ionization cross section

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ionization cross section

Bethe [1] in 1930 derived the expression for the inner-shell ionization cross-section from the first Born approximation. As a result, it is only expected to be
valid if the incident ionizing electron energy is suficiently high, e.g. in TEM and SEM conditions. The total ionization cross section derived by Bethe (called Bethe cross section) can be given by,

total ionization cross section -------- [4791a]

               n_i – The number of electrons in a shell or subshell (e.g., n_i = 2 for a K-shell, n_i = 8 for an L-shell, and n_i = 18 for an M-shell
               b_s, c _s – The constants for a particular inner-shell and from least-squares fitting to ionization data
               E_c – Ionization energy of given element A in the K-, L-, or M-shell (keV)
               E₀ -- Accelerating voltage of the incident electrons.

Reducing the acceleration voltage increases the ionization cross-section, which is determined by the overvoltage ratio. If we apply overvoltage into Equation [4791a], and then, we can have,

ionization cross sections -------- [4791b]

Here, the dimensions are ionizations/e-/(atom/cm²).

In low-voltage electron microscopes, e.g. low-voltage SEMs, the overvoltage can affect the generation of X-rays because the ionization cross section depends on the value of overvoltage. The optimum U_i for the ionization of the i shell (i shell is K-, L, M- shell for a characteristic x-ray) is 3~5 as indicated in Figure 4791. Then, the cross section decreases at overvoltages lower than ~30. In the TEM E₀ is ≥ 100 keV and E_i is generally < 20 keV and thus, U is usually > 5. Therefore, the cross section is pretty constant with energy.

Plot of the ionization cross-section over overvoltage

Figure 4791. Plot of the ionization cross-section over U. Ionization is most probable if the beam energy is 3~5 of the critical ionization energy.

Generally EELS (Electron Energy Loss Spectroscopy) has better sensitivity than EDS (Energy-dispersive X-ray spectroscopy), due to the potential of utilizing signals generated from larger ionization cross sections (L rather than K) and the much greater collection efficiency. [2 - 3]

[1] Bethe, HA 1930 Zur Theorie des Durchgangs Schneller Korpuskularstrahlen Durch Materie Ann. der Phys. Leipzig 5 325–400.
[2] M. Isaacson, D. Johnson, Ultramicroscopy 1 (1975) 33.
[3] H. Shuman, P. Kruit, Rev. Sci. Instrument 56 (1985) 231.

Practical Electron Microscopy and Database

An Online Book, Second Edition by Dr. Yougui Liao (2006)

ionization cross section