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

International Tables for Crystallography (2006). Vol. A, ch. 4.3, pp. 68-71

## Section 4.3.3. Orthorhombic system

E. F. Bertauta

aLaboratoire de Cristallographie, CNRS, Grenoble, France

### 4.3.3. Orthorhombic system

| top | pdf |

#### 4.3.3.1. Historical note and arrangement of the tables

| top | pdf |

The synoptic table of IT (1935) contained space-group symbols for the six orthorhombic settings', corresponding to the six permutations of the basis vectors a, b, c. In IT (1952), left-handed systems like were changed to right-handed systems by reversing the orientation of the c axis, as in . Note that reversal of two axes does not change the handedness of a coordinate system, so that the settings , , and are equivalent in this respect. The tabulation thus deals with the possible right-handed settings. For further details see Section 2.2.6.4.

An important innovation of IT (1952) was the introduction of extended symbols for the centred groups A, B, C, I, F. These symbols are systematically developed in Table 4.3.2.1. Settings which permute the two axes a and b are listed side by side so that the two C settings appear together, followed by the two A and the two B settings.

In crystal classes mm2 and 222, the last symmetry element is the product of the first two and thus is not independent. It was omitted in the short Hermann–Mauguin symbols of IT (1935) for all space groups of class mm2, but was restored in IT (1952). In space groups of class 222, the last symmetry element cannot be omitted (see examples below).

For the new double' glide plane symbol e', see the Foreword to the Fourth Edition (IT 1995) and Section 1.3.2, Note (x) .

#### 4.3.3.2. Group–subgroup relations

| top | pdf |

The present section emphasizes the use of the extended and full symbols for the derivation of maximal subgroups of types I and IIa; maximal orthorhombic subgroups of types IIb and IIc cannot be recognized by inspection of the synoptic Table 4.3.2.1.

#### 4.3.3.2.1. Maximal non-isomorphic k subgroups of type IIa (decentred)

| top | pdf |

• (i) Extended symbols of centred groups A, B, C, I

By convention, the second line of the extended space-group symbol is the result of the multiplication of the first line by the centring translation (cf. Table 4.1.2.3 ). As a consequence, the product of any two terms in one line is equal to the product of the corresponding two terms in the other line.

 (a) Class 222 The extended symbol of I222 (23) is the twofold axes intersect and one obtains . Maximal k subgroups are P222 and (plus permutations) but not . The extended symbol of is , where one obtains ; the twofold axes do not intersect. Thus, maximal non-isomorphic k subgroups are and (plus permutations), but not P222. (b) Class mm2 The extended symbol of Aea2 (41) is the following relations hold: and . Maximal k subgroups are Pba2; Pcn2 (Pnc2); (); . (c) Class mmm By convention, the first line of the extended symbol contains those symmetry elements for which the coordinate triplets are explicitly printed under Positions. From the two-line symbols, as defined in the example below, one reads not only the eight maximal k subgroups P of class mmm but also the location of their centres of symmetry, by applying the following rules: If in the symbol of the P subgroup the number of symmetry planes, chosen from the first line of the extended symbol, is odd (three or one), the symmetry centre is at 0, 0, 0; if it is even (two or zero), the symmetry centre is at for the subgroups of C groups and at for the subgroups of I groups (Bertaut, 1976).

#### Examples

 (1) According to these rules, the extended symbol of Cmce (64) is (see above). The four k subgroups with symmetry centres at 0, 0, 0 are Pmcb (Pbam); Pmna; Pbca; Pbnb (Pccn); those with symmetry centres at are Pbna (Pbcn); Pmca (Pbcm); Pmnb (Pnma); Pbcb (Pcca). These rules can easily be transposed to other settings. (2) The extended symbol of Ibam (72) is . The four subgroups with symmetry centre at 0, 0, 0 are Pbam; Pbcn; Pcan (Pbcn); Pccm; those with symmetry centre at are Pccn; Pcam (Pbcm); Pbcm; Pban.

