Cubic system

Bertaut, E. F.

doi:10.1107/97809553602060000509

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. 4.3, pp. 75-76

Section 4.3.6. Cubic system

E. F. Bertaut^a ^‡

^a Laboratoire de Cristallographie, CNRS, Grenoble, France

4.3.6. Cubic system

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4.3.6.1. Historical note and arrangement of tables

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In the synoptic tables of IT (1935) and IT (1952), for cubic space groups short and full Hermann–Mauguin symbols were listed. They agree, except that in IT (1935) the tertiary symmetry element of the space groups of class 432 was omitted; it was re-established in IT (1952).

In the present edition, the symbols of IT (1952) are retained, with one exception. In the space groups of crystal classes $[m\bar{3}]$ and $[m\bar{3}m]$ , the short symbols contain $[\bar{3}]$ instead of 3 (cf. Section 2.2.4 ). In Table 4.3.2.1, short and full symbols for all cubic space groups are given. In addition, for centred groups F and I and for P groups with tertiary symmetry elements, extended space-group symbols are listed. In space groups of classes 432 and $[\bar{4}3m]$ , the product rule (as defined below) is applied in the first line of the extended symbol.

4.3.6.2. Relations between symmetry elements

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Conventionally, the representative directions of the primary, secondary and tertiary symmetry elements are chosen as [001], [111], and [ $[1\bar{1}0]$ ] (cf. Table 2.2.4.1 for the equivalent directions). As in tetragonal and hexagonal space groups, tertiary symmetry elements are not independent. In classes 432, $[\bar{4}3m]$ and $[m\bar{3}m]$ , there are product rules $[4 \times 3 = (2){\hbox{;}}\quad \bar{4} \times 3 = (m) = 4 \times \bar{3},]$ where the tertiary symmetry element is in parentheses; analogous rules hold for the space groups belonging to these classes. When the symmetry directions of the primary and secondary symmetry elements are chosen along [001] and [111], respectively, the tertiary symmetry direction is [011], according to the product rule. In order to have the tertiary symmetry direction along [ $[1\bar{1}0]$ ], one has to choose the somewhat awkward primary and secondary symmetry directions [010] and [ $[\bar{1}1\bar{1}]$ ].

Examples

(1) In $[P\bar{4}3n]$ (218), with the choice of the 3 axis along [ $[\bar{1}1\bar{1}]$ ] and of the $[\bar{4}]$ axis parallel to [010], one finds $[\bar{4} \times 3 = n]$ , the n glide plane being in x, x, z, as shown in the space-group diagram.
(2) In $[F\bar{4}3c]$ (219), one has the same product rule as above; the centring translation $[t({1 \over 2},{1 \over 2},0)]$ , however, associates with the n glide plane a c glide plane, also located in x, x, z (cf. Table 4.1.2.3 ). In the space-group diagram and symbol, c was preferred to n.

4.3.6.3. Additional symmetry elements

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Owing to periodicity, the tertiary symmetry elements alternate; diagonal axes 2 alternate with parallel screw axes $[2_{1}]$ ; diagonal planes m alternate with parallel glide planes g; diagonal n planes, i.e. planes with glide components $[{1 \over 2}, {1 \over 2}, {1 \over 2}]$ , alternate with glide planes a, b or c (cf. Chapter 4.1 and Tables 4.1.2.2 and 4.1.2.3 ). For the meaning of the various glide planes g, see Section 11.1.2 and the entries Symmetry operations in Part 7 .

4.3.6.4. Group–subgroup relations

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4.3.6.4.1. Maximal k subgroups

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The extended symbol of $[Fm\bar{3}]$ (202) shows clearly that $[Pm\bar{3}]$ , $[Pn\bar{3}]$ , $[Pb\bar{3}\; (Pa\bar{3})]$ and $[Pa\bar{3}]$ are maximal subgroups. $[Pm\bar{3}m]$ , $[Pn\bar{3}n]$ , $[Pm\bar{3}n]$ and $[Pn\bar{3}m]$ are maximal subgroups of $[Im\bar{3}m]$ (229). Space groups with d glide planes have no k subgroup of lattice P.

4.3.6.4.2. Maximal t subgroups

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(a) Cubic subgroups

The cubic space groups of classes $[m\bar{3}]$ , 432 and $[\bar{4}3m]$ have maximal cubic subgroups of class 23 which are found by simple inspection of the full symbol.

