International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 2.1, pp. 142-174
doi: 10.1107/97809553602060000926

Chapter 2.1. Guide to the use of the space-group tables

Th. Hahn,a A. Looijenga-Vos,b M. I. Aroyo,c H. D. Flack,d K. Mommae and P. Konstantinovf

aInstitut für Kristallographie, RWTH Aachen University, 52062 Aachen, Germany,bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands,cDepartamento de Física de la Materia Condensada, Universidad del País Vasco (UPV/EHU), Bilbao, Spain,dChimie minérale, analytique et appliquée, University of Geneva, Geneva, Switzerland,eNational Museum of Nature and Science, 4–1-1 Amakubo, Tsukuba, Ibaraki 305–0005, Japan, and fInstitute for Nuclear Research and Nuclear Energy, 72 Tzarigradsko Chaussee, BG-1784 Sofia, Bulgaria

This chapter forms a guide to understanding and using the plane-group and space-group tables in Chapters 2.2 and 2.3. It explains in a systematic fashion, with many examples and figures, all entries and diagrams in the order in which they occur in the plane-group and space-group tables. Particularly detailed treatments are given to Hermann–Mauguin space-group symbols, space-group diagrams, general and special positions, reflections conditions, monoclinic space groups, and the two crystallographic space groups in one dimension (line groups).

In this part of the volume, tables and diagrams of crystallographic data for the 17 types of plane groups (Chapter 2.2[link] ) and the 230 types of space groups (Chapter 2.3[link] ) are presented. With the exception of the data for maximal subgroups and minimal supergroups (which have been transferred to Volume A1 ), the crystallographic data presented in Chapters 2.2 and 2.3 closely follow those in the fifth (2002)[link] edition of Volume A, hereafter IT A (2002)[link]. This chapter is a guide to understanding and using these data.

Only a minimum of theory is provided here, as the emphasis is on the practical use of the data. For the theoretical background to these data, the reader is referred to Parts 1[link] and 3[link] , which also include suitable references. A textbook explaining space-group symmetry and the use of the data in Chapters 2.2 and 2.3 (with exercises) is provided by Hahn & Wondratschek (1994[link]); see also Müller (2013[link]).

Section 2.1.1[link] displays, with the help of an extensive synoptic table, the classification of the 17 plane groups and 230 space groups. This is followed by an explanation of the characterization of the conventional crystallographic coordinate systems, including the symbols for the centring types of lattices and cells. Section 2.1.2[link] lists the alphanumeric and graphical symbols for symmetry elements and symmetry operations used throughout this volume. The lists are accompanied by notes and cross-references to related IUCr nomenclature reports. Section 2.1.3[link] explains in a systematic fashion, with many examples and figures, all the entries and diagrams in the order in which they occur in the plane-group and space-group tables of Chapters 2.2 and 2.3. Detailed treatments are given for the Hermann–Mauguin space-group symbols, the space-group diagrams, the general and special positions, the reflections conditions, monoclinic space groups, and the two crystallographic space groups in one dimension (which are also known as the line groups and are treated in Section 2.1.3.16[link]). Section 2.1.4[link] discusses the computer generation of the space-group tables in this and earlier editions of the volume.

2.1.1. Conventional descriptions of plane and space groups

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Th. Hahna and A. Looijenga-Vosb

2.1.1.1. Classification of space groups

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In this volume, the plane groups and space groups are classified according to three criteria:

  • (i) According to geometric crystal classes, i.e. according to the crystallographic point group to which a particular space group belongs. There are 10 crystal classes in two dimensions and 32 in three dimensions. They are described and listed in Chapter 3.2[link] and in column 4 of Table 2.1.1.1[link]. [For arithmetic crystal classes, see Chapter 1.3[link] and Table 2.1.3.3[link] in this volume, and Chapter 1.4[link] of International Tables for Crystallography, Vol. C (2004)[link].]

    Table 2.1.1.1| top | pdf |
    Crystal families, crystal systems, conventional coordinate systems and Bravais lattices in one, two and three dimensions

    Crystal familySymbolCrystal systemCrystallographic point groupsNo. of space groupsConventional coordinate systemBravais lattices
    Restrictions on cell parametersParameters to be determined
    One dimension
    [Scheme scheme1] 2 None a [{\scr p}]
    Two dimensions
    Oblique m Oblique [Scheme scheme2] 2 None a, b mp
    (monoclinic)         γ§  
    Rectangular o Rectangular [Scheme scheme3] 7 [\gamma = 90^{\circ}] a, b op
    (orthorhombic)           oc
    Square t Square [Scheme scheme4] 3 [a = b] a tp
    (tetragonal)       [\gamma = 90^{\circ}]    
    Hexagonal h Hexagonal [Scheme scheme5] 5 [a = b] a hp
            [\gamma = 120^{\circ}]    
    Three dimensions
    Triclinic a Triclinic [Scheme scheme6] 2 None [a,b,c] aP
    (anorthic)         [\alpha, \beta, \gamma]  
    Monoclinic m Monoclinic [Scheme scheme7] 13 b-unique setting [a,b,c] mP
            [\alpha = \gamma = 90^{\circ}] β § mS (mC, mA, mI)
            c-unique setting [a, b, c] mP
            [\alpha = \beta = 90^{\circ}] γ § mS (mA, mB, mI)
    Orthorhombic o Orthorhombic [Scheme scheme8] 59 [\alpha = \beta = \gamma = 90^{\circ}] a, b, c oP
    oS (oC, oA, oB)
    oI
    oF
    Tetragonal t Tetragonal [Scheme scheme9] 68 [a = b] a, c tP
            [\alpha = \beta = \gamma = 90^{\circ}]   tI
    Hexagonal h Trigonal [Scheme scheme10] 18 [a = b] a, c hP
            [\alpha = \beta = 90^{\circ},\ \gamma = 120^{\circ}]    
          7 [a = b = c] [\alpha = \beta = \gamma] (rhombohedral axes, primitive cell) [a = b] [\alpha = \beta = 90^{\circ}, \gamma = 120^{\circ}] (hexagonal axes, triple obverse cell) a, α hR
        Hexagonal [Scheme scheme11] 27 [a = b] a, c hP
            [\alpha = \beta = 90^{\circ}, \gamma = 120^{\circ}]    
    Cubic c Cubic [Scheme scheme12] 36 [a = b = c] a cP
            [\alpha = \beta = \gamma = 90^{\circ}]   cI
                cF
    The symbols for crystal families (column 2) and Bravais lattices (column 8) were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985)[link].
    Symbols surrounded by dashed or full lines indicate Laue groups; full lines indicate Laue groups which are also lattice point symmetries (holohedries).
    §These angles are conventionally taken to be non-acute, i.e. [\geq 90^{\circ}].
    For the use of the letter S as a new general, setting-independent `centring symbol' for monoclinic and orthorhombic Bravais lattices, see de Wolff et al. (1985)[link].
  • (ii) According to crystal families. The term crystal family designates the classification of the 17 plane groups into four categories and of the 230 space groups into six categories, as displayed in column 1 of Table 2.1.1.1[link]. Here all `hexagonal', `trigonal' and `rhombohedral' space groups are contained in one family, the hexagonal crystal family. The `crystal family' thus corresponds to the term `crystal system', as used frequently in the American and Russian literature.

    The crystal families are symbolized by the lower-case letters a, m, o, t, h, c, as listed in column 2 of Table 2.1.1.1[link]. If these letters are combined with the appropriate capital letters for the lattice-centring types (cf. Table 2.1.1.2[link]), symbols for the 14 Bravais lattices result. These symbols and their occurrence in the crystal families are shown in column 8 of Table 2.1.1.1[link]; mS and oS are the standard setting-independent symbols for the centred monoclinic and the one-face-centred orthorhombic Bravais lattices, cf. de Wolff et al. (1985)[link]; symbols between parentheses represent alternative settings of these Bravais lattices.

    Table 2.1.1.2| top | pdf |
    Symbols for the conventional centring types of one-, two- and three-dimensional cells

    SymbolCentring type of cellNumber of lattice points per cellCoordinates of lattice points within cell
    One dimension
    [{\scr p}] Primitive 1 0
    Two dimensions
    p Primitive 1 0, 0
    c Centred 2 0, 0; [{1 \over 2}], [{1 \over 2}]
    h Hexagonally centred 3 0, 0; [{2 \over 3}], [{1 \over 3}]; [{1 \over 3}], [{2 \over 3}]
    Three dimensions
    P Primitive 1 0, 0, 0
    C C-face centred 2 0, 0, 0; [{1 \over 2}], [{1 \over 2}], 0
    A A-face centred 2 0, 0, 0; 0, [{1 \over 2}], [{1 \over 2}]
    B B-face centred 2 0, 0, 0; [{1 \over 2}], 0, [{1 \over 2}]
    I Body centred 2 0, 0, 0; [{1 \over 2}], [{1 \over 2}], [{1 \over 2}]
    F All-face centred 4 0, 0, 0; [{1 \over 2}], [{1 \over 2}], 0; 0, [{1 \over 2}], [{1 \over 2}]; [{1 \over 2}], 0, [{1 \over 2}]
    R [\cases{\hbox{Rhombohedrally centred}\cr \hbox{(description with `hexagonal axes')}\cr \hbox{Primitive}\cr \hbox{(description with `rhombohedral axes')}\cr}] 3 [\!\openup1pt{\cases {0,{\hbox to 1pt{}} 0,{\hbox to 1pt{}} 0{\hbox{; }}{\hbox to 2pt{}}{2 \over 3},{\hbox to -1.5pt{}} {1 \over 3},{\hbox to -1.5pt{}} {1 \over 3}{\hbox{; }}{\hbox to 1pt{}}{1 \over 3},{\hbox to -1pt{}} {2 \over 3},{\hbox to -1pt{}} {2 \over 3} \hbox{ (`obverse setting')}\cr 0,{\hbox to 1pt{}} 0,{\hbox to 1pt{}} 0{\hbox{; }}{\hbox to 1.5pt{}}{1 \over 3},{\hbox to -1pt{}} {2 \over 3},{\hbox to -1.5pt{}} {1 \over 3}{\hbox{; }}{\hbox to 1pt{}}{2 \over 3}, {1 \over 3},{\hbox to -1.5pt{}} {2 \over 3} \hbox{ (`reverse setting')}\cr}}]
    1 0, 0, 0
    H§ Hexagonally centred 3 0, 0, 0; [{2 \over 3}], [{1 \over 3}], 0; [{1 \over 3}], [{2 \over 3}], 0
    The two-dimensional triple hexagonal cell h is an alternative description of the hexagonal plane net, as illustrated in Fig. 1.5.1.8[link] . It is not used for systematic plane-group description in this volume; it is introduced, however, in the sub- and supergroup entries of the plane-group tables of International Tables for Crystallography, Vol. A1 (2010)[link], abbreviated as IT A1. Plane-group symbols for the h cell are listed in Section 1.5.4[link] . Transformation matrices are contained in Table 1.5.1.1[link] .
    In the space-group tables (Chapter 2.3[link] ), as well as in IT (1935)[link] and IT (1952)[link], the seven rhombohedral R space groups are presented with two descriptions, one based on hexagonal axes (triple cell), one on rhombohedral axes (primitive cell). In the present volume, as well as in IT (1952)[link] and IT A (2002)[link], the obverse setting of the triple hexagonal cell R is used. Note that in IT (1935)[link] the reverse setting was employed. The two settings are related by a rotation of the hexagonal cell with respect to the rhombohedral lattice around a threefold axis, involving a rotation angle of 60, 180 or 300° (cf. Fig. 1.5.1.6[link] ). Further details may be found in Section 1.5.4[link] and Chapter 3.1[link] . Transformation matrices are contained in Table 1.5.1.1[link] .
    §The triple hexagonal cell H is an alternative description of the hexagonal Bravais lattice, as illustrated in Fig. 1.5.1.8[link] . It was used for systematic space-group description in IT (1935)[link], but replaced by P in IT (1952)[link]. It is used in the tables of maximal subgroups and minimal supergroups of the space groups in IT A1 (2010)[link]. Space-group symbols for the H cell are listed in Section 1.5.4[link] . Transformation matrices are contained in Table 1.5.1.1[link] .
  • (iii) According to crystal systems. This classification collects the plane groups into four categories and the space groups into seven categories. The classifications according to crystal families and crystal systems are the same for two dimensions.

