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

International Tables for Crystallography (2006). Vol. A, ch. 2.2, pp. 24-25

Section 2.2.7. Origin

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

2.2.7. Origin

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The determination and description of crystal structures and particularly the application of direct methods are greatly facilitated by the choice of a suitable origin and its proper identification. This is even more important if related structures are to be compared or if `chains' of group–subgroup relations are to be constructed. In this volume, as well as in IT (1952)[link], the origin of the unit cell has been chosen according to the following conventions (cf. Chapter 2.1[link] and Section 2.2.2[link]):

  • (i) All centrosymmetric space groups are described with an inversion centre as origin. A further description is given if a centrosymmetric space group contains points of high site symmetry that do not coincide with a centre of symmetry.

    Example: [I4_{1}/amd] (141).

  • (ii) For noncentrosymmetric space groups, the origin is at a point of highest site symmetry, as in [P{\bar 6}m2] (187). If no site symmetry is higher than 1, except for the cases listed below under (iii), the origin is placed on a screw axis, or a glide plane, or at the intersection of several such symmetry elements.

    Examples: [Pca2_{1}] (29); [P6_{1}] (169).

  • (iii) In space group [P2_{1}2_{1}2_{1}] (19), the origin is chosen in such a way that it is surrounded symmetrically by three pairs of [2_{1}] axes. This principle is maintained in the following noncentrosymmetric cubic space groups of classes 23 and 432, which contain [P2_{1}2_{1}2_{1}] as subgroup: [P2_{1}3] (198), [I2_{1}3] (199), [F4_{1}32] (210). It has been extended to other noncentrosymmetric orthorhombic and cubic space groups with [P2_{1}2_{1}2_{1}] as subgroup, even though in these cases points of higher site symmetry are available: [I2_{1}2_{1}2_{1}] (24), [P4_{3}32] (212), [P4_{1}32] (213), [I4_{1}32] (214).

There are several ways of determining the location and site symmetry of the origin. First, the origin can be inspected directly in the space-group diagrams (cf. Section 2.2.6[link]). This method permits visualization of all symmetry elements that intersect the chosen origin.

Another procedure for finding the site symmetry at the origin is to look for a special position that contains the coordinate triplet 0, 0, 0 or that includes it for special values of the parameters, e.g. position 1a: 0, 0, z in space group P4 (75), or position [3a: x,0,{1 \over 3}]; [0,x,{2 \over 3}]; [\bar{x},\bar{x},0] in space group [P3_{1}21] (152). If such a special position occurs, the symmetry at the origin is given by the oriented site-symmetry symbol (see Section 2.2.12[link]) of that special position; if it does not occur, the site symmetry at the origin is 1. For most practical purposes, these two methods are sufficient for the identification of the site symmetry at the origin.

2.2.7.1. Origin statement

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In the line Origin immediately below the diagrams, the site symmetry of the origin is stated, if different from the identity. A further symbol indicates all symmetry elements (including glide planes and screw axes) that pass through the origin, if any. For space groups with two origin choices, for each of the two origins the location relative to the other origin is also given. An example is space group Ccca (68).

In order to keep the notation as simple as possible, no rigid rules have been applied in formulating the origin statements. Their meaning is demonstrated by the examples in Table 2.2.7.1[link], which should be studied together with the appropriate space-group diagrams.

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Examples of origin statements

