Environmental Engineering Reference
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instead of the Thomson formula (2.60) [8, 106, 107]
1
! ,
D π
e 4
m e v
e
σ
C
(2.62)
ion
J
3 J
ε
where m e is the electron mass,
v e is the velocity of a bound electron, and an overline
means an average over velocities of a bound electron. In the limit
0this
formula is transformed into the Thomson formula (2.60). Since the average kinetic
energy of a bound electron is compared with the atomic ionization potential, taking
into account the velocity distribution for a bound electron leads to a remarkable
change of the ionization cross section compared with the Thomson formula (2.60).
In particular, if a bo und electron is located mostly in the Coulomb field of an atomic
core, we have m e v
D
v e
J , and (2.62) differs from the Thomson formula (2.60) by a
factor of 5/3. Accounting for this effect allows one to describe correctly the behavior
of the ionization cross section at large electron energies.
Note that according to (2.48) the ionization cross section at high electron ener-
gies
e /2
D
[78] according to the Born approximation,
whereas classical theories, including the Thomson model, give
ε
depends on this energy as ln
ε
/
ε
.The
logarithmic dependence in the quantum approach is determined [108] by large col-
lision impact parameters, whereas the ionization probability is small or is zero for
classical considerations. One can consider this as a principal contradiction between
the quantum and classical approaches. This also occurs for ionization of excited
atoms when the classical description of an excited electron holds true. This con-
tradiction was overcome by numerical analysis by Kingston [109-111], who proved
that although the quantum (Born approximation) and classical approaches are de-
scribed by different formulas, the values of the ionization cross sections for these
cases are similar for excited atoms at high collision energies. This means that the
different energy dependence for the classical and quantum ionization cross sec-
tions is not of importance.
The advantage of the Thomson model consists in its simplicity and the correct
qualitative description of the ionization process. We now use the Thomson model
for atom excitation with an electron transition in highly excited states. If the group
of excited states is characterized by the principal quantum number n and the exci-
tation energy of these states is
σ
1/
ε
ion
Δ ε
n , we assume the excitation energy
Δ ε
lies in the
range
J 0
J 0
J
1/2) 2 Δ ε
J
,
( n
( n
C
1/2) 2
where J 0 is the ionization potential of the hydrogen atom and J is the ionization
potential of a given atom. For the excitation cross section in a highly excited state
this gives
2
π
e 4
J 0
σ
D
.
(2.63)
n
ε Δ ε
n n 3
of sec-
ondary electrons in collisions of a fast electron with an atom. On the basis of the
Let us also use the Thomson model to determine the average energy
Δ ε
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