View source for Friedel's law

<font color="blue">Loi de Friedel</Font> (''Fr''). <font color="red">Friedelsches Gesetz</font> (''Ge''). <font color="black">Legge di Friedel</font>(''It''). <font color="purple">フリーデル則</font>(''Ja''). <font color="green">Ley de Friedel</font> (''Sp'').


== Definition ==

Friedel's law, or rule, states that the intensities of the ''h'', ''k'', ''l'' and <math>{\bar h}, {\bar k}, {\bar l}</math> reflections are equal. This is true either if the crystal is centrosymmetric or if no [[Resonant_scattering|resonant scattering]] is present. It is in that case not possible to tell by diffraction whether an inversion centre is present or not. The apparent symmetry of the crystal is then one of the eleven [[Laue classes]].

The reason for Friedel's rule is that the diffracted intensity is proportional to the square of the modulus of the structure factor, |''F<sub>h</sub>''|<sup>2</sup>, according to the  geometrical, or [[kinematical theory]] of diffraction. It depends similarly on the modulus of the structure factor according to the [[dynamical theory]] of diffraction. The structure factor is given by

<center>
<math>F_h = \sum_j f_j {\exp(2 \pi i} \mathbf{h} . \mathbf{r_j})</math>
</center>

where ''f<sub>j</sub>'' is the atomic scattering factor of atom ''j'', '''h''' the reflection vector and <math>{\bold r_j}</math> the position vector of atom ''j''. It follows that

<center>
<math>|F_h|^2 = F_h F_h^* = F_h F_{\bar h} = |F_{\bar h}|^2 </math>
</center>

if the atomic scattering factor, ''f<sub>j</sub>'', is real. The intensities of the ''h'', ''k'', ''l'' and <math>{\bar h}, {\bar k}, {\bar l}</math> reflections are then equal. If the crystal is absorbing, however, due to [[Resonant_scattering|resonant scattering]], the atomic scattering factor is complex and

<center>
<math>F_{\bar h} \ne F_h^*</math>.
</center>

The reflections ''h'', ''k'', ''l'' and <math>{\bar h}, {\bar k}, {\bar l}</math> are called a [[Friedel pair]]. They are used in the resolution of the phase problem for the solution of crystal structures and in the determination of absolute structure.

== History ==

Friedel's law was stated by G. Friedel (1865-1933) in 1913 [Friedel, G. (1913). ''C.R. Acad. Sci. Paris'', '''157''', 1533-1536. ''Sur les symétries cristallines que peut révéler la diffraction des rayons X''].

== See also ==
*[[Absolute structure]]
*Chapter 3.2.1.1 of ''International Tables for Crystallography, Volume A'', 6th edition

[[Category:X-rays]]<br>