# Difference between revisions of "Friedel's law"

### From Online Dictionary of Crystallography

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== Definition == | == Definition == | ||

− | Friedel's law, or rule, states that the intensities of the ''h'', ''k'', ''l'' and | + | 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. The reason is that the diffracted intensity is proportional to the the square of the modulus of the structure factor, |''F<sub>h</sub>''|<sup>2</sup>, according to the geometrical, or [[kinematical theory]]. The structure factor is given by: |

<center> | <center> | ||

− | + | <math>F_h = \Sigma_j f_j {\rm exp - 2 \pi i} {\bold h} . {\bold r_j}</math> | |

</center> | </center> | ||

− | where ''f<sub>j</sub>'' is the atomic scattering factor of atom ''j'', '''h''' the reflection vector and | + | 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''. There comes: |

<center> | <center> | ||

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

</center> | </center> | ||

− | if the atomic scattering factor, ''f<sub>j</sub>'', is real. The intensities of the ''h'', ''k'', ''l'' and | + | 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 therefore equal. If the crystal is absorbing, however, due to [[anomalous dispersion]], the atomic scattering factor is complex and |

<center> | <center> | ||

− | + | <math>F_{\bar h} \ne F_h^*</math> | |

</center> | </center> | ||

## Revision as of 17:11, 24 March 2006

Loi de Friedel (*Fr*). Friedelsche Gesetz (*Ge*). Ley de Friedel (*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. The reason is that the diffracted intensity is proportional to the the square of the modulus of the structure factor, |*F _{h}*|

^{2}, according to the geometrical, or kinematical theory. The structure factor is given by:

[math]F_h = \Sigma_j f_j {\rm exp - 2 \pi i} {\bold h} . {\bold r_j}[/math]

where *f _{j}* is the atomic scattering factor of atom

*j*,

**h**the reflection vector and [math]{\bold r_j}[/math] the position vector of atom

*j*. There comes:

[math]|F_h|^2 = F_h F_h^* = F_h F_{\bar h} = |F_{\bar h}|^2 [/math]

if the atomic scattering factor, *f _{j}*, is real. The intensities of the

*h*,

*k*,

*l*and [math]{\bar h}, {\bar k}, {\bar l}[/math] reflections are therefore equal. If the crystal is absorbing, however, due to anomalous dispersion, the atomic scattering factor is complex and

[math]F_{\bar h} \ne F_h^*[/math]

Friedel's law does not hold for absorbing crystals.

## History

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