• (ii) Extended symbols of F-centred space groups

Maximal k subgroups of the groups F222, Fmm2 and Fmmm are C, A and B groups. The corresponding centring translations are and .

The (four-line) extended symbols of these groups can be obtained from the following scheme: The second, third, and fourth lines are the result of the multiplication of the first line by the centring translations w, u and v, respectively.

The following abbreviations are used: For the location of the symmetry elements in the above scheme, see Table 4.1.2.3 . In Table 4.3.2.1, the centring translations and the superscripts u, v, w have been omitted. The first two lines of the scheme represent the extended symbols of C222, Cmm2 and Cmmm. An interchange of the symmetry elements in the first two lines does not change the group. To obtain further maximal C subgroups, one has to replace symmetry elements of the first line by corresponding elements of the third or fourth line. Note that the symbol e' is not used in the four-line symbols for Fmm2 and Fmmm in order to keep the above scheme transparent.

#### Examples

 (1) F222 (22). In the first line replace by (third line, same column) and keep . Complete the first line by the product and obtain the maximal C subgroup . Similarly, in the first line keep and replace with (fourth line, same column). Complete the first line by the product and obtain the maximal C subgroup . Finally, replace and by and and form the product , to obtain the maximal C subgroup (where can be replaced by 2). Note that C222 and are two different subgroups, as are and . (2) Fmm2 (42). A similar procedure leads to the four maximal k subgroups ; and Ccc2. (3) Fmmm (69). One finds successively the eight maximal k subgroups Cmmm; Cmma; Cmcm; Ccmm (Cmcm); Cmca; Ccma (Cmca); Cccm; and Ccca.

Maximal A- and B-centred subgroups can be obtained from the C subgroups by simple symmetry arguments.

In space groups Fdd2 (43) and Fddd (70), the nature of the d planes is not altered by the translations of the F lattice; for this reason, a two-line symbol for Fdd2 and a one-line symbol for Fddd are sufficient. There exist no maximal non-isomorphic k subgroups for these two groups.

#### 4.3.3.2.2. Maximal t subgroups of type I

| top | pdf |

• (i) Orthorhombic subgroups

The standard full symbol of a P group of class mmm indicates all the symmetry elements, so that maximal t subgroups can be read at once.

#### Example

(51) has the following four t subgroups: ; ; ; Pm2a (Pma2).

From the standard full symbol of an I group of class mmm, the t subgroup of class 222 is read directly. It is either I222 [for Immm (71) and Ibam (72)] or [for Ibca (73) and Imma (74)]. Use of the two-line symbols results in three maximal t subgroups of class mm2.

#### Example

(72) has the following three maximal t subgroups of class mm2: ; ; .

From the standard full symbol of a C group of class mmm, one immediately reads the maximal t subgroup of class 222, which is either [for Cmcm (63) and Cmce (64)] or C222 (for all other cases). For the three maximal t subgroups of class mm2, the two-line symbols are used.

#### Example

(64) has the following three maximal t subgroups of class mm2: ; (); ().

Finally, Fmmm (69) has maximal t subgroups F222 and Fmm2 (plus permutations), whereas Fddd (70) has F222 and Fdd2 (plus permutations).

• (ii) Monoclinic subgroups

These subgroups are obtained by substituting the symbol `l' in two of the three positions. Non-standard centred cells are reduced to primitive cells.

#### Examples

 (1) (20) has the maximal t subgroups C211 (C2), C121 (C2) and . The last one reduces to (). (2) Ama2 (40) has the maximal t subgroups Am11, reducible to Pm, A1a1 (Cc) and A112 (C2). (3) Pnma (62) has the standard full symbol , from which the maximal t subgroups , and are obtained. (4) Fddd (70) has the maximal t subgroups , and , each one reducible to .

### 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).]
International Tables for Crystallography (1995). Vol. A, fourth, revised ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. [Abbreviated as IT (1995).]
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]
Bertaut, E. F. (1976). Study of principal subgroups and of their general positions in C and I groups of class mmm–Dzh. Acta Cryst. A32, 380–387.