Friedel's law

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


 * $$F(k)=\int^{+\infty}_{-\infty}f(x)e^{i k \cdot x }dx$$

has the following properties.


 * $$F(k)=F^*(-k) \,$$

where $$F^*$$ is the complex conjugate of $$F$$.

Centrosymmetric points $$(k,-k)$$ are called Friedel's pairs.

The squared amplitude ($$|F|^2$$) is centrosymmetric:
 * $$|F(k)|^2=|F(-k)|^2 \,$$

The phase $$\phi$$ of $$F$$ is antisymmetric:
 * $$\phi(k) = -\phi(-k) \,$$.

Friedel's law is used in X-ray diffraction, crystallography and scattering from real potential within the Born approximation. Note that a twin operation (a.k.a. Opération de maclage) is equivalent to an inversion centre and the intensities from the individuals are equivalent under Friedel's law.