International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 64   | 1 | 2 |

Section 1.3.4.2.1.4. Friedel's law, anomalous scatterers

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.2.1.4. Friedel's law, anomalous scatterers

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If the wavelength λ of the incident X-rays is far from any absorption edge of the atoms in the crystal, there is a constant phase shift in the scattering, and the electron density may be considered to be real-valued. Then[\eqalign{F({\bf h}) &= {\textstyle\int\limits_{{\bb R}^{3}/{\bb Z}^{3}}} \rho\llap{$-\!$} ({\bf x}) \exp (2\pi i {\bf h} \cdot {\bf x}) \,\hbox{d}^{3} {\bf x} \cr &= \overline{{\textstyle\int\limits_{{\bb R}^{3}/{\bb Z}^{3}}} \overline{\rho\llap{$-\!$} ({\bf x})} \exp [2\pi i (-{\bf h}) \cdot {\bf x}] \,\hbox{d}^{3} {\bf x}} \cr &= \overline{F (-{\bf h})} \hbox{ since } \overline{\rho\llap{$-\!$} ({\bf x})} = \rho\llap{$-\!$} ({\bf x}).}]Thus if[F({\bf h}) = |F({\bf h})| \exp (i\varphi ({\bf h})),]then[|F(-{\bf h})| = |F({\bf h})|\quad \hbox{ and } \quad \varphi (-{\bf h}) = - \varphi ({\bf h}).]This is Friedel's law (Friedel, 1913[link]). The set [\{F_{{\bf h}}\}] of Fourier coefficients is said to have Hermitian symmetry.

If λ is close to some absorption edge(s), the proximity to resonance induces an extra phase shift, whose effect may be represented by letting [\rho\llap{$-\!$} ({\bf x})] take on complex values. Let[\rho\llap{$-\!$} ({\bf x}) = \rho\llap{$-\!$}^{\rm R} ({\bf x}) + i\rho\llap{$-\!$}^{\rm I} ({\bf x})]and correspondingly, by termwise Fourier transformation[F({\bf h}) = F^{\rm R} ({\bf h}) + iF^{\rm I} ({\bf h}).]

Since [\rho\llap{$-\!$}^{\rm R} ({\bf x})] and [\rho\llap{$-\!$}^{\rm I} ({\bf x})] are both real, [F^{\rm R} ({\bf h})] and [F^{\rm I} ({\bf h})] are both Hermitian symmetric, hence[F(-{\bf h}) = \overline{F^{\rm R} ({\bf h})} + \overline{iF^{\rm I} ({\bf h})},]while[\overline{F({\bf h})} = \overline{F^{\rm R}({\bf h})} - \overline{iF^{\rm I}({\bf h})}.]Thus [F(-{\bf h}) \neq \overline{F({\bf h})}], so that Friedel's law is violated. The components [F^{\rm R}({\bf h})] and [F^{\rm I}({\bf h})], which do obey Friedel's law, may be expressed as:[\eqalign{F^{\rm R}({\bf h}) &= {\textstyle{1 \over 2}} [F({\bf h}) + \overline{F(-{\bf h})}],\cr F^{\rm I}({\bf h}) &= {1 \over 2i}[F({\bf h}) - \overline{F(-{\bf h})}].}]

References

Friedel, G. (1913). Sur les symétries cristallines que peut révéler la diffraction des rayons Röntgen. C. R. Acad. Sci. Paris, 157, 1533–1536.








































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