Ewald's Sphere - Ewald Construction

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. 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, of incident radiation.

The incident plane wave falling on the crystal has a wave vector whose length is . The diffracted plane wave has a wave vector . If no energy is gained or lost in the diffraction process (it is elastic) then has the same length as . The difference between the wave-vectors of diffracted and incident wave is defined as scattering vector . Since and have the same length the scattering vector must lie on the surface of a sphere of radius . 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 then diffraction will occur only for reciprocal lattice points that lie on the surface of the Ewald sphere.