Why do impacting electrons have a low probability of causing ionization?

Question

In a section on ionization efficiency and ionization cross section, my mass spectrometry textbook, Mass Spectrometry by Jürgen Gross [1, pp. 38–39], says the following:

The ionization energy represents the absolute minimum energy required for ionization of the neutral concerned. This means in turn that in order to effect ionization, the impacting electrons need to carry at least this amount of energy. If this energy were then to be quantitatively transferred during the collision, ionization would take place. Obviously, such an event is of rather low probability and therefore, the ionization efficiency is close to zero with electrons carrying just the IE of the pertinent neutral. However, a slight increase in electron energy brings about a steady increase in ionization efficiency.

Why is such an event of "rather low probability"? It seems to me that if the impacting electrons are carrying at least the necessary ionization energy, as the author suggests, then shouldn't ionization occur (be a certainty or, at least, a high probability event)?

I would greatly appreciate it if people could please take the time to clarify this.

References

1. Gross, J. H. Mass Spectrometry: A Textbook, 3rd ed.; Springer International Publishing: Cham, Switzerland, 2017. ISBN 978-3-319-54397-0.

Answer 1

Ionization has a low probability of occurring if the energy of the incoming electron is exactly equal to the ionization energy. Using a classical analogy, if ionization occurred the incoming electron would be left with no energy, ie, stationary. The incoming, now stationary, electron would not scatter away but rather remain trapped by the nuclear attraction, such that no net ionization would have occurred. It would also imply (classically) a colinear collision with the departing electron, a seemingly unlikely event. On the other hand, if the incoming electron has a significant energy surplus, such that it has sufficient energy to escape the nuclear attraction (scatter), then ionization is more likely to occur.

There are obviously more accurate non-classical ways to describe the process, but the analogy illustrates the problem. There are also intermediate scenarios, such as those exploited in Auger spectroscopy.