    For three dimensions, this applies to the triclinic, monoclinic, orthorhombic, tetragonal and cubic systems. The only complication exists in the hexagonal crystal family, for which several subdivisions into systems have been proposed in the literature. In this volume [as well as in International Tables for X-ray Crystallography (1952)[link], hereafter IT (1952), and the subsequent editions of IT], the space groups of the hexagonal crystal family are grouped into two `crystal systems' as follows: all space groups belonging to the five crystal classes 3, [\bar{3}], 32, 3m and [\bar{3}m], i.e. having 3, [3_{1}], [3_{2}] or [\bar{3}] as principal axis, form the trigonal crystal system, irrespective of whether the Bravais lattice is hP or hR; all space groups belonging to the seven crystal classes 6, [\bar{6}, 6/m], 622, 6mm, [\bar{6}]2m and [6/mmm], i.e. having 6, [6_{1}], [6_{2}], [6_{3}], [6_{4}], [6_{5}] or [\bar{6}] as principal axis, form the hexagonal crystal system; here the lattice is always hP (cf. Chapter 1.3[link] ). The crystal systems, as defined above, are listed in column 3 of Table 2.1.1.1[link].

A different subdivision of the hexagonal crystal family is in use, mainly in the French literature. It consists of grouping all space groups based on the hexagonal Bravais lattice hP (lattice point symmetry [6/mmm]) into the `hexagonal' system and all space groups based on the rhombohedral Bravais lattice hR (lattice point symmetry [\bar{3}m]) into the `rhombohedral' system. In Chapter 1.3[link] , these systems are called `lattice systems'. They were called `Bravais systems' in earlier editions of this volume.

The theoretical background for the classification of space groups is provided in Chapter 1.3[link] .

2.1.1.2. Conventional coordinate systems and cells

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A plane group or space group usually is described by means of a crystallographic coordinate system, consisting of a crystallographic basis (basis vectors are lattice vectors) and a crystallographic origin (origin at a centre of symmetry or at a point of high site symmetry). The choice of such a coordinate system is not mandatory, since in principle a crystal structure can be referred to any coordinate system; cf. Chapters 1.3[link] and 1.5[link] .

The selection of a crystallographic coordinate system is not unique. Conventionally, a right-handed set of basis vectors is taken such that the symmetry of the plane or space group is displayed best. With this convention, which is followed in the present volume, the specific restrictions imposed on the cell parameters by each crystal family become particularly simple. They are listed in columns 6 and 7 of Table 2.1.1.1[link]. If within these restrictions the smallest cell is chosen, a conventional (crystallographic) basis results. Together with the selection of an appropriate conventional (crystallographic) origin (cf. Sections 2.1.3.2[link] and 2.1.3.7[link]), such a basis defines a conventional (crystallographic) coordinate system and a conventional cell. The conventional cell of a point lattice or a space group, obtained in this way, turns out to be either primitive or to exhibit one of the centring types listed in Table 2.1.1.2[link]. The centring type of a conventional cell is transferred to the lattice which is described by this cell; hence, we speak of primitive, face-centred, body-centred etc. lattices. Similarly, the cell parameters are often called lattice parameters; cf. Chapters 1.3[link] and 3.1[link] for further details.

In the triclinic, monoclinic and orthorhombic crystal systems, additional conventions (for instance cell reduction or metrical conventions based on the lengths of the cell edges) are needed to determine the choice and the labelling of the axes. Reduced bases are treated in Chapter 3.1[link] , orthorhombic settings in Section 2.1.3.6[link], and monoclinic settings and cell choices in Section 2.1.3.15[link] (cf. Section 1.5.4[link] for a detailed treatment of alternative settings of space groups).

In this volume, all space groups within a crystal family are referred to the same kind of conventional coordinate system, with the exception of the hexagonal crystal family in three dimensions. Here, two kinds of coordinate systems are used, the hexagonal and the rhombohedral systems. In accordance with common crystallographic practice, all space groups based on the hexagonal Bravais lattice hP (18 trigonal and 27 hexagonal space groups) are described only with a hexagonal coordinate system (primitive cell), whereas the seven space groups based on the rhombohedral Bravais lattice hR (the so-called `rhombohedral space groups', cf. Section 1.4.1[link] ) are treated in two versions, one referred to `hexagonal axes' (triple obverse cell) and one to `rhombohedral axes' (primitive cell); cf. Table 2.1.1.2[link]. In practice, hexagonal axes are preferred because they are easier to visualize.

Table 2.1.1.2[link] contains only those conventional centring symbols which occur in the Hermann–Mauguin space-group symbols. There exist, of course, further kinds of centred cells which are unconventional, see for example the synoptic tables of plane (Table 1.5.4.3[link] ) and space (Table 1.5.4.4[link] ) groups discussed in Chapter 1.5[link] . The centring type of a cell may change with a change of the basis vectors; in particular, a primitive cell may become a centred cell and vice versa. Examples of relevant transformation matrices are contained in Table 1.5.1.1[link] .

2.1.2. Symbols of symmetry elements

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Th. Hahna and M. I. Aroyoc

As already introduced in Section 1.2.3[link] , a `symmetry element' (of a given structure or object) is defined as a concept with two components; it is the combination of a `geometric element' (that allows the fixed points of a reduced symmetry operation to be located and oriented in space) with the set of symmetry operations having this geometric element in common (`element set'). The element set of a symmetry element is represented by the so-called `defining operation', which is the simplest symmetry operation from the element set that suffices to identify the geometric element. The alphanumeric and graphical symbols of symmetry elements and the related symmetry operations used throughout the tables of plane (Chapter 2.2[link] ) and space groups (Chapter 2.3[link] ) are listed in Tables 2.1.2.1[link] to 2.1.2.7[link][link][link][link][link][link]. For detailed discussion of the definition and symbols of symmetry elements, cf. Section 1.2.3[link] , de Wolff et al. (1989[link], 1992[link]) and Flack et al. (2000[link]).

Table 2.1.2.1| top | pdf |
Symbols for symmetry elements and for the corresponding symmetry operations in one, two and three dimensions

