Difference between revisions of "Friedel's law"
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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: |
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| − | + | <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, |Fh|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 fj 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, fj, 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.