Ewald's sphere

The Ewald sphere is a geometric construct used in X-ray crystallography which neatly demonstrates the relationship between:

• the wavelength of the incident and diffracted x-ray beams,
• the diffraction angle for a given reflection,
• the reciprocal lattice of the crystal

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It was conceived by Paul Peter Ewald, a German physicist and crystallographer.

Ewald's sphere can be used to find the maximum resolution available for a given x-ray wavelength and the unit cell dimensions. It is often simplified to the two-dimensional "Ewald's circle" model or may be referred to as the Ewald sphere.

Ewald Construction

A crystal can be described as a lattice of points of equal symmetry. The requirement for constructive interference in a diffraction experiment means that in momentum or reciprocal space the values of momentum transfer where constructive interference occurs also form a lattice (the reciprocal lattice). For example, the reciprocal lattice of a simple cubic real-space lattice is also a simple cubic structure. The aim of the Ewald sphere is to determine which lattice planes (represented by the grid points on the reciprocal lattice) will result in a diffracted signal for a given wavelength, λ, of incident radiation.

The incident plane wave falling on the crystal has a wave vector K_i whose length is 2π / λ. The diffracted plane wave has a wave vector K_f. If no energy is gained or lost in the diffraction process (it is elastic) then K_f has the same length as K_i. The amount the beam is diffracted by is defined by the scattering vector ΔK = K_f − K_i. Since K_i and K_f have the same length the scattering vector must lie on the surface of a sphere of radius 2π / λ. This sphere is called the Ewald sphere.

The reciprocal lattice points are the values of momentum transfer where the Bragg diffraction condition is satisfied and for diffraction to occur the scattering vector must be equal to a reciprocal lattice vector. Geometrically this means that if the origin of reciprocal space is placed at the tip of K_i then diffraction will occur only for reciprocal lattice points that lie on the surface of the Ewald sphere.