SymbolSymmetry element and its orientationDefining symmetry operation with glide or screw vector
m [\Bigg\{] Reflection plane, mirror plane Reflection through the plane
Reflection line, mirror line (two dimensions) Reflection through the line
Reflection point, mirror point (one dimension) Reflection through the point
a, b or c `Axial' glide plane Glide reflection through the plane, with glide vector
a [\perp [010]] or [\perp [001]] [{1 \over 2}{\bf a}]
b [\perp [001]] or [\perp [100]] [{1 \over 2}{\bf b}]
c [\left\{\vphantom{\matrix{a\cr b\cr c\cr d\cr}}\right.] [\perp [100]] or [\perp [010]] [{1 \over 2}{\bf c}]
[\perp [1\bar{1}0]] or [\perp [110]] [{1 \over 2}{\bf c}]
[\!\openup1pt\matrix{\perp [100] \hbox{ or} \perp [010] \hbox{ or} \perp [\bar{1}\bar{1}0]\cr \perp [1\bar{1}0] \hbox{ or} \perp [120] \hbox{ or} \perp [\bar{2}\bar{1}0]\cr}] [\!\left.\openup1pt\matrix{{1 \over 2}{\bf c}\cr {1 \over 2}{\bf c}\cr}\right\} \hbox{ hexagonal coordinate system}]
e `Double' glide plane (in centred cells only) Two glide reflections through one plane, with perpendicular glide vectors
[\perp [001]] [{1 \over 2}{\bf a}] and [{1 \over 2}{\bf b}]
[\perp [100]] [{1 \over 2}{\bf b}] and [{1 \over 2}{\bf c}]
[\perp [010]] [{1 \over 2}{\bf a}] and [{1 \over 2}{\bf c}]
[\perp [1\bar{1}0]]; [\perp [110]] [{1 \over 2}({\bf a} + {\bf b})] and [{1 \over 2}{\bf c}]; [{1 \over 2}({\bf a} - {\bf b})] and [{1 \over 2}{\bf c}]
[\perp [01\bar{1}]]; [\perp [011]] [{1 \over 2}({\bf b} + {\bf c})] and [{1 \over 2}{\bf a}]; [{1 \over 2}({\bf b} - {\bf c})] and [{1 \over 2}{\bf a}]
[\perp [\bar{1}01]]; [\perp [101]] [{1 \over 2}({\bf a} + {\bf c})] and [{1 \over 2}{\bf b}]; [{1 \over 2}({\bf a} - {\bf c})] and [{1 \over 2}{\bf b}]
n `Diagonal' glide plane Glide reflection through the plane, with glide vector
[\perp [001]]; [\perp [100]]; [\perp [010]] [{1 \over 2}({\bf a} + {\bf b})]; [{1 \over 2}({\bf b} + {\bf c})]; [{1 \over 2}({\bf a} + {\bf c})]
[\perp [1\bar{1}0]] or [\perp [01\bar{1}]] or [\perp [\bar{1}01]] [{1 \over 2}({\bf a} + {\bf b} + {\bf c})]
[\perp [110]]; [\perp [011]]; [\perp [101]] [{1 \over 2}(- {\bf a} + {\bf b} + {\bf c})]; [{1 \over 2}({\bf a} - {\bf b} + {\bf c})]; [{1 \over 2}({\bf a} + {\bf b} - {\bf c})]
d § `Diamond' glide plane Glide reflection through the plane, with glide vector
[\perp [001]]; [\perp [100]]; [\perp [010]] [{1 \over 4}({\bf a} \pm {\bf b})]; [{1 \over 4}({\bf b} \pm {\bf c})]; [{1 \over 4}(\pm {\bf a} + {\bf c})]
[\perp [1\bar{1}0]]; [\perp [01\bar{1}]]; [\perp [\bar{1}01]] [{1 \over 4}({\bf a} + {\bf b} \pm {\bf c})]; [{1 \over 4}(\pm {\bf a} + {\bf b} + {\bf c})]; [{1 \over 4}({\bf a} \pm {\bf b} + {\bf c})]
[\perp [110]]; [\perp [011]]; [\perp [101]] [{1 \over 4}(- {\bf a} + {\bf b} \pm {\bf c})]; [{1 \over 4}(\pm {\bf a} - {\bf b} + {\bf c})]; [{1 \over 4}({\bf a} \pm {\bf b} - {\bf c})]
g Glide line (two dimensions) Glide reflection through the line, with glide vector
[\perp [01]]; [\perp [10]] [{1 \over 2}{\bf a}]; [{1 \over 2}{\bf b}]
1 None Identity
2, 3, 4, 6 [\left\{\vphantom{\matrix{a\cr b\cr c\cr d\cr}}\right.] n-fold rotation axis, n Counter-clockwise rotation of [360/n] degrees around the axis
n-fold rotation point, n (two dimensions) Counter-clockwise rotation of [360/n] degrees around the point
[\bar{1}] Centre of symmetry, inversion centre Inversion through the point
[\bar{2} = m], [\bar{3},\bar{4},\bar{6}] Rotoinversion axis, [\bar{n}], and inversion point on the axis †† Counter-clockwise rotation of [360/n] degrees around the axis, followed by inversion through the point on the axis ††
[2_{1}] n-fold screw axis, [n_{p}] Right-handed screw rotation of [360/n] degrees around the axis, with screw vector (pitch) ([p/n]) t; here t is the shortest lattice translation vector parallel to the axis in the direction of the screw
[3_{1}, 3_{2}]
[4_{1}, 4_{2}, 4_{3}]
[6_{1}, 6_{2}, 6_{3}, 6_{4}, 6_{5}]
In the rhombohedral space-group symbols [R3c] (161) and [R\bar{3}c] (167), the symbol c refers to the description with `hexagonal axes'; i.e. the glide vector is [{1 \over 2}{\bf c}], along [001]. In the description with `rhombohedral axes', this glide vector is [{1 \over 2}({\bf a} + {\bf b} + {\bf c})], along [111], i.e. the symbol of the glide plane would be n: cf. Table 1.5.4.4[link] .
Glide planes `e' occur in orthorhombic A-, C- and F-centred space groups, tetragonal I-centred and cubic F- and I-centred space groups. The geometric element of an e-glide plane is a plane shared by glide reflections with perpendicular glide vectors, with at least one glide vector along a crystal axis [cf. Section 1.2.3[link] and de Wolff et al. (1992)].
§Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance [{1 \over 4}({\bf a} + {\bf b})] and [{1 \over 4}({\bf a} - {\bf b})]. The second power of a glide reflection d is a centring vector.
Only the symbol m is used in the Hermann–Mauguin symbols, for both point groups and space groups.
††The inversion point is a centre of symmetry if n is odd.

Table 2.1.2.2| top | pdf |
Graphical symbols of symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions)

DescriptionGraphical symbolGlide vector(s) of the defining operation(s) of the glide plane (in units of the shortest lattice translation vectors parallel and normal to the projection plane)Symmetry element represented by the graphical symbol
[\left.\openup3pt\matrix{\hbox{Reflection plane, mirror plane}\hfill\cr \hbox{Reflection line, mirror line (two dimensions)}\cr}\right\}] [Scheme scheme14] None m
[\left.\openup3pt\matrix{\hbox{`Axial' glide plane}\hfill\cr\hbox{Glide line (two dimensions)}\cr}\right\}] [Scheme scheme15] [\!\openup2pt\matrix{{1 \over 2} \hbox{parallel to line in projection plane}\cr {1 \over 2} \hbox{parallel to line in figure plane}\hfill\cr}] [\!\matrix{a,\ b \hbox{ or } c\cr g\hfill\cr}]
`Axial' glide plane [Scheme scheme16] [{1 \over 2}] normal to projection plane a, b or c
`Double' glide plane [Scheme scheme17] [\!\openup2pt\matrix{Two\hbox{ glide vectors:}\hfill\cr{1 \over 2}\hbox{ parallel to line in, and}\hfill\cr{1 \over 2} \hbox{ normal to projection plane}\hfill}] e
`Diagonal' glide plane [Scheme scheme18] [\!\openup2pt\matrix{One\hbox{ glide vector with }two\hbox{ components:}\hfill\cr{1 \over 2}\hbox{ parallel to line in, and}\hfill\cr{1 \over 2}\hbox{ normal to projection plane}\hfill}] n
`Diamond' glide plane (pair of planes) [Scheme scheme19] [{1 \over 4}] parallel to line in projection plane, combined with [{1 \over 4}] normal to projection plane (arrow indicates direction parallel to the projection plane for which the normal component is positive) d
The graphical symbols of the `e'-glide planes are applied to the diagrams of seven orthorhombic A-, C- and F-centred space groups, five tetragonal I-centred space groups, and five cubic F- and I-centred space groups.
Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance [{1 \over 4}({\bf a} + {\bf b})] and [{1 \over 4}({\bf a} - {\bf b})]. The second power of a glide reflection d is a centring vector.

Table 2.1.2.3| top | pdf |
Graphical symbols of symmetry planes parallel to the plane of projection

DescriptionGraphical symbolGlide vector(s) of the defining operation(s) of the glide plane (in units of the shortest lattice translation vectors parallel to the projection plane)Symmetry element represented by the graphical symbol
Reflection plane, mirror plane [Scheme scheme20] None m
`Axial' glide plane [Scheme scheme21] [{1 \over 2}] in the direction of the arrow a, b or c
`Double' glide plane [Scheme scheme22] [\!\matrix{Two\hbox{ glide vectors:}\hfill\cr{1 \over 2}\hbox{ in either of the directions of the two arrows}}] e
`Diagonal' glide plane [Scheme scheme23] [\!\matrix{One\hbox{ glide vector with }two\hbox{ components}\cr{1 \over 2}\hbox{ in the direction of the arrow}\hfill}] n
`Diamond' glide plane§ (pair of planes) [Scheme scheme24] [{1 \over 2}] in the direction of the arrow; the glide vector is always half of a centring vector, i.e. one quarter of a diagonal of the conventional face-centred cell d
The symbols are given at the upper left corner of the space-group diagrams. A fraction h attached to a symbol indicates two symmetry planes with `heights' h and [h + {1 \over 2}] above the plane of projection; e.g. [{1 \over 8}] stands for [h = {1 \over 8}] and [{5 \over 8}]. No fraction means [h = 0] and [{1 \over 2}] (cf. Section 2.1.3.6[link]).
The graphical symbols of the `e'-glide planes are applied to the diagrams of seven orthorhombic A-, C- and F-centred space groups, five tetragonal I-centred space groups, and five cubic F- and I-centred space groups.
§Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance [{1 \over 4}({\bf a} + {\bf b})] and [{1 \over 4}({\bf a} - {\bf b})]. The second power of a glide reflection d is a centring vector.

Table 2.1.2.4| top | pdf |
Graphical symbols of symmetry planes inclined to the plane of projection (in cubic space groups of classes [\overline{4}{3m}] and [m\overline{3}m] only)

DescriptionGraphical symbol for planes normal toGlide vector(s) (in units of the shortest lattice translation vectors) of the defining operation(s) of the glide plane normal toSymmetry element represented by the graphical symbol
[011] and [[01\bar{1}]][101] and [[10\bar{1}]][011] and [[01\bar{1}]][101] and [[10\bar{1}]]
Reflection plane, mirror plane [Scheme scheme25] [Scheme scheme31] None None m
`Axial' glide plane [Scheme scheme26] [Scheme scheme32] [{1 \over 2}] along [100] [\left.\!\matrix{{1 \over 2}\hbox{ along }[010]\hfill\cr\noalign{\vskip 38pt}\cr{1 \over 2}\hbox{ along }[10\bar{1}]\cr\hbox{ or along }[101]\hfill\cr}\right\}] a or b
`Axial' glide plane [Scheme scheme27] [Scheme scheme33] [{1 \over 2}] along [[01\bar{1}]] or along [011]
`Double' glide plane [in space groups [I\bar{4}3m] (217) and [Im\bar{3}m] (229) only] [Scheme scheme28] [Scheme scheme34] Two glide vectors: [{1 \over 2}] along [100] and [{1 \over 2}] along [[01\bar{1}]] or [{1 \over 2}] along [011] Two glide vectors: [{1 \over 2}] along [010] and [{1 \over 2}] along [[10\bar{1}]] or [{1 \over 2}] along [101] e
`Diagonal' glide plane [Scheme scheme29] [Scheme scheme35] One glide vector: [{1 \over 2}] along [[11\bar{1}]] or along [111] One glide vector: [{1 \over 2}] along [[11\bar{1}]] or along [111] n
`Diamond' glide plane§ (pair of planes) [Scheme scheme30] [Scheme scheme36] [{1 \over 2}] along [[1\bar{1}1]] or along [111] [\left.\matrix{{1 \over 2}\hbox{ along }[\bar{1}11]\hbox { or}\cr \hbox{along }[111]\cr\noalign{\vskip 40pt} {1 \over 2}\hbox{ along }[\bar{1}\bar{1}1]\hbox{ or}\cr \hbox{ along }[1\bar{1}1]}\right\}] d
[{1 \over 2}] along [[\bar{1}\bar{1}1]] or along [[\bar{1}11]]
The symbols represent orthographic projections. In the cubic space-group diagrams, complete orthographic projections of the symmetry elements around high-symmetry points, such as [0,0,0]; [{1 \over 2},0,0]; [{1 \over 4},{1 \over 4},0], are given as `inserts'.
In the space groups [F\bar{4}3m] (216), [Fm\bar{3}m] (225) and [Fd\bar{3}m] (227), the shortest lattice translation vectors in the glide directions are [{\bf t}(1, {1 \over 2}, \bar{{1 \over 2}}\,)] or [{\bf t}(1, {1 \over 2}, {1 \over 2}\,)] and [{\bf t}(\,{1 \over 2}, 1, \bar{{1 \over 2}}\,)] or [{\bf t}(\,{1 \over 2}, 1, {1 \over 2}\,)], respectively.
§Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance [{1 \over 4}({\bf a} + {\bf b})] and [{1 \over 4}({\bf a} - {\bf b})]. The second power of a glide reflection d is a centring vector.
The glide vector is half of a centring vector, i.e. one quarter of the diagonal of the conventional body-centred cell in space groups [I\bar{4}3d] (220) and [Ia\bar{3}d] (230).

