frommeasurements [9] of small-angle scattering
of inert gas atoms with kiloelectronvolt energies in inert gases. As is seen, criterion
(2.12) holds true in this range of atom interaction.
If we apply the hard-sphere model for a certain interaction potential, we can
estimate the hard-sphere radius R 0 from the relation
D
U ( R 0 )
ε
,
where
is the energy of colliding particles. One can determine this value more
accurately as a result of expansion of the cross section over a small parameter
1/ n
ε
1. Within the framework of this expansion, we have the dependence of
the scattering angle
#
as a function of the distance
of closest approach r 0 at a given value of the impact parameter
for a given impact parameter
[10, 11]:
2ln2 p u (1
2arcsin p u
u )
U ( r 0 )
ε
# D
nu /2 u
D
,
(2.14)
C
1
where the distance of closest approach r 0 depends weakly on the collision impact
parameter
.
Formula (2.14) allows us to determine certain integral cross sections for parti-
cle scattering in a sharply varied repulsive interaction potential. In particular, for
Ta b l e 2 . 1 The parameter n for the dependence (2.13) of the interaction potential of two inert
gas atoms and the distance R 0 , where the repulsive interaction potential U ( R 0 ) D 0.3 eV [9].
The values of R 0 are given in parentheses and are expressed in angstroms.
Interacting atoms He
Ne
Ar
Kr
Xe
He
5.9 (1.58)
5.6 (1.87)
5.2 (2.31)
5.5 (2.48)
5.2 (2.50)
Ne
-
7.6 (2.07)
6.6 (2.42)
7.6 (2.59)
6.8 (2.64)
Ar
-
-
6.1 (2.85)
6.9 (3.16)
5.9 (3.44)
Kr
-
-
-
7.7 (2.99)
7.1 (3.08)
Xe
-
-
-
-
6.4 (3.18)
 
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