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

International Tables for Crystallography (2006). Vol. A, ch. 4.1, pp. 56-60

## Section 4.1.2. Additional symmetry elements

E. F. Bertauta

aLaboratoire de Cristallographie, CNRS, Grenoble, France

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In space groups, due to periodicity', symmetry elements occur that are not recorded in the Hermann–Mauguin symbols. These additional symmetry elements are products of a symmetry translation and a symmetry operation . This product is and its geometrical representation is found in the space-group diagrams (cf. Sections 8.1.2 and 11.1.1 ).2

Two cases have to be distinguished:

• (i) Symmetry operations of the same nature

The symmetry operations and are of the same nature and only the locations of their symmetry elements differ. This occurs when the translation vector t is perpendicular to the symmetry element of (symmetry plane or symmetry axis); it also holds when is an inversion or a rotoinversion (see below).

Table 4.1.2.1 summarizes the symmetry elements, located at the origin, and the location of those additional symmetry elements' which are generated by periodicity in the interior of the unit cell. Additional' axes , 6, , , , , do not occur. The first column of Table 4.1.2.1 specifies , the second column the translation vector t, the third the location of the symmetry element of . The last column indicates space groups and plane groups with representative diagrams. Other orientations of the symmetry axes and symmetry planes can easily be derived from the table.

 Table 4.1.2.1| top | pdf | Location of additional symmetry element, if the translation vector t is perpendicular to the symmetry axis along or to the symmetry plane in
 The symmetry centre at 0, 0, 0 is included. The table is restricted to integral translations (for centring translations, see Table 4.1.2.3). The symbol ↺ indicates cyclic permutation.
Symmetry element at the originTranslation vector tLocation of additional symmetry elementRepresentative plane and space groups (numbers)
1, 0, 0
0, 1, 0
1, 1, 0
1, 0, 0
1, 1, 0
1, 0, 0
m, a, b, n, d, e 0, 0, 1
1, 0, 0 ↺ (2)
1, 1, 0 ↺
1, 1, 1
(147)
0, 1, 0 (81)
0, 1, 0 (174)
1, 1, 0

#### Example

Let be a threefold rotation with Seitz symbol and axis along 0, 0, z. The product with the translation , perpendicular to the axis, is and again is a threefold rotation, for ; its location is .

Table 4.1.2.1 also deals with certain powers of symmetry operations , namely with for operations of order four and with for operations of order six. These powers give rise to their own additional symmetry elements', as illustrated by the following list and by the example below (operations of order 2 or 3 obviously do not have to be considered).

#### Example

in 0, 0, z; the powers to be considered are The axes and 2 at 0, 0, z create additional symmetry elements:

If is an inversion operation with its centre of symmetry at point M, the operation creates an additional centre at the endpoint of the translation vector , drawn from M (cf. Table 4.1.2.1, where M is in 0, 0, 0).

• (ii) Symmetry operations of different nature

The symmetry operations and are of a different nature and have different symbols, corresponding to rotation and screw axes, to mirror and glide planes, to screw axes of different nature, and to glide planes of different nature, respectively.3

In this case, the translation vector t has a component parallel to the symmetry axis or symmetry plane of . This parallel component determines the nature and the symbol of the additional symmetry element, whereas the normal component of t is responsible for its location, as explained in Section 11.1.1 . If the normal component is zero, symmetry element and additional symmetry element coincide geometrically. Note that such additional symmetry elements with glide or screw components exist even in symmorphic space groups.

Integral and centring translations: In primitive lattices, only integral translations occur and Tables 4.1.2.1 and 4.1.2.2 are relevant. For centred lattices, Tables 4.1.2.1 and 4.1.2.2 remain valid for the integral translations, whereas Table 4.1.2.3 has to be considered for the centring translations, which cause further additional symmetry elements'.

