Ewald sphere

The Ewald sphere is a sphere of the radius defined as the reciprocal of the wavelength of the incident wave 1/λ, and is drawn with the point as the center, the point being at the length 1/λ from a certain reciprocal lattice point along the direction of the incident wave to a specimen crystal. The Ewald sphere explains what Bragg reflections occur using the relation between the incident wave vector and the reciprocal lattice points. All of the reciprocal lattice points on the Ewald sphere satisfy the Bragg condition. For a high energy incident electron (100 or 200 keV), the Ewald sphere can be approximated as a flat plane because the radius of the Ewald sphere is much larger than the distance between the reciprocal lattice points. As a result, the reflections, which appear as cross sections between the Ewald sphere and the reciprocal lattice points, can be indexed easier than the indexing in X-ray diffraction.



The reciprocal lattice points of gold (Au) for [001] incidence (lattice spacing d = 0.204 nm) and the Ewald spheres. The small sphere is the Ewald sphere for X-ray of MoKa (λ = 0.07109 nm) and the large sphere (arc) is the Ewald sphere for an electron beam of 200 kV (λ = 0.002508 nm).
The Ewald sphere for the electron beam is displayed up to ±10°, which corresponds to a usual limitation of the acceptance angle of an electron microscope. Blue reciprocal lattice points approximately satisfy the Bragg condition.

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