Ewald's sphere

With the help of the Ewald sphere (named after Paul Peter Ewald ), the Laue condition for constructive interference in the scattering of a crystal can be represented graphically. In this case, the construction of the real and the associated reciprocal space. The following are the crystallographic definition of the reciprocal lattice is used ( instead of the usual in solid-state physics ).

The ball is constructed as follows (see the illustration): In the center of the Ewald sphere, the origin of the real space in which to be measured crystal is located (in the picture drawn green ). The radius of the Ewald sphere is 1 / λ, where λ is the wavelength of the X-ray beam. Therefore, all wave vectors lie on the surface of this sphere ( red in the picture shown). The origin of belonging to this lattice reciprocal lattice (dots in the figure) is placed in the intersection of the Ewald sphere with the passing through the crystal primary X-ray ( blue in the picture shown). The X-ray therefore always travels along a spherical diameter. Rotations of the crystal about the origin of real space lead to a corresponding rotation of the reciprocal lattice around the origin of the reciprocal space. Reciprocal lattice and crystal keep it in the same orientation. The crystal is rotated such that a still further point of the reciprocal lattice is on the surface of the Ewald sphere, the corresponding wave vector additionally fulfills the condition

This is the Laue conditions. Exactly in this case is held in the direction of scattering.

This construction serves to illustrate many measurement methods in crystallography. From it can be seen, for example, that only the points of the reciprocal lattice, which are smaller at a distance from the origin, can satisfy the Bragg condition ( in the picture by the black circle, able sphere with radius 2 / λ, shown). So also vividly clear why at long wavelengths ( ie, smaller wave number k) can not take place at the crystal diffraction: There is no more possible vectors that can satisfy the Bragg condition, since the Ewald sphere is too small.