Thermal X-ray radiation (PDF) - SRON

3.6.5 Collisional ionisation
Collisional ionisation occurs when during the interaction of a free electron with an
atom or ion the free electron transfers a part of its energy to one of the bound
electrons, which is then able to escape from the ion. A necessary condition is that
the kinetic energy E of the free electron must be larger than the binding energy I
of the atomic shell from which the bound electron escapes.
Q (m 2 )
0 2×10 −26 4×10 −26 6×10 −26
Fe I
Fe XXV
Fe XXVI
10 100 1000
Energy (keV)
Figure 3.14: Illustration of the Lotz formula for three iron ions. Shown is the K-shell
ionisation cross section. Note that Fei and Fexxv have two electrons in this shell,
while Fexxvi only has one (therefore its cross section is half that of Fexxv).
The Lotz formula
There are several formulae that describe the effective cross section Q as a function
of E. A formula that gives a correct order of magnitude estimate of the cross section
σ of this process and that has the proper asymptotic behaviour (first calculated by
Bethe and Born) is the formula of Lotz (1968):
where
and further
uI 2 Q = C ln u, (3.27)
u ≡ E/I (3.28)
C = an s , (3.29)
where n s is the number of electrons in the shell and the normalisation a = 4.5 ×
10 −24 m 2 keV 2 . See Fig. 3.14 for an illustration of the shape and scaling of this
formula.
exercise 3.7. Show that for E → I, Q → 0 and in particular Q ∼ E−I. Determine
the constant of proportionality.
It is further evident that for E → ∞ the cross section Q ∼ (ln E)/E → 0. This
shows that energetic electrons are not very efficient in ionising an atom or ion. They
pass too fast.
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