Patterson symmetry

Kopskyacute, V.; Litvin, D. B.

doi:10.1107/97809553602060000647

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. E. ch. 1.2, p. 8 | 1 | 2 |

Section 1.2.5. Patterson symmetry

V. Kopský^a and D. B. Litvin^b ^*

^a Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and ^bDepartment of Physics, 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

| top | pdf |

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) and Wilson (2004).] 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. 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, 1.2.1.2 and 1.2.1.3).

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

(a) Layer groups.

Laue class	Lattice type	Patterson symmetry (with subperiodic group number)
$[\bar{1}]$	p	p $[\bar{1}]$ (L2)
112/m	p	p 112/m (L6)
2/m11	p , c	p 2/m11 (L14), c2/m11 (L18)
mmm	p , c	pmmm (L37), cmmm (L47)
4/m	p	p 4/m (L51)
4/mmm	p	p 4/mmm (L61)
$[\bar{3}]$	p	p $[\bar{3}]$ (L66)
$[\bar{3}]$ 1m	p	p $[\bar{3}]$ 1m (L71)
$[\bar{3}]$ m 1	p	p $[\bar{3}]$ m 1 (L72)
6/m	p	p 6/m (L75)
6/mmm	p	p 6/mmm (L80)

(b) Rod groups.

Laue class	Lattice type	Patterson 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)

Laue class	Lattice type	Patterson 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 lamda technique. Z. Kristallogr. 180, 237–242.Google Scholar

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.Google Scholar

International Tables for Crystallography (2006). Vol. E. ch. 1.2, p. 8