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 <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:
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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 only true if there is no [[anomalous scattering]]. It is in that case not possible to tell by diffraction whether an inversion center is present or not. The apparent symmetry of the crystal is then one of the eleven [[Laue classes]].
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The reason for Friedel's rule 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:
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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
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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 [[anomalous dispersion]], the atomic scattering factor is complex and
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Friedel's law does not hold for absorbing crystals.
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Friedel's law therefore does not hold for absorbing crystals. 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.
== History ==
== History ==
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== See also ==
== See also ==
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[[Absolute structure]]
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[[Absolute structure]]<br>
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Section 3.1 of ''International Tables of crystallography, Volume A''<br>

Revision as of 07:37, 25 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. This is only true if there is no anomalous scattering. It is in that case not possible to tell by diffraction whether an inversion center 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 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 then 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 therefore does not hold for absorbing crystals. The reflections h, k, l and {\bar h}, {\bar k}, {\bar l} are called a Friedel pair. They are used in the resolution of the phase problem for the solution of crystal structures.

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
Section 3.1 of International Tables of crystallography, Volume A