Special positions of given symmetry

Wilson, A. J. C.; Karen, V. L.; Mighell, A.

doi:10.1107/97809553602060000623

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 9.7, pp. 900-902

Section 9.7.2. Special positions of given symmetry

A. J. C. Wilson,^a ^‡ V. L. Karen^b and A. Mighell^b

^a St John's College, Cambridge CB2 1TP, England, and ^bNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

9.7.2. Special positions of given symmetry

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As noted by Kitajgorodskij, in many crystal structures molecules with inherent symmetry may occupy Wyckoff special positions, so that molecular and crystallographic symmetry elements coincide, and this may affect the relative frequencies of occurrence of structures with particular space groups. Tables of the frequency of occurrence of space groups have been published by many authors, from Nowacki (1942) onwards. Some typical recent papers are Brock & Dunitz (1994), Donohue (1985), Mighell, Himes & Rodgers (1983), Padmaya, Ramakumar & Viswamitra (1990), Wilson (1988, 1990, 1993b,c), but many of them hardly go beyond recognizing the fact that structures frequently made use of molecular symmetry – Wilson (1988) explicitly chose to ignore it. The early work of Belsky, Zorky and their colleagues did not attract much attention outside Russian-speaking areas. Recently, however, there has been a spate of interest (Wilson, 1991, 1993b,c,d; Brock & Dunitz, 1994; Belsky, Zorkaya & Zorky, 1995). Earlier lack of results is partly due to the fact that the Cambridge Structural Database (Section 9.7.3) did not provide a search program that would distinguish between occupation of a general position and multiple occupation of special positions of the required symmetry (Wilson, 1993d, Section 3). Belsky, Zorkaya & Zorky (1995) were able to make this distinction, and their paper is the source of many of the statistics quoted without special citation here.

It would be interesting to know which space groups possess positions with the symmetry of each of the 32 point groups 1, $[\overline {1}]$ , 2, m, 2/m, $[\ldots ]$ , $[m{\overline 3}m]$ . Volume A of International Tables for Crystallography (Hahn, 2005) enumerates the symmetry of all the special positions of a given space group, but does not readily answer the reverse question: which space groups contain special positions of given point group $[{\cal G}]$ ? Some general points may be noted.

(i) Special positions of symmetry $[{\cal G}]$ will be found in the symmorphic, but not other, space groups of the geometric class $[{\cal G}]$ . Thus, for example, there are special positions of symmetry mmm in Pmmm, Cmmm, Fmmm, Immm, but not in any other space group in the geometric class mmm.
(ii) A `family tree' of point groups is given in Fig. 10.1.3.2 of Volume A of International Tables for Crystallography (Hahn, 2005). Special positions of symmetry $[{\cal G}]$ may be sought in space groups of the geometric classes linked to $[{\cal G}]$ by a line (possibly zigzag) having a generally upwards direction. Thus, to take the same example, special positions of symmetry mmm are found in certain space groups of 4/mmm (P4/mmm, P4/mbm, P4₂/mmc, P4₂/mcm, P4₂/mmm, I4/mmm, I4/mcm), in 6/mmm (P6/mmm), in $[m\overline {3}]$ ( $[Pm\overline {3}]$ , $[Im\overline {3}]$ ), and in $[m\overline {3}m]$ ( $[Pm\overline {3}m]$ , $[Fm\overline {3}m]$ ).

(iii) Obviously, the higher up the tree the symmetry $[{\cal G}]$ is, the fewer will be the space groups in which it can occur – special positions of symmetry $[m\overline {3}m]$ can occur only in the three symmorphic space groups of the corresponding geometric class. The lower symmetries (2, m, $[\overline {1}]$ , $[\overline {3}]$ ), with nothing below them but 1, can be traced upwards along many branches, and so can occur in many space groups, but not all are equally favoured. Special positions of symmetry 2 can be sought in all higher geometric classes except $[\overline {6}]$ , 3m, and $[\overline {3}]$ , but those of symmetry 3 could occur only in the classes of the trigonal, hexagonal, and cubic systems. An approximate count⁵ (Table 9.7.2.1) shows that special positions of symmetry 2 occur in 167 space groups, of m in 99, of $[\overline {1}]$ in 38, and of 3 in 57. The only other special positions with space-group frequencies of this order are 2/m (39), 222 (30), and mm (57).

Table 9.7.2.1 | top | pdf |
Statistics of the use of Wyckoff positions of specified symmetry $[{\cal G}]$ in the homomolecular organic crystals, based on the data by Belsky, Zorkaya & Zorky (1995)

$[{\cal G}]$	Space groups with positions of symmetry $[{\cal G}]$	Space groups actually occurring	Space groups using such positions
1	230	116	57
2	167	79	38
m	99	42	25
$[\overline {1} ]$	38	28	10
3	57	18	7
4	24	6	4
$[\overline {4}]$	29	17	8
222	50	15	5
mm 2	57	18	5
2/m	39	21	6
6	5	1	0
$[\overline {6}]$	8	3	1
32	22	5	1
3m	22	8	5
$[\overline {3}]$	14	7	4
422	9	1	1
$[\overline {4}]$ 2m, $[\overline {4}]$ m2	19	5	3
4mm	8	3	0
4/m	7	2	1
mmm	16	3	2
23	12	3	2
622	2	1	0
$[\overline {6}]$ 2m, $[\overline {6}]$ m2	5	1	1
6mm	2	0	0
6/m	2	1	0
$[\overline {3}]$ m	8	3	1
4/mmm	4	2	1
432	5	0	0
$[\overline {4}]$ 3m	6	2	1
m $[\overline {3}]$	5	2	1
6/mmm	1	0	0
m $[\overline {3}]$ m	3	2	2

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International Tables for Crystallography (2006). Vol. C. ch. 9.7, pp. 900-902