Friedel's law

The Friedel's law goes back to Georges Friedel and comes in crystallography in crystal structure analysis using X-rays used.

It states that the diffraction image of a crystal is always centrosymmetric, regardless of whether the crystal has a center of symmetry itself. Or more precisely: the intensities of the reflections (hkl ) and ( hkl ) are the same, even if the crystal is not centrosymmetric:

These two reflections come from the two sides of the same lattice planes and are also referred to as Friedel pairs.

Derivation

The Bragg reflections of a crystal are described by the Miller indices hkl. The intensity of a Bragg reflection Ihkl calculated as the square modulus of the structure factor Fhkl:

For the structure factor Fhkl applies:

Wherein the sum over all n atoms with atomic form factors of a crystal extending fn the base, at the locations xn, yn, zn are.

For this centrosymmetric lying reflex (hkl ) shall apply accordingly:

This also applies:

It follows for the intensities:

The transformation (1 ) is valid only under the assumption that the fn are all real quantities, which is usually fulfilled.

Note

The Friedel's law is a concrete application of a property of the Fourier transforms of real functions f (x):

Of the Fourier transform

The real function f (x) the following applies:

Follow

Because of Friedel 's law, one can not distinguish with X-ray diffraction, a point group that has no center of symmetry of the same point group with an additional center of symmetry. So, you can not determine all of the 32 point groups crystallographically, but only the 11 so-called Laue groups.

Deviations from Friedel 's law

The basis of Friedel 's law that the atomic form factors are real quantities. For wavelengths near the absorption edge of an atom of the atomic form factor but gets a significant imaginary component. Therefore, the relationship no longer holds. The Friedel's law is then only satisfied when the point group has a center of symmetry. The difference is, Bijvoet difference. By analyzing these differences one can solve the phase problem.

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