Ewald ball

Ewald construction

With the help of the Ewald sphere (named after Paul Peter Ewald ), the Laue condition for constructive interference in the scattering on a crystal can be clearly illustrated. The construction links the (real) local space and the reciprocal space . In the following, the crystallographic definition of the reciprocal lattice is used ( instead of in the solid state physics usual ). ${\displaystyle |{\vec {k}}|={\frac {1}{\lambda }}}$ ${\displaystyle |{\vec {k}}|={\frac {2\pi }{\lambda }}}$

The sphere is constructed as follows (see the illustration): In the center of the Ewald sphere is the origin of the real space in which the crystal to be measured is located (shown in green in the picture). The radius of the Ewald sphere is 1 / λ, where λ is the wavelength of the incident beam (X-ray in the picture). Therefore, all wave vectors are having on the surface of this sphere (in the picture shown in red). The origin of the reciprocal lattice belonging to this crystal lattice (points in the picture) is placed at the point of intersection of the Ewald sphere with the primary X-ray beam passing through the crystal (shown in blue in the picture). The X-ray beam therefore always runs along the diameter of a sphere. Rotations of the crystal around the origin of the real space lead to a corresponding rotation of the reciprocal lattice around the origin of the reciprocal space. The reciprocal lattice and crystal retain the same orientation. If the crystal is rotated so that another point of the reciprocal lattice lies on the surface of the Ewald sphere, the corresponding wave vector also fulfills the condition ${\displaystyle {\vec {k}}}$ ${\displaystyle |{\vec {k}}|={\frac {1}{\lambda }}}$ ${\displaystyle {\vec {k_{s}}}}$

${\displaystyle \Delta {\vec {k}}={\vec {k_{s}}}-{\vec {k_{i}}}={\vec {G}}}$ (a reciprocal lattice vector).

This is the Laue condition. It is precisely in this case that elastic scattering takes place in the direction of . ${\displaystyle {\vec {k_{s}}}}$

This construction is used to illustrate many measuring methods in crystallography. It can be seen, for example, that only the points of the reciprocal lattice that are at a smaller distance from the origin can meet the Laue condition (represented in the picture by the black circle, the layer sphere with radius 2 / λ). This also makes it clear why at large wavelengths (i.e. small wave number k) no diffraction can take place on the crystal: There are no longer any possible vectors that can meet the Laue condition because the Ewald sphere becomes too small. ${\displaystyle 2|{\vec {k_{s}}}|}$ ${\displaystyle \lambda }$ ${\displaystyle {\vec {G}}}$

See also

• X-ray diffraction
• Electron diffraction
• Rotary crystal method

literature

• PP Ewald : On the theory of X-ray interference in crystals . In: Physics. Z . tape 14 , 1913, pp. 465-472 .
• Martin J. Buerger : Crystallography . 1st edition. Walter de Gruyter, Berlin 1977, ISBN 3-11-004286-X .

Web links

Ewald sphere (IUCr, engl.)