In solid state physics, in particular crystallography, the Ewald construction is a method for reconstructing a crystal structure by examining and interpreting an x-ray diffraction pattern.
To every crystallographic spatial structure, there corresponds exactly one so-called reciprocal lattice. The reciprocal lattice is the set of all allowed values of crystal momentum that a particle (or wave) inside the periodic potential of the lattice may have. For example, the lattice dual to a simple cubic spatial structure is a simple cubic structure (in momentum or k-space). These allowed values of momentum form a cubic lattice.
The Ewald construction for plane waves of wave-vector q incident on a crystal (that is to say, going in the direction that q points in, with wavelength 2 Pi / |q|) tells you to pick any point in the lattice, and draw a sphere of radius |q| about the point at the tip of vector q of the points in the dual lattice. In general, this sphere will only have the origin you chose on its surface, and will not have any other points of the lattice on its surface. If it does, then denote by K the vector corresponding to this second point. If one then performs a diffraction pattern experiment with waves of wave vector q incident on the crystal, one will then see a large peak in the spectrum of the diffraction pattern at K.