Example numberSpace group (No.)Origin statementMeaning of last symbol in E4–E11
E1 [P\bar{1}\ (2)] at [\bar{1}]  
E2 [P2/m\ (10)] at centre [(2/m)]  
E3 P222 (16) at 222  
E4 Pcca (54) at [\bar{1}] on 1ca [c \perp [010]], [ a \perp [001]]
E5 Cmcm (63) at centre [(2/m)] at [2/mc2_{1}] [2 \ \| \ [100]], [ m \perp [100]], [ c \perp [010]], [ 2_{1} \ \| \ [001]]
E6 Pcc2 (27) on cc2; short for: on 2 on cc2 [c \perp [100]], [ c \perp [010]], [ 2 \ \| \ [001]]
E7 P4bm (100) on 41g; short for: on 4 on 41g [4 \ \| \ [001]], [ g \perp [1\bar{1}0]] and [g \perp [110]]
E8 [P4_{2}mc\ (105)] on 2mm on [4_{2}mc] [4_{2} \ \| \ [001]], [ m \perp [100]] and [m \perp [010],] [c \perp\! [1\bar{1}0]\; \hbox{and}\;c \perp [110]]
E9 [P4_{3}2_{1}2\ (96)] on 2[110] at [2_{1} 1 (1,2)] [2_{1} \ \| \ [001]], 1 in [[1\bar{1}0]] and [2 \ \| \ [110]]
E10 [P3_{1} 21\ (152)] on 2[110] at [3_{1} (1,1,2)1] [3_{1} \ \| \ [001]], [ 2 \ \| \ [110]]
E11 [P3_{1} 12\ (151)] on 2[210] at [3_{1} 1(1,1,2)] [3_{1} \ \| \ [001]], [ 2 \ \| \ [210]]

These examples illustrate the following points:

  • (i) The site symmetry at the origin corresponds to the point group of the space group (examples E1–E3) or to a subgroup of this point group (E4–E11).

    The presence of a symmetry centre at the origin is always stated explicitly, either by giving the symbol [\bar{1}] (E1 and E4) or by the words `at centre', followed by the full site symmetry between parentheses (E2 and E5). This completes the origin line, if no further glide planes or screw axes are present at the origin.

  • (ii) If glide planes or screw axes are present, as in examples E4–E11, they are given in the order of the symmetry directions listed in Table 2.2.4.1[link]. Such a set of symmetry elements is described here in the form of a `point-group-like' symbol (although it does not describe a group). With the help of the orthorhombic symmetry directions, the symbols in E4–E6 can be interpreted easily. The shortened notation of E6 and E7 is used for space groups of crystal classes mm2, 4mm, [\bar{4}2m], 3m, 6mm and [\bar{6}2m] if the site symmetry at the origin can be easily recognized from the shortened symbol.

  • (iii) For the tetragonal, trigonal and hexagonal space groups, the situation is more complicated than for the orthorhombic groups. The tetragonal space groups have one primary, two secondary and two tertiary symmetry directions. For hexagonal groups, these numbers are one, three and three (Table 2.2.4.1[link]). If the symmetry elements passing through the origin are the same for the two (three) secondary or the two (three) tertiary directions, only one entry is given at the relevant position of the origin statement [example E7: `on 41g' instead of `on 41(g, g)']. An exception occurs for the site-symmetry group 2mm (example E8), which is always written in full rather than as 2m1.

    If the symmetry elements are different, two (three) symbols are placed between parentheses, which stand for the two (three) secondary or tertiary directions. The order of these symbols corresponds to the order of the symmetry directions within the secondary or tertiary set, as listed in Table 2.2.4.1[link]. Directions without symmetry are indicated by the symbol 1. With this rule, the last symbols in the examples E9–E11 can be interpreted.

    Note that for some tetragonal space groups (Nos. 100, 113, 125, 127, 129, 134, 138, 141, 142) the glide-plane symbol g is used in the origin statement. This symbol occurs also in the block Symmetry operations of these space groups; it is explained in Sections 2.2.9[link] and 11.1.2[link] .

  • (iv) To emphasize the orientation of the site-symmetry elements at the origin, examples E9 and E10 start with `on 2[110]' and E11 with `on 2[210]'. In E8, the site-symmetry group is 2mm. Together with the space-group symbol this indicates that 2 is along the primary tetragonal direction, that the two symbols m refer to the two secondary symmetry directions [100] and [010], and that the tertiary set of directions does not contribute to the site symmetry.

    For monoclinic space groups, an indication of the orientation of the symmetry elements is not necessary; hence, the site symmetry at the origin is given by non-oriented symbols. For orthorhombic space groups, the orientation is obvious from the symbol of the space group.

  • (v) The extensive description of the symmetry elements passing through the origin is not retained for the cubic space groups, as this would have led to very complicated notations for some of the groups.

References

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]








































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