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 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). 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. Account is taken of the fact that the Laue class combines in two ways with the hexagonal translation lattice, namely as and as .

 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
P
P C
mmm P C I F   Pmmm (47) Cmmm (65) Immm (71) Fmmm (69)
P   I
P   I
P       R
P       R
P
P
P
P   I F
P   I F

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 ). 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.