Table 2.1.2.5| top | pdf |
Graphical symbols of symmetry axes normal to the plane of projection and symmetry points in the plane of the figure

DescriptionAlphanumeric symbolGraphical symbolScrew vector of the defining operation of the screw axis (in units of the shortest lattice translation vector parallel to the axis)Symmetry elements represented by the graphical symbol
[\!\left.\matrix{\hbox{Twofold rotation axis}\hfill\cr \hbox{Twofold rotation point (two dimensions)}\cr}\right\}] 2 [Scheme scheme37] None 2
Twofold screw axis: `2 sub 1' 21 [Scheme scheme38] [{1 \over 2}] [2_{1}]
[\!\left.\matrix{\hbox{Threefold rotation axis}\hfill\cr \hbox{Threefold rotation point (two dimensions)}\cr}\right\}] 3 [Scheme scheme39] None 3
Threefold screw axis: `3 sub 1' 31 [Scheme scheme40] [{1 \over 3}] [3_{1}]
Threefold screw axis: `3 sub 2' 32 [Scheme scheme41] [{2 \over 3}] [3_{2}]
[\!\left.\openup3pt\matrix{\hbox{Fourfold rotation axis}\hfill\cr \hbox{Fourfold rotation point (two dimensions)}\cr}\right\}] 4 [Scheme scheme42] None 4
Fourfold screw axis: `4 sub 1' 41 [Scheme scheme43] [{1 \over 4}] [4_{1} ]
Fourfold screw axis: `4 sub 2' 42 [Scheme scheme44] [{1 \over 2}] [4_{2}]
Fourfold screw axis: `4 sub 3' 43 [Scheme scheme45] [{3 \over 4}] [4_{3} ]
[\!\left.\openup3pt\matrix{\hbox{Sixfold rotation axis}\hfill\cr \hbox{Sixfold rotation point (two dimensions)}\cr}\right\}] 6 [Scheme scheme46] None 6
Sixfold screw axis: `6 sub 1' 61 [Scheme scheme47] [{1 \over 6}] [6_{1}]
Sixfold screw axis: `6 sub 2' 62 [Scheme scheme48] [{1 \over 3}] [6_{2}]
Sixfold screw axis: `6 sub 3' 63 [Scheme scheme49] [{1 \over 2}] [6_{3}]
Sixfold screw axis: `6 sub 4' 64 [Scheme scheme50] [{2 \over 3}] [6_{4}]
Sixfold screw axis: `6 sub 5' 65 [Scheme scheme51] [{5 \over 6}] [6_{5}]
[\!\left.\openup3pt\matrix{\hbox{Centre of symmetry, inversion centre: `1 bar'}\hfill\cr\hbox{Reflection point, mirror point (one dimension)}\cr}\right\}] [\bar 1] [Scheme scheme52] None [\bar{1}]
Inversion axis: `3 bar' [\bar 3] [Scheme scheme53] None [\bar{3}, \bar{1}, 3]
Inversion axis: `4 bar' [\bar 4] [Scheme scheme54] None [\bar{4}, 2]
Inversion axis: `6 bar' [\bar 6] [Scheme scheme55] None [\bar{6}, 3]
Twofold rotation axis with centre of symmetry [2/m] [Scheme scheme56] None [2, \bar{1}]
Twofold screw axis with centre of symmetry [2_1/m] [Scheme scheme57] [{1 \over 2}] [2_{1}, \bar{1}]
Fourfold rotation axis with centre of symmetry [4/m] [Scheme scheme58] None [4, \bar{4}, \bar{1}]
`4 sub 2' screw axis with centre of symmetry [4_2/m] [Scheme scheme59] [{1 \over 2}] [4_{2}, \bar{4},\bar{1}]
Sixfold rotation axis with centre of symmetry [6/m] [Scheme scheme60] None [6,\bar{6},\bar{3},\bar{1}]
`6 sub 3' screw axis with centre of symmetry [6_3/m] [Scheme scheme61] [{1 \over 2}] [6_{3}, \bar{6},\bar{3},\bar{1}]

Notes on the `heights' h of symmetry points [\bar{1}], [\bar{3}], [\bar{4}] and [\bar{6}]:

(1) Centres of symmetry [\bar{1}] and [\bar{3}], as well as inversion points [\bar{4}] and [\bar{6}] on [\bar{4}] and [\bar{6}] axes parallel to [001], occur in pairs at `heights' h and [h + {1 \over 2}]. In the space-group diagrams, only one fraction h is given, e.g. [{1 \over 4}] stands for [h = {1 \over 4}] and [{3 \over 4}]. No fraction means [h = 0] and [{1 \over 2}]. In cubic space groups, however, because of their complexity, both fractions are given for vertical [\bar{4}] axes, including [h = 0] and [{1 \over 2}].

(2) Symmetries [4/m] and [6/m] contain vertical [\bar{4}] and [\bar{6}] axes; their [\bar{4}] and [\bar{6}] inversion points coincide with the centres of symmetry. This is not indicated in the space-group diagrams.

(3) Symmetries [4_{2}/m] and [6_{3}/m] also contain vertical [\bar{4}] and [\bar{6}] axes, but their [\bar{4}] and [\bar{6}] inversion points alternate with the centres of symmetry; i.e. [\bar{1}] points at h and [h + {1 \over 2}] interleave with [\bar{4}] or [\bar{6}] points at [h + {1 \over 4}] and [h + {3 \over 4}]. In the tetragonal and hexagonal space-group diagrams, only one fraction for [\bar{1}] and one for [\bar{4}] or [\bar{6}] is given. In the cubic diagrams, all four fractions are listed for [4_{2}/m]; e.g. [Pm\bar{3}n] (223): [\bar{1}]: [0, {1 \over 2}]; [\bar{4}]: [{1 \over 4}, {3 \over 4}].


Table 2.1.2.6| top | pdf |
Graphical symbols of symmetry axes parallel to the plane of projection

DescriptionGraphical symbolScrew vector of the defining operation of the screw axis (in units of the shortest lattice translation vector parallel to the axis)Symmetry elements represented by the graphical symbol
Twofold rotation axis [Scheme scheme62] None 2
Twofold screw axis: `2 sub 1' [Scheme scheme63] [{1 \over 2}] [2_{1}]
Fourfold rotation axis [Scheme scheme64] [Scheme scheme70] None 4
Fourfold screw axis: `4 sub 1' [Scheme scheme65] [{1 \over 4}] [4_{1} ]
Fourfold screw axis: `4 sub 2' [Scheme scheme66] [{1 \over 2}] [4_{2} ]
Fourfold screw axis: `4 sub 3' [Scheme scheme67] [{3 \over 4}] [4_{3} ]
Inversion axis: `4 bar' [Scheme scheme68] None [\bar{4}, 2]
Inversion point on `4 bar' axis [Scheme scheme69] None None
The symbols for horizontal symmetry axes are given outside the unit cell of the space-group diagrams. Twofold axes always occur in pairs, at `heights' h and [h + {1 \over 2}] above the plane of projection; here, a fraction h attached to such a symbol indicates two axes with heights h and [h + {1 \over 2}]. No fraction stands for [h = 0] and [{1 \over 2}]. The rule of pairwise occurrence, however, is not valid for the horizontal fourfold axes in cubic space groups; here, all heights are given, including [h = 0] and [{1 \over 2}]. This applies also to the horizontal [\bar{4}] axes and the [\bar{4}] inversion points located on these axes.

Table 2.1.2.7| top | pdf |
Graphical symbols of symmetry axes inclined to the plane of projection (in cubic space groups only)

DescriptionGraphical symbolScrew vector of the defining operation of the screw axis (in units of the shortest lattice translation vector parallel to the axis)Symmetry elements represented by the graphical symbol
Twofold rotation axis [Scheme scheme71] [Scheme scheme77] None 2
Twofold screw axis: `2 sub 1' [Scheme scheme72] [{1 \over 2}] [2_{1}]
Threefold rotation axis [Scheme scheme73] [Scheme scheme78] None 3
Threefold screw axis: `3 sub 1' [Scheme scheme74] [{1 \over 3}] [3_{1}]
Threefold screw axis: `3 sub 2' [Scheme scheme75] [{2 \over 3}] [3_{2}]
Inversion axis: `3 bar' [Scheme scheme76] None [\bar{3}, 3,\bar{1}]
The dots mark the intersection points of axes with the plane at [h = 0]. In some cases, the intersection points are obscured by symbols of symmetry elements with height [h \geq 0]; examples: [Fd\bar{3}] (203), origin choice 2; [Pn\bar{3}n] (222), origin choice 2; [Pm\bar{3}n] (223); [Im\bar{3}m] (229); [Ia\bar{3}d] (230).

The alphanumeric symbols shown in Table 2.1.2.1[link] correspond to those symmetry elements and symmetry operations which occur in the conventional Hermann–Mauguin symbols of point groups and space groups. Further so-called `additional symmetry elements' are described in Sections 1.4.2.3[link] and 1.5.4.1[link] , and Tables 1.5.4.3[link] and 1.5.4.4[link] show additional symmetry operations that appear in the so-called `extended Hermann–Mauguin symbols' (cf. Section 1.5.4[link] ). The symbols of symmetry elements (symmetry operations), except for glide planes (glide reflections), are independent of the choice and the labelling of the basis vectors and of the origin. The symbols of glide planes (glide reflections), however, may change with a change of the basis vectors. For this reason, the possible orientations of glide planes and the glide vectors of the corresponding operations are listed explicitly in columns 2 and 3 of Table 2.1.2.1[link].

