Electron microscopy
 
Non-centrosymmetric Space Groups
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Two centrosymmetric reciprocal lattice points (hkl) and (-h-k-l) are called Friedel (or Bijvoet) pair. Friedel's law describes the relationship between the structure amplitude of the two centrosymmetric reciprocal lattice points. According to the Friedel's law, regardless of whether the crystal structure is centrosymmetric or not, if there is no anomalous scattering the reciprocal space is always centrosymmetric, given by,
         |F(hkl)| = |F(-h-k-l)| ------------------------ [1676]
The square of |F| represents the intensity of the peak of the corresponding reciprocal lattice point. Therefore, it is difficult to distinguish if the space group of a material is centrosymmetric or non-centrosymmetric.

However, there are still different ways to distinguish if the crystal belongs to centrosymmetric or non-centrosymmetric space groups:
         i) Distinguished by its symmetry elements. For instance, if there is -x,-y,-z in its symmetry elements, the crystal is centrosymmetric; otherwise, non-centrosymmetric. For instance, Pca21 (see page1850) and Pbca (see page3016) space groups are non-centrosymmetric and centrosymmetric, respectively.
         ii) Distringuished by the physical properties. For instance, a ferroelectric material must belong to a non-centrosymmetric space group.
         iii) Distringuished by CBED, which can be sensitive enough to detect the polarization of crystals.
         iv) Distringusihed by the intensity statistics of X-ray diffraction.

Table 1676. Other characteristics of non-centrosymmetric space groups.

Difference between centrosymmetric, non-centrosymmetric, and chiral space groups page1675
Table of centrosymmetric, non-centrosymmetric, and chiral space groups page1677

 

 

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