Von Laue condition for scattering and Ewald construction


Ewald came up with a geometrical construction to help visualize which Bragg planes are in the correct orientation to diffract.
In the Ewald sphere, we have two origins (which can make you uncomfortable until you realize that it is just a geometrical construction that makes the mathematics of diffraction easy to picture). The origin of the crystal is at the center of the Ewald sphere, and the incoming X-rays are diffracted from that crystal. The origin of reciprocal space is at the point where the incoming X-ray beam would exit the Ewald sphere. If we rotate the crystal, we rotate the Bragg planes and so we rotate reciprocal space in the same direction. Since diffraction from a crystal is confined to points on the reciprocal lattice (corresponding to planes that can be specified by integer indices), we can think of rotating the reciprocal lattice when we rotate the crystal. The following figure shows this schematically, illustrating planes of points in the reciprocal lattice. The planes of points in the reciprocal lattice intersect the Ewald sphere to give a circle of points in the diffracting condition. When the planes are aligned perpendicular to the X-ray beam, these circles on the Ewald sphere will project onto circles of spots surrounding the direct beam position but, as we rotate the crystal (and the reciprocal lattice) the circles on the Ewald sphere will be distorted and will project into what are called lines of spots.

Direct lattice and reciprocal lattice

The Ewald construction

When executing the function ewald(k), which can be downloaded in the download section , we can see selection of Bragg planes and representations of vectors G K and K' in the reciprocal lattice (lower right window), Bragg planes and waves phases(upper left window), and the interference phase(upper right window).

Here are some examples of Bragg scattering from different planes:




in this movie we can see "scanning" of the points in the reciprocal lattice and the positive diffraction from different Bragg planes

Movie






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