Friedel's law

The Friedel law goes on Georges Friedel back and comes in crystallography in the crystal structure analysis using X-rays are used.

It says that the diffraction pattern of a crystal is always centrosymmetrical , regardless of whether the crystal itself has a center of symmetry. Or more precisely: The intensities of the reflections hkl and h k l are the same, even if the crystal is not centrosymmetric:

${\displaystyle I_{hkl}=I_{{\bar {h}}{\bar {k}}{\bar {l}}}}$

These two reflections come from the two sides of the same set of lattice planes ( hkl ) and are also referred to as Friedel pairs .

Derivation

The Bragg reflections of a crystal are described by the Laue indices hkl . The intensity I _{hkl of} a Bragg reflex results from the square of the magnitude of the structure factor F _hkl :

${\displaystyle I_{hkl}=F_{hkl}\cdot F_{hkl}^{*},}$where * means conjugate complex .

The following applies to the structure _factor F _hkl :

${\displaystyle F_{hkl}=\sum _{n}f_{n}e^{2\pi i(hx_{n}+ky_{n}+lz_{n})}}$,

where the sum extends over all n atoms with the atomic shape factor f _{n of} the base of a crystal, which lie at the points x _n , y _n , z _n .

For the centrally symmetrical reflex ( h k l ) the following applies accordingly:

${\displaystyle F_{{\bar {h}}{\bar {k}}{\bar {l}}}=\sum _{n}f_{n}e^{2\pi i(-hx_{n}-ky_{n}-lz_{n})}=^{(1)}(\sum _{n}f_{n}e^{2\pi i(hx_{n}+ky_{n}+lz_{n})})^{*}=F_{hkl}^{*}.}$

This also applies:

${\displaystyle F_{{\bar {h}}{\bar {k}}{\bar {l}}}^{*}=F_{hkl}.}$

From this it follows for the intensities:

${\displaystyle I_{hkl}=F_{hkl}\cdot F_{hkl}^{*}=F_{{\bar {h}}{\bar {k}}{\bar {l}}}^{*}\cdot F_{{\bar {h}}{\bar {k}}{\bar {l}}}=I_{{\bar {h}}{\bar {k}}{\bar {l}}}}$

The transformation (1) only applies under the assumption that the f _{n are} all real quantities, which is usually the case.

annotation

Friedel's law is a concrete application of a property of the Fourier transforms of real functions f (x):

For the Fourier transform

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

of the real function f (x):

${\displaystyle F(k)=F^{*}(-k)\,}$.

consequences

Due to Friedel's law, X-ray diffraction cannot distinguish a point group that has no center of symmetry from the same point group with an additional center of symmetry. Therefore, not all 32 point groups can be determined crystallographically, only the 11 so-called Lau groups .

Deviations from Friedel's Law

Friedel's law is based on the fact that the atomic form factors are real quantities. For wavelengths near the absorption edge of an atom, however, the atomic shape factor has a clear imaginary component. Therefore, the relationship no longer applies . Friedel's law is only fulfilled if the point group has a center of symmetry. The difference is called the Bijvoet difference. The phase problem can be solved by evaluating these differences . ${\displaystyle F_{hkl}=F_{{\bar {h}}{\bar {k}}{\bar {l}}}^{*}}$ ${\displaystyle F_{hkl}^{2}-F_{{\bar {h}}{\bar {k}}{\bar {l}}}^{2}}$

Individual evidence

1. ↑ Lothar Spieß, etc: Modern X-ray diffraction . Vieweg and Teubner, 2009, ISBN 978-3-8351-0166-1 .

literature

• Schwarzenbach D. Kristallographie Springer Verlag, Berlin 2001, ISBN 3-540-67114-5
• Walter Borchardt-Ott (2002): Crystallography. Jumper. ISBN 3-540-43964-1
• Werner Massa (2002): Crystal structure determination. Teubner. ISBN 3-519-23527-7