International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 2.2, p. 19

Section 2.2.5. Patterson symmetry

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

2.2.5. Patterson symmetry

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The entry Patterson symmetry in the headline gives the space group of the Patterson function P(x, y, z). With neglect of anomalous dispersion, this function is defined by the formula [P(x,y,z) = {1 \over V} \sum_{h} \sum_{k} \sum_{l} |F(hkl)|^{2} \cos 2\pi (hx + ky + lz).] The Patterson function represents the convolution of a structure with its inverse or the pair-correlation function of a structure. A detailed discussion of its use for structure determination is given by Buerger (1959)[link]. The space group of the Patterson function is identical to that of the `vector set' of the structure, and is thus always centrosymmetric and symmorphic.1

The symbol for the Patterson space group of a crystal structure can be deduced from that of its space group in two steps:

  • (i) Glide planes and screw axes have to be replaced by the corresponding mirror planes and rotation axes, resulting in a symmorphic space group.

  • (ii) If this symmorphic space group is not centrosymmetric, inversions have to be added.

There are 7 different Patterson symmetries in two dimensions and 24 in three dimensions. They are listed in Table 2.2.5.1[link]. Account is taken of the fact that the Laue class [\bar{3}m] combines in two ways with the hexagonal translation lattice, namely as [\bar{3}m1] and as [\bar{3}1m].

Table 2.2.5.1| top | pdf |
Patterson symmetries for two and three dimensions

Laue classLattice typePatterson symmetry (with space-group number)
Two dimensions
2 p         p2 (2)        
2mm p c       p2mm (6) c2mm (9)      
4 p         p4 (10)        
4mm p         p4mm (11)        
6 p         p6 (16)        
6mm p         p6mm (17)        
Three dimensions
[\bar{1}] P         [P\bar{1}\ (2)]        
[2/m] P C       [P2/m\ (10)] [C2/m\ (12)]      
mmm P C I F   Pmmm (47) Cmmm (65) Immm (71) Fmmm (69)  
[4/m] P   I     [P4/m\ (83)]   [I4/m\ (87)]    
[4/mmm] P   I     [P4/mmm\ (123)]   [I4/mmm\ (139)]    
[\bar{3}] P       R [P\bar{3}\ (147)]       [R\bar{3}\ (148)]
[\!\left\{\matrix{\bar{3}m1\hfill\cr \bar{3}1m\hfill\cr}\right.] P       R [P\bar{3}m1\ (164)]       [R\bar{3}m\ (166)]
P         [P\bar{3}1m\ (162)]        
[6/m] P         [P6/m\ (175)]        
[6/mmm] P         [P6/mmm\ (191)]        
[m\bar{3}] P   I F   [Pm\bar{3}\ (200)]   [Im\bar{3}\ (204)] [Fm\bar{3}\ (202)]  
[m\bar{3}m] P   I F   [Pm\bar{3}m\ (221)]   [Im\bar{3}m\ (229)] [Fm\bar{3}m\ (225)]  

Note: For the four orthorhombic space groups with A cells (Nos. 38–41), the standard symbol for their Patterson symmetry, Cmmm, is added (between parentheses) after the actual symbol Ammm in the space-group tables.

The `point group part' of the symbol of the Patterson symmetry represents the Laue class to which the plane group or space group belongs (cf. Table 2.1.2.1[link] ). In the absence of anomalous dispersion, the Laue class of a crystal expresses the point symmetry of its diffraction record, i.e. the symmetry of the reciprocal lattice weighted with I(hkl).

References

Buerger, M. J. (1959). Vector space. New York: Wiley.








































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