Practical Electron Microscopy and Database

A partial cross-section specifies the probability of a specific shell scattering through angles up to a specific angle, β, and with energy losses covering an energy range, Δ, above an ionization energy. For instance, the partial cross-section, σ_K(β,Δ), of K-shell, E_K, can be given by, [1]
          ---------------------------------------- [954a]
where,
         α_K(β) -- the energy-differential cross-section, which can be modeled theoretically, e.g. by the first Born approximation [2] which is based on a hydrogenic model with Zener screening constant or based on Hartree–Slater model [1].

For elements of low atomic number, σ_K(β,Δ) can be measured by EELS [1, 3-5],
          ---------------------------------------- [954b]
where,
         I_K(β,Δ) -- the electron signal recorded, by an EELS spectrometer system with entrance semi-angle β, from K-shell losses between E_K and E_K + Δ (from the energy window Δ),
         I₁(β,Δ) -- the plamon signal recorded under the same conditions (from the same energy window with an energy loss range of 0 and Δ, which includes the entire zero-loss peak),
         n -- the number of atoms per unit volume of the sample,
         t -- the sample thickness.

This approximation in Equation 954b is accurate for Δ ≥ 50 eV. [1,5]

Based on Equation 954b, the Areal Density, N, namely the number of atoms per unit area of the TEM sample through the electron beam, can be obtained by, [1]
          ---------------------------------------- [954c]

In practice, σ_K(β,Δ) obtained by Equations 954a or 954b can be applied into Equation 954c for Areal Density, N, calculation.

For large collection angles, the partial ionization cross-section, σ(β,Δ), becomes the total ionization cross-section, σ(Δ).

[1] R. F. Egerton, K-shell ionization cross-sections for use in microanalysis, 4(2), (1979) 169-179.
[2] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297.
[3] M. Isaacson and D. Johnson, Ultramicroscopy 1 (1975) 33.
[4] C. CoUiex, V.E. Cosslett, R.D. Leapman and P. Trebbia, Ultramicroscopy 1 (1976) 301.
[5] R.F. Egerton, Ultra.microscopy 3 (1978) 243.