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


This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.


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]

          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]


                    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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