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Ewald Sphere of Diffraction

Diffraction, which mathematically corresponds to a Fourier transform, results in spots (reflections) at well-defined positions. Each set of parallel lattice planes is represented by spots which have a distance of 1/d (d: interplanar spacing) from the origin and which are perpendicular to the reflecting set of lattice plane. The two basic lattice planes (blue lines) of the two-dimensional rectangular lattice shown below are transformed into two sets of spots (blue). The diagonals of the basic lattice (green lines) have a smaller interplanar distance and therefore cause spots that are farther away from the origin than those of the basic lattice. The complete set of all possible reflections of a crystal constitutes its reciprocal lattice.

The diffraction event can be described in reciprocal space by the Ewald sphere construction (figure below). A sphere with radius 1/λ is drawn through the origin of the reciprocal lattice. Now, for each reciprocal lattice point that is located on the Ewald sphere of reflection, the Bragg condition is satisfied and diffraction arises.

point 0: origin of reciprocal lattice

k0: wave vector of the incident wave

kD: wave vector of a diffracted wave

ZOLZ: Zero Order Laue Zone

FOLZ(SOLZ): First (Second) Order Laue Zone

Due to the small wavelength of electrons (e.g., λ = 1.97 pm for 300 keV electrons), the radius of the Ewald sphere (1/λ) is quite large. Furthermore, the lattice points in the reciprocal lattice of thin samples are elongated so that the Ewald sphere intersects several of the rods (see figure). Because of that, diffraction occurs even if the Bragg condition is not exactly satisfied, and many reflections appear simultaneously. In fact, ED patterns correspond to 2D cuttings of the reciprocal lattice. The rod shape is due to the fact that TEM specimens are very thin in real space, leading to an elongation of the reflections along this direction in reciprocal space.
If the interplanar distance in direction of observation is large (that means a small distance between ZOLZ and FOLZ in reciprocal space), higher order Laue zones (HOLZ) can be observed as well.

A general introduction into diffraction is given in an interactive tutorial by Proffen and Neder.

 

ED:Basics | Bragg law |ED vs. XRD | Examples

 

ETH Zürich | ETH chemistry department | ETH inorganic chemistry

modified: 6 February, 2015 by F. Krumeich | © ETH Zürich and the authors