Examples

$[Ia\bar{3}]$ (206), full symbol $[I2_{1}/a\bar{3}]$ , contains $[I2_{1}3]$ . $[P2_{1}3]$ is a maximal subgroup of $[P4_{1}32]$ (213) and its enantiomorph $[P4_{3}32]$ (212). A more difficult example is $[I\bar{4}3d]$ (220) which contains $[I2_{1}3]$ .³

The cubic space groups of class $[m\bar{3}m]$ have maximal subgroups which belong to classes 432 and $[\bar{4}3m]$ .

Examples

$[F4/m\bar{3}2/c]$ (226) contains F432 and $[F\bar{4}3c]$ ; $[I4_{1}/a\bar{3}2/d]$ (230) contains $[I4_{1}32]$ and $[I\bar{4}3d]$ .
(b) Tetragonal subgroups

In the cubic space groups of classes 432 and $[\bar{4}3m]$ , the primary and tertiary symmetry elements are relevant for deriving maximal tetragonal subgroups.

Examples

The groups P432 (207), $[P4_{2}32]$ (208), $[P4_{3}32]$ (212) and $[P4_{1}32]$ (213) have maximal tetragonal t subgroups of index [3]: P422, $[P4_{2}22, P4_{3}2_{1}2]$ and $[P4_{1}2_{1}2]$ . I432 (211) gives rise to I422 with the same cell. F432 (209) also gives rise to I422, but via F422, so that the final unit cell is $[a\sqrt{2}/2, a\sqrt{2}/2, a]$ .

In complete analogy, the groups $[P\bar{4}3m]$ (215) and $[P\bar{4}3n]$ (218) have maximal subgroups $[P\bar{4}2m]$ and $[P\bar{4}2c]$ .⁴

For the space groups of class $[m\bar{3}m]$ , the full symbols are needed to recognize their tetragonal maximal subgroups of class . The primary symmetry planes of the cubic space group are conserved in the primary and secondary symmetry elements of the tetragonal subgroup: m, n and d remain in the tetragonal symbol; a remains a in the primary and becomes c in the secondary symmetry element of the tetragonal symbol.

Example

$[P4_{2}/n\;\bar{3}\;2/m]$ (224) and $[I4_{1}/a\;\bar{3}\;2/d]$ (230) have maximal subgroups $[P4_{2}/n\;2/n\;2/m]$ and $[I4_{1}/a\;2/c\;2/d]$ , respectively, $[F4_{1}/d\;\bar{3}\;2/c]$ (228) gives rise to $[F4_{1}/d\;2/d\;2/c]$ , which is equivalent to $[I4_{1}/a\;2/c\;2/d]$ , all of index [3].
(c) Rhombohedral subgroups⁵

Here the secondary and tertiary symmetry elements of the cubic space-group symbols are relevant. For space groups of classes 23, $[m\bar{3}]$ , 432, the maximal R subgroups are R3, $[R\bar{3}]$ and R32, respectively. For space groups of class $[\bar{4}3m]$ , the maximal R subgroup is when the tertiary symmetry element is m and R3c otherwise. Finally, for space groups of class $[m\bar{3}m]$ , the maximal R subgroup is $[R\bar{3}m]$ when the tertiary symmetry element is m and $[R\bar{3}c]$ otherwise. All subgroups are of index [4].
(d) Orthorhombic subgroups

Maximal orthorhombic space groups of index [3] are easily derived from the cubic space-group symbols of classes 23 and $[m\bar{3}]$ .⁵ Thus, P23, F23, I23, $[P2_{1}3]$ , $[I2_{1}3]$ (195–199) have maximal subgroups P222, F222, I222, $[P2_{1}2_{1}2_{1}]$ , $[I2_{1}2_{1}2_{1}]$ , respectively. Likewise, maximal subgroups of $[Pm\bar{3}]$ , $[Pn\bar{3}]$ , $[Fm\bar{3}]$ , $[Fd\bar{3}]$ , $[Im\bar{3}]$ , $[Pa\bar{3}]$ , $[Ia\bar{3}]$ (200–206) are Pmmm, Pnnn, Fmmm, Fddd, Immm, Pbca, Ibca, respectively. The lattice type (P, F, I) is conserved and only the primary symmetry element has to be considered.

References

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]Google Scholar

International Tables for Crystallography (1995). Vol. A, fourth, revised ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. [Abbreviated as IT (1995).]Google Scholar

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 4.3, pp. 75-76