International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

International Tables for Crystallography (2010). Vol. E, ch. 1.2, p. 8   | 1 | 2 |

Section 1.2.5. Patterson symmetry

V. Kopskýa and D. B. Litvinb*

aFreelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and bDepartment of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail:  u3c@psu.edu

1.2.5. Patterson symmetry

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The entry Patterson symmetry in the headline gives the subperiodic group of the Patterson function, where Friedel's law is assumed, i.e. with neglect of anomalous dispersion. [For a discussion of the effect of dispersion, see Fischer & Knof (1987[link]) and Wilson (2004[link]).] The symbol for the Patterson subperiodic group can be deduced from the symbol of the subperiodic group in two steps:

  • (i) Glide planes and screw axes are replaced by the corresponding mirror planes and rotation axes.

  • (ii) If the resulting symmorphic subperiodic group is not centrosymmetric, inversion is added.

There are 13 different Patterson symmetries for the layer groups, ten for the rod groups and two for the frieze groups. These are listed in Table 1.2.5.1[link]. The `point-group part' of the symbol of the Patterson symmetry represents the Laue class to which the subperiodic group belongs (cf. Tables 1.2.1.1[link], 1.2.1.2[link] and 1.2.1.3[link]).

Table 1.2.5.1| top | pdf |
Patterson symmetries for subperiodic groups

(a) Layer groups.

Laue classLattice typePatterson symmetry (with subperiodic group number)
[\bar{1}] p p[\bar{1}] (L2)
112/m p p112/m (L6)
2/m11 p, c p2/m11 (L14), c2/m11 (L18)
mmm p, c pmmm (L37), cmmm (L47)
4/m p p4/m (L51)
4/mmm p p4/mmm (L61)
[\bar{3}] p p[\bar{3}] (L66)
[\bar{3}]1m p p[\bar{3}]1m (L71)
[\bar{3}]m1 p p[\bar{3}]m1 (L72)
6/m p p6/m (L75)
6/mmm p p6/mmm (L80)

(b) Rod groups.

Laue classLattice typePatterson symmetry (with subperiodic group number)
[\bar{1}] [{\scr p}] [\scr p][\bar{1}] (R2)
2/m11 [{\scr p}] [\scr p]2/m11 (R6)
112/m [{\scr p}] [\scr p]112/m (R11)
mmm [{\scr p}] [\scr p]mmm (R20)
4/m [{\scr p}] [\scr p]4/m (R28)
4/mmm [{\scr p}] [\scr p]4/mmm (R39)
[\bar{3}] [{\scr p}] [\scr p][\bar{3}] (R48)
[\bar{3}]m [{\scr p}] [\scr p][\bar{3}]1m (R51)
6/m [{\scr p}] [\scr p]6/m (R60)
6/mmm [{\scr p}] [\scr p]6/mmm (R73)

(c) Frieze groups.

Laue classLattice typePatterson symmetry (with subperiodic group number)
2 [{\scr p}] [\scr p]211 (F2)
2mm [{\scr p}] [\scr p]2mm (F6)

References

Fischer, K. F. & Knof, W. E. (1987). Space groups for imaginary Patterson and for difference Patterson functions used in the lambda technique. Z. Kristallogr. 180, 237–242.
Wilson, A. J. C. (2004). Arithmetic crystal classes and symmorphic space groups. In International Tables for Crystallography, Vol. C, Mathematical, Physical and Chemical Tables, edited by E. Prince, ch. 1.4. Dordrecht: Kluwer Academic Publishers.








































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