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

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

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 $\varphi$ of $F$ is antisymmetric:

• $\varphi (k)=-\varphi (-k)$.

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

References

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