In 1992, following a proposal of the Commission on Crystallographic Nomenclature (de Wolff et al., 1992[link]), the International Union of Crystallography introduced the symbol `e' and graphical symbols for the designation of the so-called `double' glide planes. The double- or e-glide plane occurs only in centred cells and its geometric element is a plane shared by glide reflections with perpendicular glide vectors related by a centring translation (for details on e-glide planes, cf. Section 1.2.3[link] ). The introduction of the symbol e for the designation of double-glide planes (cf. de Wolff et al., 1992[link]) results in the modification of the Hermann–Mauguin symbols of five ortho­rhombic groups:

Space group No. 39 41 64 67 68
New symbol: Aem2 Aea2 Cmce Cmme Ccce
Former symbol: Abm2 Aba2 Cmca Cmma Ccca

Since the introduction of its use in IT A (2002)[link] the new symbol is the standard one; it is indicated in the headline of these space groups, while the former symbol is given underneath.

The graphical symbols of symmetry planes are shown in Tables 2.1.2.2[link] to 2.1.2.4[link][link]. Like the alphanumeric symbols, the graphical symbols and their explanations (columns 2 and 3) are independent of the projection direction and the labelling of the basis vectors. They are, therefore, applicable to any projection diagram of a space group. The alphanumeric symbols of glide planes (column 4), however, may change with a change of the basis vectors. For example, the dash-dotted n glide in the hexagonal description becomes an a, b or c glide in the rhombohedral description. In monoclinic space groups, the `parallel' vector of a glide plane may be along a lattice translation vector that is inclined to the projection plane.

The `e'-glide graphical symbols are applied to the diagrams of seven orthorhombic A-, C- and F- centred space groups, five tetragonal I-centred space groups, and five cubic F- and I-centred space groups. The `double-dotted-dash' symbol for e glides `normal' and `inclined' to the plane of projection was introduced in 1992 (de Wolff et al., 1992[link]), while the `double-arrowed' graphical symbol for e-glide planes oriented `parallel' to the projection plane had already been used in IT (1935[link]) and IT (1952[link]).

The graphical symbols of symmetry axes and their descriptions are shown in Tables 2.1.2.5[link]–2.1.2.7[link][link]. The screw vectors of the defining operations of screw axes are given in units of the shortest lattice translation vectors parallel to the axes. The symbols in the last column of the tables indicate the symmetry elements that are represented by the graphical symbols in the symmetry-element diagrams of the space groups. Two main cases may be distinguished:

  • (i) graphical symbols of symmetry elements that in the space-group diagrams represent just one symmetry element. Thus, the graphical symbol of a fourfold rotation axis or an inversion centre represent the symmetry element 4 or [\bar{1}]. Similarly, the graphical symbols of symmetry planes (Tables 2.1.2.2[link]–2.1.2.4[link][link]) represent just one symmetry element (namely, mirror or glide plane) in the space-group diagrams;

  • (ii) graphical symbols of symmetry elements that in the space-group diagrams represent more than one symmetry element. For example, the graphical symbol described in Table 2.1.2.5[link] as `Inversion axis: 3 bar' [(\bar 3)], [Scheme scheme153] represents in the diagrams the three different symmetry elements [\bar 3], 3, [\bar 1].

The last six entries of Table 2.1.2.5[link] are combinations of symbols of symmetry axes with that of a centre of inversion. When displayed on the space-group diagrams, the combined graphical symbols represent more than one symmetry element. For example, the symbol for a fourfold rotation axis with a centre of inversion (4/m), [Scheme scheme158] represents the symmetry elements [\bar 4], 4 and [\bar 1].

The meaning of a graphical symbol on the space-group dia­grams is often confused with the set of symmetry elements that constitute the site-symmetry group associated with the symmetry element displayed. As an example, consider the rotoinversion axis [\bar 6] (described as `Inversion axis: 6 bar' in Table 2.1.2.5[link]). The site-symmetry group [\bar 6] can be decomposed into three symmetry elements: [\bar 6], 3 and m (cf. de Wolff et al., 1989[link]). However, the graphical symbol of [\bar 6] in the diagrams represents the two symmetry elements [\bar 6] and 3, as the symmetry element `m' (that `belongs' to [\bar 6]) is represented by a separate graphical symbol.

2.1.3. Contents and arrangement of the tables

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Th. Hahna and A. Looijenga-Vosb

2.1.3.1. General layout

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The presentation of the plane-group and space-group data in Chapters 2.2[link] and 2.3[link] follows the style of the previous editions of International Tables. The data for most of the space groups are displayed on one page or on two facing pages. A typical distribution of the data is shown below and is illustrated by the example of Cccm (66) provided inside the front and back covers.

Left-hand page:
(1) Headline
(2) Diagrams for the symmetry elements and the general position (for graphical symbols of symmetry elements see Section 2.1.2[link])
(3) Origin
(4) Asymmetric unit
(5) Symmetry operations
   
Right-hand page:
(6) Headline in abbreviated form
(7) Generators selected; this information is the basis for the order of the entries under Symmetry operations and Positions
(8) General and special Positions, with the following columns:
  Multiplicity
  Wyckoff letter
  Site symmetry, given by the oriented site-symmetry symbol
  Coordinates
  Reflection conditions
  Note: In a few space groups, two special positions with the same reflection conditions are printed on the same line
(9) Symmetry of special projections (not given for plane groups)

It is important to note that the symmetry data are displayed in the same sequence for all the space groups. The actual distribution of the data between pages can vary depending on the amount and nature of the data that are shown.

The symmetry data for the ten space groups of the crystal class [m\bar{3}m] [[Pm\bar{3}m] (221) to [Ia\bar{3}d] (230)] are displayed on four pages. Additional general-position diagrams in tilted projection are shown on the fourth page, providing a three-dimensional-style view of these complicated general-position diagrams.

2.1.3.2. Space groups with more than one description

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For several space groups, more than one description is available. Three cases occur:

  • (i) Two choices of origin (cf. Section 2.1.3.7[link])

    For all centrosymmetric space groups, the tables contain a description with a centre of symmetry as origin. Some centrosymmetric space groups, however, contain points of high site symmetry that do not coincide with a centre of symmetry. For these 24 cases, a further description (including diagrams) with a high-symmetry point as origin is provided. Neither of the two origin choices is considered standard. Noncentrosymmetric space groups and all plane groups are described with only one choice of origin.

    Examples

    • (1) Pnnn (48)

      Origin choice 1 at a point with site symmetry 222

      Origin choice 2 at a centre with site symmetry [\bar{1}].

    • (2) [Fd\bar{3}m] (227)

      Origin choice 1 at a point with site symmetry [\bar{4}3m]

      Origin choice 2 at a centre with site symmetry [\bar{3}m].

  • (ii) Monoclinic space groups

    Two complete descriptions are given for each of the 13 monoclinic space groups, one for the setting with `unique axis b', followed by one for the setting with `unique axis c'.

    Additional descriptions in synoptic form are provided for the following eight monoclinic space groups with centred lattices or glide planes:[\eqalign{&C2\ (5), Pc\ (7), Cm\ (8), Cc\ (9), C2/m\ (12), P2/c\ (13),\hfill\cr &P2_{1}/c\ (14), C2/c\ (15)\hfill}]These synoptic descriptions consist of abbreviated treatments for three `cell choices', here called `cell choices 1, 2 and 3'. Cell choice 1 corresponds to the complete treatment, mentioned above; for comparative purposes, it is repeated among the synoptic descriptions which, for each setting, are printed on two facing pages. The cell choices and their relations are explained in Section 2.1.3.15[link].

  • (iii) Rhombohedral space groups

    The seven rhombohedral space groups R3 (146), [R\bar{3}] (148), R32 (155), R3m (160), R3c (161), [R\bar{3}m] (166) and [R\bar{3}c] (167) are described with two coordinate systems, first with hexagonal axes (triple hexagonal cell) and second with rhombohedral axes (primitive rhombohedral cell). The same space-group symbol is used for both descriptions. For convenience, the relations between the cell parameters a, c of the triple hexagonal cell and the cell parameters a′ and α′ of the primitive rhombohedral cell (cf. Table 2.1.1.1[link]) are listed:[\eqalign{a &= a' \sqrt{2} \sqrt{1 - \cos \alpha'} = 2a' \sin {\alpha' \over 2}\cr c &= a' \sqrt{3} \sqrt{1 + 2 \cos \alpha'}\phantom{\Big(_{2_{2\over2}}}\cr {c \over a} &= \sqrt{{3 \over 2}} \sqrt{{1 + 2 \cos \alpha' \over 1 - \cos \alpha'}} = \sqrt{{9 \over 4 \sin^{2} (\alpha'/2)} - 3}\phantom{\Big(_{2_{2\over2}}}\cr a' &= {\textstyle{1 \over 3}} \sqrt{3a^{2} + c^{2}}\phantom{\Big(_{2\over2}^{2\over2}}\cr \sin {\alpha' \over 2} &= {3 \over 2\sqrt{3 + (c^{2}/a^{2})}} \hbox{ or } \cos \alpha' = {(c^{2}/a^{2}) - {3 \over 2} \over (c^{2}/a^{2}) + 3}.\phantom{\bigg)^{2\over2}}}]

    The hexagonal triple cell is given in the obverse setting (centring points [{2 \over 3}, {1 \over 3}, {1 \over 3};\; {1 \over 3}, {2 \over 3}, {2 \over 3}\,]). In IT (1935)[link], the reverse setting (centring points [{1 \over 3}, {2 \over 3}, {1 \over 3}; \;{2 \over 3}, {1 \over 3}, {2 \over 3}\,]) was employed; cf. Table 2.1.1.2[link].

Coordinate transformations between different space-group descriptions are treated in detail in Section 1.5.3[link] .

2.1.3.3. Headline

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The description of each plane group or space group starts with a headline consisting of two (sometimes three) lines which contain the following information, when read from left to right.

First line

  • (1) The short international (Hermann–Mauguin) symbol for the plane or space group. These symbols will be further referred to as Hermann–Mauguin symbols. A detailed discussion of space-group symbols is given in Section 1.4.1[link] and Chapter 3.3[link] ; for convenience, a summary is given in Section 2.1.3.4[link].

    Note on standard monoclinic space-group symbols: In order to facilitate recognition of a monoclinic space-group type, the familiar short symbol for the b-axis setting (e.g. [P\,2_{1}/c] for No. 14 or [C\,2/c] for No. 15) has been adopted as the standard symbol for a space-group type. It appears in the headline of every description of this space group and thus does not carry any information about the setting or the cell choice of this particular description. No other short symbols for monoclinic space groups are used in this volume (cf. Section 2.1.3.15[link]).

  • (2) The Schoenflies symbol for the space group (cf. Section 1.4.1[link] ).

    Note: No Schoenflies symbols exist for the plane groups.

