Electron microscopy
 
Laue Groups & Centrosymmetric Space Groups
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Laue groups are the 11 characteristic centrosymmetric point groups (in yellow) as listed in Table 1677a. The Laue groups are obtained by adding a center of symmetry to each point group.

Table 1677a. Centrosymmetric, non-centrosymmetric, and chiral space groups.

Crystal system Laue group
Point group
Space group
International notation Schoenflies notation
Triclinic -1 1 C1 P1
-1 Ci P1
Monoclinic 2/m 2 C2 P2, P21, C2
m C1h Pm, Pc, Cm, Cc
2/m C2h P2/m, P21/m, C2/m, P2/c, P21/c, C2/c
Orthorhombic mmm 222 D2 P222, P2221, P21212, P212121, C222, C2221, I222, I212121, F222
mm2 C2v Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2, Cmm2, Cmc21, Ccc2, Amm2, Abm2, Ama2, Aba2, Imm2, Iba2, Ima2, Fmm2, Fdd2
mmm D2h Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma, Cmcm, Cmca, Cmmm, Cccm, Cmma, Ccca, Immm, Ibam, Ibca, Imma, Fmmm, Fddd
Tetragonal   4 C4 P4, P41, P42, P43, I4, I41
  422 D4 P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212, I422, I4122
  -4 S4  
  4mm C4v  
  -42m D2d  
  4/m C4h  
  4/mmm D4h  
Trigonal   3 C3  
  32 D3  
  3m C3v  
  -3 S6  
  -3m D3d  
Hexagonal   6 C6  
  622 D6  
  -6 C3h  
  6mm C6v  
  -6m2 D3h  
  6/m C6h  
  6/mmm D6h  
Cubic   23 T  
  432 O  
  -43m Td  
  m-3 Th  
  m-3m Oh  
Red: chiral; green: non-centrosymmetric; and yellow: centrosymmetric.

Table 1677b. Other characteristics of non-centrosymmetric space groups.

Difference between centrosymmetric, non-centrosymmetric, and chiral space groups page1675

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)| ------------------------ [1677]
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 (i.e. contains the center of inversion); 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.

 

 

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