Lauegruppe
The diffraction pattern of crystals in normal X-ray diffraction (as in neutron and electron diffraction ) is always centrosymmetric. This is independent of whether the point group of the crystal has a center of symmetry or not ( Friedel's law ). Therefore, with simple crystallographic methods, one cannot distinguish a point group that does not have a center of symmetry from the one that also has this symmetry element.
If you add an inversion center to each of the 32 crystal classes or point groups , you get 11 different Taue classes or Tau groups . The namesake is the physicist Max von Laue .
Only the Lau groups can be distinguished with simple crystallographic diffraction methods. One of the first steps in crystal structure analysis is to determine the Lau group. Only then can the space group be derived.
The Lauegruppen are arranged according to the crystal system (indicated in the Hermann Mauguin symbolism ):
| Crystal system | Lauegruppe | Point groups |
|---|---|---|
| triclinic | 1 | 1 , 1 |
| monoclinic | 2 / m | 2 / m , m , 2 |
| orthorhombic | mmm | mmm , mm 2, 222 |
| tetragonal | 4 / m | 4 / m , 4 , 4 |
| 4 / mmm | 4 / mmm , 4 2 m , 4 mm , 422 | |
| trigonal | 3 | 3 , 3 |
| 3 m | 3 m , 3 m , 32 | |
| hexagonal | 6 / m | 6 / m , 6 , 6 |
| 6 / mmm | 6 / mmm , 6 m 2, 6 mm , 622 | |
| cubic | m 3 | m 3 , 23 |
| m 3 m | m 3 m , 4 3 m , 432 |
The determination of the actual point group from the symmetry of the X-ray diffraction pattern is possible if one takes advantage of the anomalous scattering by using X-rays whose wavelength is close to an absorption edge of an atomic species of the material under investigation. Then the X-ray radiation diffracted by the corresponding atom has an additional phase shift, which means that Friedel's law no longer applies. This results in small deviations from the inversion symmetry of the diffraction pattern, so that not only the real point group can be observed, but the phase problem can also be solved. Bijvoet used this effect in the course of his studies on sodium rubidium tartrate.
Since radiation of a suitable wavelength is often not available, attempts are usually made to provide evidence of a missing center of symmetry through other physical effects, such as the piezo effect or frequency doubling .
literature
- D. Schwarzenbach: Crystallography. Springer Verlag, Berlin 2001, ISBN 3-540-67114-5 .
- Will Kleber , Hans-Joachim Bautsch , Joachim Bohm , Detlef Klimm: Introduction to crystallography . 19th edition. Oldenbourg Wissenschaftsverlag, 2010, ISBN 978-3-486-59075-3 .