Friedel's law

Friedel's law, named after Georges Friedel, is a property of Fourier transforms of real functions.^[1]

Given a real function ${\displaystyle f(x)}$, its Fourier transform

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

has the following properties.

• ${\displaystyle F(k)=F^{*}(-k)\,}$

where ${\displaystyle F^{*}}$ is the complex conjugate of ${\displaystyle F}$.

Centrosymmetric points ${\displaystyle (k,-k)}$ are called Friedel's pairs.

The squared amplitude (${\displaystyle |F|^{2}}$) is centrosymmetric:

• ${\displaystyle |F(k)|^{2}=|F(-k)|^{2}\,}$

The phase ${\displaystyle \phi }$ of ${\displaystyle F}$ is antisymmetric:

• ${\displaystyle \phi (k)=-\phi (-k)\,}$.

Friedel's law is used in X-ray diffraction, crystallography and scattering from real potentials 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 individual reflections are equal under Friedel's law.^[2][3][4]