Practical Electron Microscopy and Database

The periodic potential in crystals causes the amplitude of the accelerating electron (e.g. 200 keV in TEM) to be transferred back-and-forth between the transmitted and diffracted wavefunctions. This transfer process can be explained by dynamical theory. The crystal has a periodic potential that is weak compared with an electron energy of 200 keV. At the Laue condition ( s= 0), the physical distance between two back-and-forth transfers is called the “extinction distance.”

Assuming the TEM specimen is a perfect crystal, in two beam condition the intensity (I_g) of diffracted electron beam can be given by (based on dynamical theory),

              ------------------------- [4134a]
             ------------------------- [4134b]

             ------------------------- [4134c]

where,
           s_eff -- Effective deviation parameter
           ξ_g -- Extinction distance
           V -- Volume of the unit cell
           λ -- Electron wavelength
           Γ_g -- Structure factor of the unit cell for diffraction g
           s -- Deviation parameter
           t -- Crystal thickness

Equation 4134a for dynamical theory is valid when the TEM sample is thick and s is about zero, but is not valid in kinematical theory. From Equation 4134c we can know that the extinction distance decreases with increase of scattering (increase of Γ_g). When s is equal to zero (exact Bragg condition) we have s_eff = 1/ξ_g, meaning the transmitted (I₀) and diffracted (I_g) intensities has a periodicity of ξ_g in TEM specimen depth indicated in Figure 4134 in two beam condition. Here, I₀ = 1 - I_g in two beam condition. Equation 4134a also indicates that the intensity of these low order diffraction disks fluctuates with thickness.

Figure 4134. The diffracted intensity (I_g) showing a periodicity of ξ_g in TEM
specimen depth in two beam condition. I_g is the intensity of transmitted beam.