Sequence of space-group types

Wondratschek, H.

doi:10.1107/97809553602060000516

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

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International Tables for Crystallography (2006). Vol. A. ch. 8.3, p. 736

Section 8.3.4. Sequence of space-group types

H. Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: hans.wondratschek@physik.uni-karlsruhe.de

8.3.4. Sequence of space-group types

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The sequence of space-group entries in the space-group tables follows that introduced by Schoenflies (1891) and is thus established historically. Within each geometric crystal class, Schoenflies has numbered the space-group types in an obscure way. As early as 1919, Niggli (1919) considered this Schoenflies sequence to be unsatisfactory and suggested that another sequence might be more appropriate. Fedorov (1891) used a different sequence in order to distinguish between symmorphic, hemisymmorphic and asymmorphic space groups.

The basis of the Schoenflies symbols and thus of the Schoenflies listing is the geometric crystal class. For the present Tables, a sequence might have been preferred in which, in addition, space-group types belonging to the same arithmetic crystal class were grouped together. It was decided, however, that the long-established sequence in the earlier editions of International Tables should not be changed.

In Table 8.3.4.1, those geometric crystal classes are listed in which the Schoenflies sequence separates space groups belonging to the same arithmetic crystal class. The space groups are rearranged in such a way that space groups of the same arithmetic crystal class are grouped together. The arithmetic crystal classes are separated by rules spanning the first three columns of the table and the geometric crystal classes are separated by rules spanning the full width of the table. In all cases not listed in Table 8.3.4.1, the Schoenflies sequence, as used in these Tables, does not break up arithmetic crystal classes. Nevertheless, some rearrangement would be desirable in other arithmetic crystal classes too. For example, the symmorphic space group should always be the first entry of each arithmetic crystal class.

Table 8.3.4.1| top | pdf |
Listing of space-group types according to their geometric and arithmetic crystal classes

No.	Hermann–Mauguin symbol	Schoenflies symbol	Geometric crystal class
10		$[C_{2h}^{1}]$
11	$[P2_{1}/m]$	$[C_{2h}^{2}]$
13		$[C_{2h}^{4}]$
14	$[P2_{1}/c]$	$[C_{2h}^{5}]$
12		$[C_{2h}^{3}]$
15		$[C_{2h}^{6}]$
149	P312	$[D_{3}^{1}]$	32
151	$[P3_{1}12]$	$[D_{3}^{3}]$
153	$[P3_{2}12]$	$[D_{3}^{5}]$
150	P321	$[D_{3}^{2}]$
152	$[P3_{1}21]$	$[D_{3}^{4}]$
154	$[P3_{2}21]$	$[D_{3}^{6}]$
155	R32	$[D_{3}^{7}]$
156	P3m1	$[C_{3v}^{1}]$	3m
158	P3c1	$[C_{3v}^{3}]$
157	P31m	$[C_{3v}^{2}]$
159	P31c	$[C_{3v}^{4}]$
160	R3m	$[C_{3v}^{5}]$
161	R3c	$[C_{3v}^{6}]$
195	P23	$[T^{1}]$	23
198	$[P2_{1}3]$	$[T^{4}]$
196	F23	$[T^{2}]$
197	I23	$[T^{3}]$
199	$[I2_{1}3]$	$[T^{5}]$
200	$[Pm\bar{3}]$	$[T_{h}^{1}]$	$[m\bar{3}]$
201	$[Pn\bar{3}]$	$[T_{h}^{2}]$
205	$[Pa\bar{3}]$	$[T_{h}^{6}]$
202	$[Fm\bar{3}]$	$[T_{h}^{3}]$
203	$[Fd\bar{3}]$	$[T_{h}^{4}]$
204	$[Im\bar{3}]$	$[T_{h}^{5}]$
206	$[Ia\bar{3}]$	$[T_{h}^{7}]$
207	P432	$[O^{1}]$	432
208	$[P4_{2}32]$	$[O^{2}]$
213	$[P4_{1}32]$	$[O^{7}]$
212	$[P4_{3}32]$	$[O^{6}]$
209	F432	$[O^{3}]$
210	$[F4_{1}32]$	$[O^{4}]$
211	I432	$[O^{5}]$
214	$[I4_{1}32]$	$[O^{8}]$
215	$[P\bar{4}3m]$	$[T_{d}^{1}]$	$[\bar{4}3m]$
218	$[P\bar{4}3n]$	$[T_{d}^{4}]$
216	$[F\bar{4}3m]$	$[T_{d}^{2}]$
219	$[F\bar{4}3c]$	$[T_{d}^{5}]$
217	$[I\bar{4}3m]$	$[T_{d}^{3}]$
220	$[I\bar{4}3d]$	$[T_{d}^{6}]$

References

Fedorov, E. S. (1891). The symmetry of regular systems of figures. (In Russian.) [English translation by D. & K. Harker (1971). Symmetry of crystals, pp. 50–131. American Crystallographic Association, Monograph No. 7.]Google Scholar

Niggli, P. (1919). Geometrische Kristallographie des Diskontinuums. Leipzig: Borntraeger. [Reprint: Sändig, Wiesbaden (1973).]Google Scholar

Schoenflies, A. (1891). Krystallsysteme und Krystallstructur. Leipzig: Teubner. [Reprint: Springer, Berlin (1984).]Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 8.3, p. 736