Environmental Engineering Reference
In-Depth Information
From (2.127) we have for the energy
Δ
E
that an electron loses in one collision
Z
1
Z
1
2
e
2
3
c
3
R
r
4
3
c
3
j
r
ω
j
e
2
8
π
ω
2
dt
2
Δ
E
D
D
S
ω
d
ω
,
S
ω
D
,
1
0
where
S
ω
d
ω
is the energy emitted during collision in the frequency range from
ω
to
,weobtain
from this for the differential cross section of emission in electron-atomic particle
collision in the classical case
ω
C
d
ω
. Transferring to the rate of the radiative transition
S
ω
/
„
ω
Z
d
σ
S
ω
„
ω
ω
D
d
σ
,
(2.128)
d
0
where
d
is the differential cross section of electron scattering, and this formula
holds true if the electron energy variation in this process is relatively small, that is,
σ
„
ω
ε
,
where
is the energy of a scattered electron.
We now consider bremsstrahlung as a result of electron scattering by atoms
ε
e
C
A
!
e
C
A
C„
ω
.
(2.129)
On the basis of (2.128) and following [161, 162], we use the pulse approximation in
which
R
r
D
Δ
v
δ
(
v
), which gives
Z
1
dt
2
π
D
Δ
v
e
i
ω
t
R
r
ω
D
R
r
,
2
π
1
since
. Because the change of the electron
velocity after elastic scattering by an atom is
ωτ
1 for a typical collision time
τ
Δ
v
D
2
v
sin(
#
/2), where
#
is the
scattering angle, we obtain for the radiated energy per unit frequency
e
2
3
c
3
jR
r
ω
j
4
e
2
2
8
π
v
2
S
ω
D
D
c
2
(1
cos
#
).
3
π
This gives for the differential cross section of bremsstrahlung in an electron-atom
collision on the basis of (2.128)
Z
1
4
e
2
2
c
2
σ
(
v
) ,
d
σ
S
ω
„
ω
v
ω
D
d
σ
D
(2.130)
d
3
π
„
ω
0
σ
D
R
(1
#
)
d
where
is the diffusion cross section of electron-atom scat-
tering. In deriving this formula we assumed that the electron scattering proceeds
in a narrow range
cos
σ
Δ
r
a
0
of electron distances from an atom. Together with the
