Ewald sphere (crystallography)

The Ewald sphere is a mathematical construct that relates the geometry of the incident and scattered wave vectors to the reciprocal lattice.

Consider a crystal at A being irradiated by an incident wave vector s₀, scattering a wave vector s that makes an angle 2θ with s₀, thereby satisfying Bragg’s law (see diagram below).

Since X-ray scattering is elastic, IsI = Is₀I and the two wave vectors become radii of a sphere called the Ewald sphere. The vector OP is the reciprocal lattice vector (s – s₀) that is denoted by h with the origin at O.

$\frac{OP}{IO}=\frac{\left | \textbf{\textit{h}}\right |}{IO}=sin\theta\; \; \; \; \; \; \; (24e)$

Since h = (s – s₀), eq24 becomes

$d_{nh,nk,nl}=\frac{1}{\left | \textbf{\textit{h}}\right |}\; \; \; \; \; \; \; (24f)$

From eq11, $sin\theta =\frac{\lambda }{2d_{nh,nk,nl}}$ . Substitute $sin\theta =\frac{\lambda }{2d_{nh,nk,nl}}$ and eq24f in eq24e, we have:

$IO=\frac{2}{\lambda }$

This means that IA = AO = AP = 1/λ. Therefore, to satisfy Bragg’s law that results in constructive interference of scattered X-rays, the head of the reciprocal lattice vector must lie on any point on the surface of the Ewald sphere. From eq24f and eq11,

$\left | \textbf{\textit{h}}\right |=\frac{2sin\theta }{\lambda }$

Since -1 ≤ sinθ ≤ 1,

$\left | \textbf{\textit{h}}\right |_{max}=\frac{2 }{\lambda }\; \; \; \; \; \; \; (24g)$

As the maximum magnitude of the reciprocal lattice vector is 2/λ, all reciprocal lattice vectors that potentially satisfy the Laue equations or Bragg’s law are enclosed in a sphere of radius known as the limiting sphere. The Ewald sphere is used to visualise different X-ray diffraction techniques including single crystal X-ray diffraction.

Ewald sphere (crystallography)

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