Bethe Cross Section
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Ionization cross section has the form [1],

                 Ionization cross section ---------- [4553a]

where Ec is the critical ionization energy and C0 is a constant that can be determined from other fundamental atomic data that depend on the particular transition.

Bethe went a step further [2] to include the effects of virtual photon exchange (i.e. the Breit interaction) in a perturbation expansion, retaining terms up to β(v2/c2). The resulting ionization cross section is given by,

                  Bethe cross section ---------- [4553b]

contains two additional terms that do not appear in Equation [4553a]. These terms are sometimes referred to as the “relativistic rise", and it is straightforward to verify that they provide an increasingly positive contribution to the cross section with increasing impact energy.

As a result of Equation 4553a, it is expected to be valid if the incident ionizing electron energy is suficiently high, e.g. at accelerating voltages (E0) of the incident electrons in TEM and SEM. The total ionization cross section derived by Bethe can be given by,

total ionization cross section -------- [4553c]

               ni – The number of electrons in a shell or subshell (e.g., ni = 2 for a K-shell, ni = 8 for an L-shell, and ni = 18 for an M-shell
               bs, c s – The constants for a particular inner-shell and from least-squares fitting to ionization data
               Ec – Ionization energy of given element A in the K-, L-, or M-shell (keV)             

If we apply overvoltage into Equation [4791a], and then, we can have,

ionization cross sections -------- [4553d]

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

Combining Equations 4553b and 4553c, Bethe cross section equation becomes,

                  Bethe cross section equation --------- [4553e]


[1] An Overview of Relativistic Distorted-Wave Cross Sections Christopher J. Fontes, Hong Lin Zhang and Joseph Abdallah, Jr., CP730, Atomic Processes in Plasmas: 14th APS Topical Conference on Atomic Processes in Plasmas, edited by J. S. Cohen, S. Mazevet, and D. P. Kilcrease.
[2] H. Bethe, Z. Physik 76, 293 (1932).



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