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

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

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.