Ewald's sphere

The Ewald sphere is a geometric construct used in electron, neutron, and X-ray crystallography which demonstrates the relationship between:

It was conceived by Paul Peter Ewald, a German physicist and crystallographer. Ewald himself spoke of the sphere of reflection.

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.

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. Another example, the reciprocal lattice of an FCC crystal real-space lattice is a BCC structure, and vice versa. 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, ${\displaystyle \lambda }$, of incident radiation.

The incident plane wave falling on the crystal has a wave vector ${\displaystyle K_{i}}$ whose length is ${\displaystyle 2\pi /\lambda }$. The diffracted plane wave has a wave vector ${\displaystyle K_{f}}$. If no energy is gained or lost in the diffraction process (it is elastic) then ${\displaystyle K_{f}}$ has the same length as ${\displaystyle K_{i}}$. The difference between the wave-vectors of diffracted and incident wave is defined as scattering vector ${\displaystyle \Delta {K}=K_{f}-K_{i}}$. Since ${\displaystyle K_{i}}$ and ${\displaystyle K_{f}}$ have the same length the scattering vector must lie on the surface of a sphere of radius ${\displaystyle 2\pi /\lambda }$. This sphere is called the Ewald sphere.

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