 Table 4.1.2.2| top | pdf | Additional symmetry elements and their locations, if the translation vector t is inclined to the symmetry axis or symmetry plane
 The table is restricted to integral translations and thus is valid for P lattices and for integral translations in centred lattices (for centring translations see Table 4.1.2.3).
Symmetry element at the originTranslation vector tAdditional symmetry elementRepresentative plane and space groups (numbers)
SymbolLocationSymbolScrew or glide componentLocation
Tetragonal, rhombohedral and cubic coordinate systems
2 x, x, 0 1, 0, 0 P422 (89)
0, 1, 0     R32 (155)
P432 (207)
m x, x, z 1, 0, 0 g p4mm (11)
0, 1, 0     P4mm (99)
R3m (160)
(215)
c x, x, z 1, 0, 0 n (112)
0, 1, 0     R3c (161)
(218)
Hexagonal coordinate system
2 x, 0, 0 1, 1, 0 P321 (150)
0, 1, 0     R32 (155)
2 x, 2x, 0 0, 1, 0 P312 (149)
1, 1, 0       P622 (177)
m x, 2x, z 0, 1, 0 b P3m1 (156)
1, 1, 0       p3m1 (14)
R3m (160)
c x, 2x, z 0, 1, 0 n P3c1 (158)
1, 1, 0       (188)
R3c (161)
m x, 0, z 1, 1, 0 a P31m (157)
0, 1, 0     p31m (15)
c x, 0, z 1, 1, 0 n P31c (159)
0, 1, 0     (190)
Rhombohedral and cubic coordinate systems
3 x, x, x 1, 0, 0 R3 (146)
0, 1, 0
0, 0, 1
3 x, x, x 2, 0, 0 P23 (195)
0, 2, 0
0, 0, 2
 Table 4.1.2.3| top | pdf | Additional symmetry elements due to a centring vector t and their locations
Symmetry element at the originAdditional symmetry elementsRepresentative space groups (numbers)
F
SymbolLocationSymbolLocationSymbolLocationSymbolLocationSymbolLocationSymbol
m 0, y, z b n 0, y, z c n b, n, c, e Cmmm, Ammm, Bmmm (65)
c   n   b   m   b     Immm (71), Fmmm (69)
b   m   c   n   c     Cccm, Amaa, Bbmb (66) Ibca (73)
e       e             Aem2 (39)
d, d, d Fddd (70)
m x, 0, z a c n x, 0, z n a, c, n, e As above
a   m   n   c   c
c   n   m   a   a
e           e         Fmm2 (42)
d, d, d
m x, y, 0 n x, y, 0 b a n n, b, a, e As above
b   a   m   n   a
a   b   n   m   b
e   e                 Cmme (67)
d, d, d
m x, x, z x, x, z x, x, z g, g, g I4mm (107), (216)
c           n, g, g (219)
e               e     I4cm (108)
(220)
2 x, 0, 0 2 C222, A222, B222 (21)
2 0, y, 0 2 I222 (23)
2 0, 0, z 2 F222 (22)
2 2 2 C422 (P422) (89), I422 (97)
4 0, 0, z 4 F432 (209)
0, 0, z (210)
0, 0, 0 Immm (71), Fmmm (69)

#### 4.1.2.1. Integral translations

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Table 4.1.2.2 lists representative symmetry elements, corresponding to , and their associated glide planes and screw axes, corresponding to . The upper part of the table contains the diagonal twofold axes and symmetry planes that appear as tertiary symmetry elements in tetragonal and cubic space groups and as secondary symmetry elements in rhombohedral space groups (referred to rhombohedral axes). The middle part lists the twofold axes and symmetry planes that are secondary and tertiary symmetry elements in trigonal and hexagonal space groups and secondary symmetry elements in rhombohedral space groups (referred to hexagonal axes). The lower part illustrates the occurrence of threefold screw axes in rhombohedral and cubic space groups for the orientation [111].

Note that integral translations do not produce additional glide or screw components in triclinic, monoclinic and orthorhombic groups.

#### Example

The operation in a rhombohedral or cubic space group represents a screw rotation with axis along [111]. Indeed, the third power of is the translation t(1, 1, 1), i.e. the periodicity along the threefold axis. The translation t(1, 0, 0) is decomposed uniquely into the screw component parallel to and the location component perpendicular to the threefold axis. The location of the axis is then found to be , which can also be expressed as or .

For 2, m and c, the locations of the symmetry elements at the origin and within the cell can be interchanged.

#### Example

According to Table 4.1.2.2, the c plane located in x, x, z implies an n plane in . Vice versa, an n plane in x, x, z implies a c plane in .

In the rhombohedral space groups R3c (161) and (167) and in their cubic supergroups, diagonal n planes in x, x, z and, by symmetry, in z, x, x and x, z, x coexist with c planes in , a planes in and b planes in , respectively (cf. Section 4.3.5 ).

Note that the symbol of a glide plane depends on the reference frame. Thus, the above-mentioned n planes in the rhombohedral description become c planes in the hexagonal description of R3c and ; similarly, the a, b and c planes become n planes; cf. Sections 1.3.1 and 1.4.4 .

#### 4.1.2.2. Centring translations4

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The general rules given under (i) and (ii) remain valid. In lattices C, A, B, I and F, a centring vector t with a component parallel to the symmetry element leads to an additional symmetry element of a different kind. When the centring vector t is perpendicular to the symmetry element or when the symmetry element is an inversion centre or a rotoinversion axis, the additional symmetry element is of the same kind.

