Environmental Engineering Reference
In-Depth Information
above criterion
„
ω
ε
, this uses the condition
v
e
2
/
„
,thatis,theelectron
energy is small compared with typical atomic energies.
We now consider bremsstrahlung in an electron-ion collision that proceeds ac-
cording to the scheme
A
C
!
A
C
C„
ω
e
C
e
C
.
(2.131)
Assuming that an electron moves along classical trajectories, we represent (2.128)
in the form
Z
Z
1
1
d
σ
S
ω
„
ω
8
π
e
2
ω
4
2
2
ω
D
2
π
d
D
„
ω
j
r
ω
j
π
d
,
(2.132)
d
3
c
3
0
0
where
is the collision impact parameter. Considering scattering of a slow electron
2
) by an ion of charge
e
, we find below the dependence of the differ-
ential cross section of bremsstrahlung on the relevant parameters. The potential
energy of the electron at a distance
r
min
of closest approach exceeds significantly its
initial kinetic energy at given collision impact parameters
m
e
e
4
/
(
ε
„
, and hence according
to (2.6) we have the connection between these parameters
r
min
2
/
e
2
,where
ε
is the kinetic energy of the electron far from the ion. Typical collision parameters
for a given frequency
D
ε
ω
of an emitted photon are estimated as
ω
v
/
r
m
in
,where
v
is the electron velocity at the distance of closest approach, that is,
v
p
e
2
/
m
e
r
min
.
Thus, we have
e
2/3
m
1/3
e
8/3
m
1/3
e
2/3
m
1/3
r
min
ω
2
r
min
,
,
r
ω
.
ω
2/3
ω
2/3
ε
ω
5/3
e
e
e
From this consideration we have for the differential cross section of bremsstrahlung
as a result of electron-ion scattering [45]
2
e
2
4
r
2
min
ω
e
4
„
ω
ω
2
r
3
min
ε
e
6
d
σ
ω
ω
ω
c
3
.
d
„
ω
c
3
2
„
ω
m
e
c
3
ε
Note that this method of estimation of the cross section of electron-ion scattering
does not allow us to determine the numerical factor in the expression for the cross
section of bremsstrahlung, and this requires more cumbrous operations. Such an
action is given, for example, in [163] and for this cross section gives
e
6
3
p
3
m
e
c
3
d
σ
8
π
ω
D
.
(2.133)
d
ε
„
ω
This consideration holds tr
ue in
the classical limit, so large collision momenta
l
D
p
2
/
m
e
is the initial electron velocity far from the ion)
give the main contribution to the cross section. This leads to the criterion
D
m
e
v
0
/
„
1(
v
0
ε
„
ω
m
e
e
4
/
2
. Another requirement corresponds to the criterion
„
r
min
,whichgives
m
e
v
3
0
ω
.
e
2