  • (3) The short international (Hermann–Mauguin) symbol for the point group to which the plane or space group belongs (cf. Section 1.4.1[link] and Chapter 3.3[link] ).

  • (4) The name of the crystal system (cf. Table 2.1.1.1[link]).

Second line

  • (5) The sequential number of the plane or space group, as introduced in IT (1952)[link].

  • (6) The full international (Hermann–Mauguin) symbol for the plane or space group.

    For monoclinic space groups, the headline of every description contains the full symbol appropriate to that description.

  • (7) The Patterson symmetry (see Section 2.1.3.5[link]).

Third line

This line is used, where appropriate, to indicate origin choices, settings, cell choices and coordinate axes (see Section 2.1.3.2[link]). For five orthorhombic space groups, an entry `Former space-group symbol' is given; cf. Section 2.1.2[link].

2.1.3.4. International (Hermann–Mauguin) symbols for plane groups and space groups

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(For more details, cf. Section 1.4.1[link] and Chapter 3.3[link] .)

Current symbols. Both the short and the full Hermann–Mauguin symbols consist of two parts: (i) a letter indicating the centring type of the conventional cell, and (ii) a set of characters indicating symmetry elements of the space group (modified point-group symbol).

  • (i) The letters for the centring types of cells are listed in Table 2.1.1.2[link]. Lower-case letters are used for two dimensions (nets), capital letters for three dimensions (lattices).

  • (ii) The one, two or three entries after the centring letter refer to the one, two or three kinds of symmetry directions of the lattice belonging to the space group. These symmetry directions were called Blickrichtungen by Heesch (1929)[link]. Symmetry directions occur either as singular directions (as in the monoclinic and orthorhombic crystal systems) or as sets of symmetry-equivalent symmetry directions (as in the higher-symmetry crystal systems). Only one representative of each set is required. The (sets of) symmetry directions and their sequence for the different lattices are summarized in Table 2.1.3.1[link]. According to their position in this sequence, the symmetry directions are referred to as `primary', `secondary' and `tertiary' directions.

    Table 2.1.3.1| top | pdf |
    Lattice symmetry directions for two and three dimensions

    Directions that belong to the same set of equivalent symmetry directions are collected between braces. The first entry in each set is taken as the representative of that set.

    LatticeSymmetry direction (position in Hermann–Mauguin symbol)
    PrimarySecondaryTertiary
    Two dimensions
    Oblique [\matrix{\hbox{Rotation}\hfill\cr \hbox{point}\hfill\cr\hbox{in plane}\hfill\cr}]    
    Rectangular [10] [01]
    Square [\left\{\matrix{[10]\cr [01]\cr}\right\}] [\left\{\matrix{[1\bar{1}]\cr [11]\cr}\right\}]
    Hexagonal [\left\{\matrix{[10]\cr [01]\cr [\bar{1}\bar{1}]\cr}\right\}] [\left\{\matrix{[1\bar{1}]\cr [12]\cr [\bar{2}\bar{1}]\cr}\right\}]
    Three dimensions
    Triclinic None    
    Monoclinic [010] (`unique axis b')  
      [001] (`unique axis c')  
    Orthorhombic [100] [010] [001]
    Tetragonal [001] [\left\{\matrix{[100]\cr [010]\cr}\right\}] [\left\{\matrix{[1\bar{1}0]\cr [110]\cr}\right\}]
    Hexagonal [001] [\left\{\matrix{[100]\cr [010]\cr [\bar{1}\bar{1}0]\cr}\right\}] [\left\{\matrix{[1\bar{1}0]\cr [120]\cr [\bar{2}\bar{1}0]\cr}\right\}]
    Rhombohedral (hexagonal axes) [001] [\left\{\matrix{[100]\cr [010]\cr [\bar{1}\bar{1}0]\cr}\right\}]  
    Rhombohedral (rhombohedral axes) [111] [\left\{\matrix{[1\bar{1}0]\cr [01\bar{1}]\cr [\bar{1}01]\cr}\right\}]  
    Cubic [\left\{\matrix{[100]\cr [010]\cr [001]\cr}\right\}] [\left\{\matrix{[111]\cr [1\bar{1}\bar{1}]\cr [\bar{1}1\bar{1}]\cr [\bar{1}\bar{1}1]\cr}\right\}] [\left\{\matrix{[1\bar{1}0]\ [110]\cr [01\bar{1}]\ [011]\cr [\bar{1}01]\ [101]\cr}\right\}]
    For the full Hermann–Mauguin symbols see Sections 2.1.3.4[link] and 1.4.1[link] .

    This sequence of lattice symmetry directions is transferred to the sequence of positions in the corresponding Hermann–Mauguin space-group symbols. Each position contains one or two characters designating symmetry elements (axes and planes) of the space group (cf. Section 2.1.2[link]) that occur for the corresponding lattice symmetry direction. Symmetry planes are represented by their normals; if a symmetry axis and a normal to a symmetry plane are parallel, the two characters (symmetry symbols) are separated by a slash, as in [P6_{3}/m] or [P2/m] (`two over m').

Short and full Hermann–Mauguin symbols differ only for the plane groups of class m, for the monoclinic space groups, and for the space groups of crystal classes mmm, [4/mmm], [\bar{3}m], [6/mmm], [m\bar{3}] and [m\bar{3}m]. In the full symbols, symmetry axes and symmetry planes for each symmetry direction are listed; in the short symbols, symmetry axes are suppressed as much as possible. Thus, for space group No. 62, the full symbol is [P2_{1}/n\;2_{1}/m\;2_{1}/a] and the short symbol is Pnma. For No. 194, the full symbol is [P6_{3}/m\;2/m\; 2/c] and the short symbol is [P6_{3}/mmc]. For No. 230, the full symbol is [I4_{1}/a\;\bar{3}\;2/d] and the short symbol is [Ia\bar{3}d].

Many space groups contain more kinds of symmetry elements than are indicated in the full symbol (`additional symmetry operations and elements', cf. Sections 1.4.2.4[link] and 1.5.4.1[link] ). A listing of additional symmetry operations is given in Tables 1.5.4.3[link] and 1.5.4.4[link] under the heading Extended full symbols. Note that a centre of symmetry is never explicitly indicated (except for space group [P\bar{1}]); its presence or absence, however, can be readily inferred from the space-group symbol.

Changes in Hermann–Mauguin space-group symbols as compared with the 1952 and 1935 editions of International Tables. Extensive changes in the space-group symbols were applied in IT (1952)[link] as compared with the original Hermann–Mauguin symbols of IT (1935)[link], especially in the tetragonal, trigonal and hexagonal crystal systems. Moreover, new symbols for the c-axis setting of monoclinic space groups were introduced. All these changes are recorded on pp. 51 and 543–544 of IT (1952)[link]. In the present edition, the symbols of the 1952 edition are retained, except for the following four cases (cf. Section 3.3.4[link] ).

  • (i) Two-dimensional groups. Short Hermann–Mauguin symbols differing from the corresponding full symbols in IT (1952) are replaced by the full symbols for the plane groups listed in Table 2.1.3.2[link].

    Table 2.1.3.2| top | pdf |
    Changes in Hermann–Mauguin symbols for two-dimensional groups

    No.IT (1952)Present edition
    6 pmm p2mm
    7 pmg p2mg
    8 pgg p2gg
    9 cmm c2mm
    11 p4m p4mm
    12 p4g p4gm
    17 p6m p6mm

    For the two-dimensional point group with two mutually perpendicular mirror lines, the symbol mm is changed to 2mm.

    For plane group No. 2, the entries `1' at the end of the full symbol are omitted: No. 2: Change from p211 to p2.

    With these changes, the symbols of the two-dimensional groups follow the rules that were introduced in IT (1952)[link] for the space groups.

  • (ii) Monoclinic space groups. Additional full Hermann–Mauguin symbols are introduced for the eight monoclinic space groups with centred lattices or glide planes (Nos. 5, 7–9, 12–15) to indicate the various settings and cell choices. A complete list of symbols, including also the a-axis setting, is contained in Table 1.5.4.4[link] ; further details are given in Section 2.1.3.15[link].

    For standard short monoclinic space-group symbols see Sections 2.1.3.3[link] and 2.1.3.15[link].

  • (iii) Cubic space groups. The short symbols for all space groups belonging to the two cubic crystal classes [m\bar{3}] and [m\bar{3}m] now contain the symbol [\bar{3}] instead of 3. This applies to space groups Nos. 200–206 and 221–230, as well as to the two point groups [m\bar{3}] and [m\bar{3}m].

    Examples

    No. 205: Change from Pa3 to [Pa\bar{3}]

    No. 230: Change from Ia3d to [Ia\bar{3}d].

    With this change, the centrosymmetric nature of these groups is apparent also in the short symbols.

  • (iv) Glide-plane symbol e. For the introduction of the `double glide plane' e into five space-group symbols, see Section 2.1.2[link].

2.1.3.5. Patterson symmetry

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H. D. Flackd

The entry Patterson symmetry in the headline gives the symmetry of the `vector set' generated by the operation of the space group on an arbitrary set of general positions. More prosaically, it may be described as the symmetry of the set of the interatomic vectors of a crystal structure with the selected space group. The Patterson symmetry is a crystallographic space group denoted by its Hermann–Mauguin symbol. It is in fact one of the 24 centrosymmetric symmorphic space groups (see Section 1.3.3.3[link] ) in three dimensions and one of 7 in two dimensions. For each of the 230 space groups, the Patterson symmetry has the same Bravais-lattice type as the space group itself and its point group is the lowest-index centrosymmetric supergroup of the point group of the space group. The `point-group part' of the symbol of the Patterson symmetry represents the Laue class to which the plane group or space group belongs (cf. Table 2.1.2.1[link]). By way of examples: space group No. 100, P4bm, has a Bravais lattice of type tP and point group 4mm. The centrosymmetric supergroup of 4mm (see Fig. 3.2.1.3[link] ) is 4/mmm, so the Patterson symmetry is P4/mmm; space group No. 66, Cccm, has a Bravais lattice of type oC and point group mmm. This point group is centrosymmetric, so the Patterson symmetry is Cmmm.

Note: For the four space groups Amm2 (38), Aem2 (39), Ama2 (40) and Aea2 (41), the standard symbol for their Patterson symmetry, Cmmm, is added (between parentheses) after the actual symbol Ammm in the space-group tables.

