Friedel's law

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== Definition ==
== Definition ==
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Friedel's law, or rule, states that the intensities of the ''h'', ''k'', ''l'' and ''-h'', ''-k'', ''-l'' 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:
+
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>
-
''F<sub>h</sub>'' = &Sigma;<sub>''j''</sub> ''f<sub>j</sub>'' exp - 2 &pi; i '''h . r<sub>j</sub>'''
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<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 '''r<sub>j</sub>''' the position vector of atom ''j''. There comes:
+
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>
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|''F<sub>h</sub>''|<sup>2</sup> = ''F<sub>h</sub> F<sub>h</sub>*'' = ''F<sub>h</sub> F<sub>-h</sub>'' = |''F<sub>-h</sub>''|<sup>2</sup>  
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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 ''-h'', ''-k'', ''-l'' reflections are therefore equal. If the crystal is absorbing, however, due to [[anomalous dispersion]], the atomic scattering factor is complex 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>
-
''F<sub>-h</sub> &ne; F<sub>h</sub>*''.
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<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 {\bar h}, {\bar k}, {\bar l} 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:

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

where fj is the atomic scattering factor of atom j, h the reflection vector and {\bold r_j} the position vector of atom j. There comes:

|F_h|^2 = F_h F_h^* = F_h F_{\bar h} = |F_{\bar h}|^2

if the atomic scattering factor, fj, is real. The intensities of the h, k, l and {\bar h}, {\bar k}, {\bar l} reflections are therefore equal. If the crystal is absorbing, however, due to anomalous dispersion, the atomic scattering factor is complex and

F_{\bar h} \ne F_h^*

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.

See also

Absolute structure