Ewald sphere

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 visualize different X-ray diffraction techniques including single crystal X-ray diffraction.


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