The Patterson symmetry is intimately related to the symmetry of the Patterson function (see Flack, 2015[link]). The latter, P|F|2(uvw), is the inverse Fourier transform of the squared structure-factor amplitudes. Patterson functions possess the crystallographic symmetry of the symmorphic space-group representative of the arithmetic crystal class (see Section 1.3.4.4.1[link] ) to which the space group belongs. Table 2.1.3.3[link] lists these crystallographic symmetries of the Patterson function and the Patterson symmetries for the space groups and plane groups. However, further symmetry is also present, as desribed below for the three common forms of the Patterson function:

  • (a) P|F|2(uvw): The most general form of the Patterson function, the complex P|F|2(uvw), is the complex Fourier transform of |F(hkl)|2. The full symmetry of P|F|2(uvw) can be described in terms of the 1651 two-colour (Shubnikov) space groups (Fischer & Knof, 1987[link]; Wilson, 1993[link]; Shubnikov & Belov, 1964[link]; cf. also Chapter 3.6[link] ). The real and imaginary parts of P|F|2(uvw) have the same symmetry as PA(uvw) and PD(uvw), respectively, described below. P|F|2(uvw) is real for centrosymmetric space groups and noncentrosymmetric ones in the absence of any resonant-scattering contribution.

    Table 2.1.3.3| top | pdf |
    Patterson symmetries and symmetries of Patterson functions for space groups and plane groups

    The space-group types of each row form an arithmetic crystal class. (In three instances the row is typeset on two lines.) The arithmetic crystal class is identified by its representative symmorphic space group for which both the Hermann–Mauguin symbol and the space-group-type number are shown in bold. A set of space groups with sequential numbers is indicated by the symbols of the first and last space group of the sequence separated by a dash.

    The column `Patterson symmetry' indicates the symmetry of the set of interatomic vectors of crystal structures described in the space groups given in the column `Space-group types'. The Patterson symmetry is also the symmetry of PA(uvw) and the real part of the complex P|F|2(uvw). The Patterson symmetry is given in the headline of each space-group table in Chapter 2.3.

    The crystallographic symmetry of both PD(uvw) and the imaginary part of the complex P|F|2(uvw) is that of the symmorphic space group of crystal structures described in the space groups given in the column `Space-group types'. To this crystallographic symmetry, the noncrystallographic operation of a centre of antisymmetry needs to be added to give [P_D(u v w) = -P_D(\bar u\bar v\bar w)]. The full symmetry of PD(uvw) is not shown in this volume.

    The setting and origin choice of the chosen space group should also be used for the space group of the Patterson symmetry and the symmorphic space group.

    Similar remarks apply to the plane groups listed in part (b) of the table.

    (a) Space groups.

    Space-group typesPatterson symmetry
    Hermann–Mauguin symbolsNos.
    Crystal family triclinic (anorthic), Bravais-lattice type aP
    P1 1 [P\bar 1]
    [{\bi P}\bar{\bf 1}] 2 [P\bar 1]
    Crystal family monoclinic, Bravais-lattice type mP
    P2P21 3–4 P2/m
    PmPc 6–7 P2/m
    P2/mP21/m, 10–11 P2/m
     P2/cP21/c 13–14 P2/m
    Crystal family monoclinic, Bravais-lattice type mS
    C2 5 C2/m
    CmCc 8–9 C2/m
    C2/m, C2/c 12, 15 C2/m
    Crystal family orthorhombic, Bravais-lattice type oP
    P222P212121 16–19 Pmmm
    Pmm2Pnn2 25–34 Pmmm
    PmmmPnma 47–62 Pmmm
    Crystal family orthorhombic, Bravais-lattice type oS
    C2221, C222 20, 21 Cmmm
    Cmm2Ccc2 35–37 Cmmm
    Amm2Aea2 38–41 Ammm
    CmcmCmce, Cmmm, 63–64, 65 Cmmm
     CccmCcce 66–68 Cmmm
    Crystal family orthorhombic, Bravais-lattice type oF
    F222 22 Fmmm
    Fmm2Fdd2 42–43 Fmmm
    FmmmFddd 69–70 Fmmm
    Crystal family orthorhombic, Bravais-lattice type oI
    I222I212121 23–24 Immm
    Imm2Ima2 44–46 Immm
    ImmmImma 71–74 Immm
    Crystal family tetragonal, Bravais-lattice type tP
    P4P43 75–78 P4/m
    [{\bi P}\bar{\bf 4}] 81 P4/m
    P4/mP42/n 83–86 P4/m
    P422P43212 89–96 P4/mmm
    P4mmP42bc 99–106 P4/mmm
    [{\bi P}\bar{\bf 4}{\bf 2}{\bi m}][P\bar 42_1c] 111–114 P4/mmm
    [{\bi P}\bar{\bf 4}{\bi m}{\bf 2}][P\bar 4n2] 115–118 P4/mmm
    P4/mmmP42/ncm 123–138 P4/mmm
    Crystal family tetragonal, Bravais-lattice type tI
    I4, I41 79–80 I4/m
    [{\bi I}\bar{\bf 4}] 82 I4/m
    I4/mI41/a 87–88 I4/m
    I422I4122 97–98 I4/mmm
    I4mmI41cd 107–110 I4/mmm
    [{\bi I}\bar{\bf 4}{\bi m}{\bf 2}][I\bar 4c2] 119–120 I4/mmm
    [{\bi I}\bar{\bf 4}{\bf 2}{\bi m}][I\bar 42d] 121–122 I4/mmm
    I4/mmmI41/acd 139–142 I4/mmm
    Crystal family hexagonal, Bravais-lattice type hP
    P3P32 143–145 [P\bar 3]
    [{\bi P}\bar{\bf 3}] 147 [P\bar 3]
    P312, P3112, P3212 149, 151, 153 [P\bar 31m]
    P321, P3121, P3221 150, 152, 154 [P\bar 3m1]
    P3m1, P3c1 156, 158 [P\bar 3m1]
    P31m, P31c 157, 159 [P\bar 31m]
    [{\bi P}\bar{\bf 3}{\bf 1}{\bi m}][P\bar 31c] 162–163 [P\bar 31m]
    [{\bi P}\bar{\bf 3}{\bi m}{\bf 1}][P\bar 3c1] 164–165 [P\bar 3m1]
    P6P63 168–173 P6/m
    [{\bi P}\bar{\bf 6}] 174 P6/m
    P6/mP63/m 175–176 P6/m
    P622P6322 177–182 P6/mmm
    P6mmP63mc 183–186 P6/mmm
    [{\bi P}\bar{\bf 6}{\bi m}{\bf 2}][P\bar 6c2] 187–188 P6/mmm
    [{\bi P}\bar{\bf 6}{\bf 2}{\bi m}][P\bar 62c] 189–190 P6/mmm
    P6/mmmP63/mmc 191–194 P6/mmm
    Crystal family hexagonal, Bravais-lattice type hR
    R3 146 [R\bar 3]
    [{\bi R}\bar{\bf 3}] 148 [R\bar 3]
    R32 155 [R\bar 3m]
    R3mR3c 160–161 [R\bar 3m]
    [{\bi R}\bar{\bf 3}{\bi m}][R\bar 3c] 166–167 [R\bar 3m]
    Crystal family cubic, Bravais-lattice type cP
    P23, P213 195, 198 [Pm\bar 3]
    [{\bi Pm}\bar{\bf 3}][Pn\bar 3], [Pa\bar 3] 200–201, 205 [Pm\bar 3]
    P432P4232, 207–208, [Pm\bar 3m]
     P4332–P4132 212–213 [Pm\bar 3m]
    [{\bi P}\bar{\bf 4}{\bf 3}{\bi m}], [P\bar 43n] 215, 218 [Pm\bar 3m]
    [{\bi Pm}\bar{\bf 3}{\bi m}][Pn\bar 3 m] 221–224 [Pm\bar 3m]
    Crystal family cubic, Bravais-lattice type cF
    F23 196 [Fm\bar 3]
    [{\bi Fm}\bar{\bf 3}][Fd\bar 3] 202–203 [Fm\bar 3]
    F432F4132 209–210 [Fm\bar 3m]
    [{\bi F}\bar{\bf 4}{\bf 3}{\bi m}][F\bar 43c] 216, 219 [Fm\bar 3m]
    [{\bi Fm}\bar{\bf 3}{\bi m}][Fd\bar 3c] 225–228 [Fm\bar 3m]
    Crystal family cubic, Bravais-lattice type cI
    I23, I213 197, 199 [Im\bar 3]
    [{\bi Im}\bar{\bf 3}], [Ia\bar 3] 204, 206 [Im\bar 3]
    I432, I4132 211, 214 [Im\bar 3m]
    [{\bi I}\bar{\bf 4}{\bf 3}{\bi m}], [I\bar 43d] 217, 220 [Im\bar 3m]
    [{\bi Im}\bar{\bf 3}{\bi m}][Ia\bar 3d] 229–230 [Im\bar 3m]

    (b) Plane groups.

    Plane-group typesPatterson symmetry
    Hermann–Mauguin symbolsNos.
    Crystal family oblique (monoclinic), Bravais-lattice type mp
    p1 1 p2
    p2 2 p2
    Crystal family rectangular (orthorhombic), Bravais-lattice type op
    pmpg 3–4 p2mm
    p2mmp2gg 6–8 p2mm
    Crystal family rectangular (orthorhombic), Bravais-lattice type oc
    cm 5 c2mm
    c2mm 9 c2mm
    Crystal family square (tetragonal), Bravais-lattice type tp
    p4 10 p4
    p4mmp4gm 11–12 p4mm
    Crystal family hexagonal, Bravais-lattice type hp
    p3 13 p6
    p3m1 14 p6mm
    p31m 15 p6mm
    p6 16 p6
    p6mm 17 p6mm
  • (b) PA(uvw): [A(hkl) = {\textstyle{1\over 2}}[|F(hkl)|^2 + |F(\bar h\bar k\bar l)|^2]], the average of the squared structure-factor amplitudes of the pair of Friedel opposites hkl and [\bar h \bar k \bar l]. Real PA(uvw) is the real cosine Fourier transform of A(hkl). The symmetry of PA(uvw) is generated by the symmorphic space-group representative of the arithmetic crystal class to which the space group belongs, combined with a centre of symmetry (inversion centre), i.e. [P_A(uvw)=P_A(\bar u\bar v\bar w)]. PA(uvw) thus possesses the Patterson symmetry of the space group whether the latter is centrosymmetric or not, and whether there is any resonant-scattering contribution or not (see Table 2.1.3.3[link]).