The first part of Table 4.1.2.3 contains pairs of symmetry planes related by a centring translation. Each box has three or four entries, which define three or four pairs of associated' planes; the cell under F contains all the planes under C, A and B. Hence, their locations are not repeated under F. Again, the locations of the two planes can be interchanged.

#### Example

The product of the C-centring translation, i.e. , and the reflection through a mirror plane m, located in 0, y, z, is a glide reflection b with glide plane in . Similarly, C centring associates a glide plane c in 0, y, z with a glide plane n in .

Note that the mirror plane and associated' glide plane coincide geometrically when the centring translation is parallel to the mirror (i.e. no normal component exists); see the first cell under A, the second under B, the third cell under C. Also, two associated' glide planes (a, b) or (b, c) or (a, c) coincide geometrically. These double' glide planes are symbolized by e'; see Table 4.1.2.3 and Section 1.3.2 , Note (x) .

Glide reflections whose square is a pure centring translation are called d; other diagonal glide planes are called g and n; in each case, the glide component is given between parentheses (cf. Sections 2.2.9 and 11.1.2 ).

The second part of Table 4.1.2.3 summarizes pairs of symmetry axes and, in the bottom line, pairs of symmetry centres related by a centring translation. For instance, the B-centring translation associates a rotation axis 2 along x, 0, 0 with a screw axis along . Here, too, the locations can be interchanged.

#### Example

The product of the translation with a twofold rotation around x, x, 0 is the operation , which occurs, for instance, in F432 (209). The square of this operation is the fractional translation . The translation is decomposed into a screw part' and a location part' perpendicular to it. The location of the additional symmetry element is then found to be which is parallel to that of the axis 2 in x, x, 0.

Inversions. The midpoint rule' given under (i) for integral translations remains valid. When M occupies successively the eight positions of inversion centres in the primitive cell (cf. Table 4.1.2.1), each of the centrings C, A, B and I creates eight supplementary centres, whereas the F centring produces supplementary centres, leading to a total of 32 inversion centres.

#### Example

For C centring, add (cf. Table 4.1.2.3) to the eight locations of symmetry centres, given in Table 4.1.2.1, in order to obtain the eight additional symmetry centres ; ; ; ; ; ; ; .

Table 4.1.2.3 contains only representative cases. For 4 and axes, only the standard orientation [001] is given. For diagonal twofold axes, only the orientation [10] is considered. When the locations of all additional symmetry elements of a chosen species are desired, it is sufficient to insert the location of one of the elements into the coordinate triplets of the general position and to remove redundancies.

#### Example

Insert the location of a axis (see Table 4.1.2.2) into the general position of a cubic space group to obtain four distinct locations of axes in P groups and sixteen in F groups.

#### 4.1.2.3. The priority rule

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When more than one kind of symmetry element occurs for a given symmetry direction, the question of choice arises for defining the appropriate Hermann–Mauguin symbol. This choice is made in order of descending priority:

m, e, a, b, c, n, d; and rotation axes before screw axes.

This priority rule is explicitly stated in IT (1952), pages 55 and 543. It is applied to the space-group symbols in IT (1952) and the present edition. There are a few exceptions, however:

 (i) For glide planes in centred monoclinic space groups, the priority rule is purposely not followed in this volume, in order to bring out the relations between the three cell choices' given for each setting (cf. Sections 2.2.16 and 4.3.2 ). (ii) For orthorhombic space groups, the priority rule is applied only to the `standard symbol'. The symbols for the other five settings are obtained from the standard symbol by the appropriate transformations, without invoking the priority rule again (cf. Table 4.3.2.1 ). (iii) Space groups I222 (23) and (24) are two distinct groups. Both contain parallel twofold rotation and screw axes and thus would receive the same symbol according to the priority rule. In I222, the three rotation axes and the three screw axes intersect, whereas in neither the three rotation axes nor the three screw axes intersect (cf. Section 4.3.3 ). (iv) For space group No. 73, the standard symbol Ibca was adopted, instead of Ibaa according to the rule, because Ibca displays the equivalence of the three symmetry directions clearly. (v) The full symbols of space groups Ibca (73) and Imma (74) were written and in IT (1952), in application of the priority rule. In the present edition, these symbols are changed to and , because both space groups contain (and not I222) as subgroup. (vi) In tetragonal space groups with both a and b glide planes parallel to [001], the preference was given to b, as in P4bm (100). (vii) In cubic space groups where tertiary symmetry planes with glide components ; ; and coexist, the tertiary symmetry element was called n in P groups (instead of a, b or c) but c in F groups, because these symmetry elements intersect the origin. (viii) Space groups I23 (197) and (199) are two distinct space groups. For this pair, the same arguments apply as given above for I222 and .

### References

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