Electron Diffraction Formation: Bloch-wave Approach
- Practical Electron Microscopy and Database -
- An Online Book -

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Electron diffraction patterns and Kikuchi patterns in both EBSD and CBED present similar geometry and can be successfully calculated using a Bloch-wave approach. The Bloch-wave is known from Bloch’s theorem for a translationally invariant scattering potential,

          Schrödinger equation ------------------- [2208a]

                      Schrödinger equation ------------------- [2208b]

where,
          cj and cg(j) -- Coefficients;
          k(j) -- Vectors.

These coefficients and vectors can be obtained by solving Schrödinger equation by limiting the wave-function expansion to a number of Fourier coefficients labeled by the respective reciprocal lattice vectors g, coupling the incident electron beam to a diffracted beam. The eigenvalues λ(j) is obtained by re-writing the Bloch-wave vector k(j) as the sum of the incident beam wave vector K in the crystal and a surface normal component as k(j) = K + λ(j)n. Equation 2208b can be re-written by,

                      Schrödinger equation ------------------- [2208c]

                      Schrödinger equation ------------------- [2208d]

where,

                    t – The depth from the entrance surface of the sample.

The exponential term exp[2πi(K + gr] in Equation 2208d indicates the contributions of plane waves moving into the directions K + g. ϕg(t) gives the a depth dependent amplitude. The plane waves in directions K + g correspond to the diffracted beams that form a spot diffraction pattern.

 

 

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