Environmental Engineering Reference
In-Depth Information
where
q
is the electron wave vector, so the electron energy
ε
is expressed through
the electron wave vector by the relation
2
q
2
/2
m
e
(
m
e
is the electron mass).
Weaccountfortheelectronmassbeingsmallincomparisonwiththeatommass,
so the reduced mass of colliding particles coincides practically with the electron
mass. Note that the partial wave method is suitable for electron scattering by a
structureless atom.
In the limit of small
q
scattering parameters (2.29) are expressed through the
zero phase only, that is given by
ε
D„
Lq
in this limit, where
L
is the scattering
length. Correspondingly, the scattering parameters (2.29) are equal in this limit of
small
q
:
δ
D
0
σ
(0)
L
2
.
f
(
#
)
D
L
,
D
4
π
(2.30)
If we approximate the effective electron-atom interaction potential as a short-range
one, the effective interaction potential
U
(
r
) is expressed through the scattering
length
L
by formula [30, 31]
2
m
e
δ
L
„
U
sh
(
r
)
D
2
π
(
r
) ,
(2.31)
if the electron's wavelength exceeds the size of the atom's potential well. The scat-
tering length coincides with the atom's effective radius [32], if the assumption is
used that an electron cannot penetrate a region that is restricted by the effective
radius. In addition, Table 2.3 contains values of the electron scattering lengths for
scattering by inert gas atoms.
If the electron-atom scattering length is negative, the zero-scattering phase
δ
0
becomes zero at low electron energy when other scattering phases
l
are small. As
a result, the electron-atom cross section (both diffusion and total) acquires a deep
minimumat low electron energies that is known as the Ramsauer effect [35, 36]. As
follows from Tables 2.3 and 2.4, the Ramsauer effect is realized in the case of elec-
tron scattering by argon, krypton, and xenon atoms. Figure 2.7 gives experimental
values of the diffusion cross section of electron scattering by xenon atoms based
on measurements [37-41], and Figure 2.8 represents the diffusion cross sections
of electron scattering by inert gas atoms which result from the sum of measure-
ments [34].
In the case where the electron-atom interaction potential is the sum of a short-
range interaction potential (2.31) and the polarization interaction potential
U
l
(
r
)
δ
D
e
2
/2
r
4
as a long-range interaction potential, these interactions may be divided
at low electron energies, and the scattering phases
α
δ
l
may be represented in the
Ta b l e 2 . 3
The electron scattering lengths for scattering by inert gas atoms [33, 34].
Atom He
Ne
Ar
Kr
Xe
L
,
a
0
1.2
0.2
-1.6
-3.5
-6.5
