## Single crystal diffractometer

The concept of a single crystal diffractometer is to fulfil thee Bragg condition for as many crystal planes as possible by turning the crystal around the three axes and the detector around its own axis in order to detect the diffracted beams. Using these axes as a coordinate system, the diffraction peaks themselves describe a lattice in three-dimensional space which corresponds to the crystal lattice. While the symmetry of diffraction point lattice and crystal lattice are the same, some other relationships are not: a small distance between two diffraction spots corresponds to a large distance in the real lattice and vice versa. Due to this reciprocal relationship between the two lattices the crystal lattice is referred to as the direct lattice, while the diffraction point lattice is called the reciprocal lattice.

The relationship between direct lattice and reciprocal lattice can be used to visualize Bragg’s law and compute the operation of a diffractometer. This is called the Ewald construction or Ewald sphere. In the centre of two Ewald construction resides the crystal around the crystal you describe a sphere of radius 1/λ, λ being the X-ray wavelength. This sphere is intersected by the primary beam which passes through its centre. The point where the undiffracted beam leaves the sphere is the turning point of the reciprocal lattice. The reciprocal lattice rotates around this point exactly the same way as the crystal rotates around the construction’s centre. The Bragg condition for a reflection iis only fulfilled when the corresponding reciprocal lattice point rests exactly on the surface of the sphere. Only then can it be observed.

The Ewald construction is perhaps the single most important construction in crystallography. It can be used as an analog computer to simulate nearly every effect in X-ray diffraction.

This is the Ewald construction with arbitrary starting parameters.

If you expand the direct lattice by a factor of 2, you shrink the reciprocal lattice by ½. Also, the glancing angle 2θ is reduced by ½.

Alternatively, if you double the wavelength you shrink the radius of the Ewald sphere by ½, which means that the glancing angle 2θ is doubled.

Displacement errors can also be emulated with the aid of the Ewald construction.