Friedel's law, named after
Georges Friedel Georges Friedel (19 July 1865 – 11 December 1933) was a French mineralogist and crystallographer.
Life
Georges was the son of the chemist Charles Friedel. Georges' grandfather was Louis Georges Duvernoy who held the chair in comparative anat ...
, is a property of
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s of real functions.
Given a real function
, its
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
:
has the following properties.
*
where
is the complex conjugate of
.
Centrosymmetric
In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point grou ...
points
are called
Friedel's pairs Friedel's law, named after Georges Friedel, is a property of Fourier transforms of real functions.
Given a real function f(x), its Fourier transform
:F(k)=\int^_f(x)e^dx
has the following properties.
*F(k)=F^*(-k) \,
where F^* is the complex co ...
.
The squared amplitude (
) is centrosymmetric:
*
The phase
of
is
antisymmetric:
*
.
Friedel's law is used in
X-ray diffraction,
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wo ...
and scattering from real potential within the
Born approximation. Note that
twin operation( ''Opération de maclage'') is equivalent to an inversion centre and the intensities from the individuals are equivalent under Friedel's law.
[Friedel G (1904). "Étude sur les groupements cristallins". Extract from '' Bullettin de la Société de l'Industrie Minérale'', Quatrième série, Tomes III et IV. Saint-Étienne: Societè de l'Imprimerie Thèolier J. Thomas et C.][Friedel G. (1923). ''Bull. Soc. Fr. Minéral.'' 46:79-95.]
References
{{Crystallography
Fourier analysis
Crystallography