  • (c) PD(uvw): [D(hkl) = |F(hkl)|^2 - |F(\bar h \bar k \bar l)|^2], the difference of the squared structure-factor amplitudes of the pair of Friedel opposites hkl and [\bar h \bar k \bar l]. Real PD(uvw) is the real sine Fourier transform of D(hkl). The crystallographic symmetry of PD(uvw) is that of the symmorphic space-group representative of the arithmetic crystal class to which the space group belongs (see Table 2.1.3.3[link]). The full symmetry of PD(uvw) is given by the type-III black-and-white space group generated by the appropriate symmorphic space group and the centre of antisymmetry, [P_D(uvw) = -P_D(\bar u \bar v \bar w)] (Fischer & Knof, 1987[link]). Thus PD(uvw) does not possess the Patterson symmetry.

2.1.3.6. Space-group diagrams

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The space-group diagrams serve two purposes: (i) to show the relative locations and orientations of the symmetry elements and (ii) to illustrate the arrangement of a set of symmetry-equivalent points of the general position.

With the exception of general-position diagrams in perspective projection for some space groups (cf. Section 2.1.3.6.8[link]), all of the diagrams are orthogonal projections, i.e. the projection direction is perpendicular to the plane of the figure. Apart from the descriptions of the rhombohedral space groups with `rhombohedral axes' (cf. Section 2.1.3.6.6[link]), the projection direction is always a cell axis. If other axes are not parallel to the plane of the figure, they are indicated by the subscript p, as [a_{p},\ b_{p}] or [c_{p}] in the case of one or two axes for monoclinic and triclinic space groups, respectively (cf. Figs. 2.1.3.1[link][link] to 2.1.3.3[link]), or by the subscript rh for the three rhombohedral axes in Fig. 2.1.3.9[link].

The graphical symbols for symmetry elements, as used in the drawings, are displayed in Tables 2.1.2.2[link] to 2.1.2.7[link].

In the diagrams, `heights' h above the projection plane are indicated for symmetry planes and symmetry axes parallel to the projection plane, as well as for centres of symmetry. The heights are given as fractions of the shortest lattice translation normal to the projection plane and, if different from 0, are printed next to the graphical symbols. Each symmetry element at height h is accompanied by another symmetry element of the same type at height [h + {1 \over 2}] (this does not apply to the horizontal fourfold axes in the diagrams for the cubic space groups). In the space-group diagrams, only the symmetry element at height h is indicated (cf. Section 2.1.2[link]).

Schematic representations of the diagrams, displaying the origin, the labels of the axes, and the projection direction [uvw], are given in Figs. 2.1.3.1[link][link][link][link][link][link][link][link] to 2.1.3.10[link] (except Fig. 2.1.3.6). The general-position diagrams are indicated by the letter [\hbox{\sf G}].

2.1.3.6.1. Plane groups

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Each description of a plane group contains two diagrams, one for the symmetry elements (left) and one for the general position (right). The two axes are labelled a and b, with a pointing downwards and b running from left to right.

2.1.3.6.2. Triclinic space groups

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For each of the two triclinic space groups, three elevations (along a, b and c) are given, in addition to the general-position diagram [{\sf G}] (projected along c) at the lower right of the set, as illustrated in Fig. 2.1.3.1[link].

[Figure 2.1.3.1]

Figure 2.1.3.1 | top | pdf |

Triclinic space groups ([\hbox{\sf G}] = general-position diagram).

The diagrams represent a reduced cell of type II for which the three interaxial angles are non-acute, i.e. [\alpha,\beta,\gamma \geq\! 90^{\circ}]. For a cell of type I, all angles are acute, i.e. [\alpha,\beta,\gamma \,\lt\,90^{\circ}]. For a discussion of the two types of reduced cells, see Section 3.1.3[link] .

2.1.3.6.3. Monoclinic space groups (cf. Sections 2.1.3.2[link] and 2.1.3.15[link])

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The `complete treatment' of each of the two settings contains four diagrams (Figs. 2.1.3.2[link] and 2.1.3.3[link]). Three of them are projections of the symmetry elements, taken along the unique axis (upper left) and along the other two axes (lower left and upper right). For the general position, only the projection along the unique axis is given (lower right).

[Figure 2.1.3.2]

Figure 2.1.3.2 | top | pdf |

Monoclinic space groups, setting with unique axis b ([\hbox{\sf G}] = general-position diagram).

[Figure 2.1.3.3]

Figure 2.1.3.3 | top | pdf |

Monoclinic space groups, setting with unique axis c ([\hbox{\sf G}] = general-position diagram).

The `synoptic descriptions' of the three cell choices (for each setting) are headed by a pair of diagrams, as illustrated in Fig. 2.1.3.4[link]. The drawings on the left display the symmetry elements and the ones on the right the general position (labelled [{\sf G}]). Each diagram is a projection of four neighbouring unit cells along the unique axis. It contains the outlines of the three cell choices drawn as heavy lines. For the labelling of the axes, see Fig. 2.1.3.4[link]. The headline of the description of each cell choice contains a small-scale drawing, indicating the basis vectors and the cell that apply to that description.

[Figure 2.1.3.4]

Figure 2.1.3.4 | top | pdf |

Monoclinic space groups, cell choices 1, 2, 3. Upper pair of diagrams: setting with unique axis b. Lower pair of diagrams: setting with unique axis c. The numbers 1, 2, 3 within the cells and the subscripts of the labels of the axes indicate the cell choice (cf. Section 2.1.3.15[link]). The unique axis points upwards from the page. [\hbox{\sf G}] = general-position diagram.

2.1.3.6.4. Orthorhombic space groups and orthorhombic settings

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The space-group tables contain a set of four diagrams for each orthorhombic space group. The set consists of three projections of the symmetry elements [along the c axis (upper left), the a axis (lower left) and the b axis (upper right)] in addition to the general-position diagram, which is given only in the projection along c (lower right). The projected axes, the origins and the projection directions of these diagrams are illustrated in Fig. 2.1.3.5[link]. They refer to the so-called `standard setting' of the space group, i.e. the setting described in the space-group tables and indicated by the `standard Hermann–Mauguin symbol' in the headline.

[Figure 2.1.3.5]

Figure 2.1.3.5 | top | pdf |

Orthorhombic space groups. Diagrams for the `standard setting' as described in the space-group tables ([\hbox{\sf G}] = general-position diagram).

For each orthorhombic space group, six settings exist, i.e. six different ways of assigning the labels a, b, c to the three ortho­rhombic symmetry directions; thus the shape and orientation of the cell are the same for each setting. These settings correspond to the six permutations of the labels of the axes (including the identity permutation); cf. Section 1.5.4.3[link] : [{\bf abc} \quad {\bf ba}\overline{{\bf c}} \quad {\bf cab} \quad \overline{{\bf c}}{\bf ba} \quad {\bf bca} \quad {\bf a}\overline{{\bf c}}{\bf b.}]The symbol for each setting, here called `setting symbol', is a shorthand notation for the (3 × 3) transformation matrix P of the basis vectors of the standard setting, a, b, c, into those of the setting considered (cf. Chapter 1.5[link] for a detailed discussion of coordinate transformations). For instance, the setting symbol cab stands for the cyclic permutation [{\bf a}' = {\bf c},\quad {\bf b}' = {\bf a},\quad {\bf c}' = {\bf b}]or [({\bf a}',{\bf b}',{\bf c}') = ({\bf a}, {\bf b}, {\bf c})\,{\bi P}=({\bf a},{\bf b}, {\bf c}) \pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr} = ({\bf c},{\bf a},{\bf b}),]where a′, b′, c′ is the new set of basis vectors. An interchange of two axes reverses the handedness of the coordinate system; in order to keep the system right-handed, each interchange is accompanied by the reversal of the sense of one axis, i.e. by an element [\bar{1}] in the transformation matrix. Thus, [{\bf ba}\overline{{\bf c}}] denotes the transformation [({\bf a}',{\bf b}',{\bf c}') = ({\bf a},{\bf b},{\bf c}) \pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr} = ({\bf b},{\bf a},\overline{{\bf c}}).]The six orthorhombic settings correspond to six Hermann–Mauguin symbols which, however, need not all be different; cf. Table 2.1.3.4[link].1

Table 2.1.3.4| top | pdf |
Numbers of distinct projections and different Hermann–Mauguin symbols for the orthorhombic space groups

The space-group numbers are given in parentheses. The space groups are listed according to point group as indicated in the column headings.

Number of distinct projections222mm2[2/m \,2/m\, 2/m]
6   [Pmc2_{1}\ (26)] [P\ 2_{1}/m\; 2/m\; 2/a\ (51)]
(22 space groups)   Pma2 (28) [P\ 2/n\;2_{1}/n\;2/a\ (52)]
    [Pca2_{1}\ (29)] [P\ 2/m\;2/n\;2_{1}/a\ (53)]
    Pnc2 (30) [P\ 2_{1}/c\;2/c\;2/a\ (54)]
    [Pmn2_{1}\ (31)] [P\ 2/b\;2_{1}/c\;2_{1}/m\ (57)]
    [Pna2_{1}\ (33)] [P\ 2_{1}/b\;2/c\;2_{1}/n\ (60)]
    [Cmc2_{1}\ (36)] [P\ 2_{1}/n\;2_{1}/m\;2_{1}/a\ (62)]
    Amm2 (38) [C\ 2/m\;2/c\;2_{1}/m\ (63)]
    Aem2 (39) [C\ 2/m\;2/c\;2_{1}/e\ (64)]
    Ama2 (40) [I\ 2_{1}/m\;2_{1}/m\;2_{1}/a\ (74)]
    Aea2 (41)  
    Ima2 (46)  
3 [P222_{1}\ (17)] Pmm2 (25) [P\ 2/c\;2/c\;2/m\ (49)]
(25 space groups) [P2_{1}2_{1}2\ (18)] Pcc2 (27) [P\ 2/b\;2/a\;2/n\ (50)]
  [C222_{1}\ (20)] Pba2 (32) [P\ 2_{1}/b\;2_{1}/a\;2/m\ (55)]
  C222 (21) Pnn2 (34) [P\ 2_{1}/c\;2_{1}/c\;2/n\ (56)]
    Cmm2 (35) [P\ 2_{1}/n\;2_{1}/n\;2/m\ (58)]
    Ccc2 (37) [P\ 2_{1}/m\;2_{1}/m\;2/n\ (59)]
    Fmm2 (42) [C\ 2/m\;2/m\;2/m\ (65)]
    Fdd2 (43) [C\ 2/c\;2/c\;2/m\ (66)]
    Imm2 (44) [C\ 2/m\;2/m\;2/e\ (67)]
    Iba2 (45) [C\ 2/c\;2/c\;2/e\ (68)]
      [I\ 2/b\;2/a\;2/m\ (72)]
2     [P\ 2_{1}/b\;2_{1}/c\;2_{1}/a\ (61)]
(2 space groups)     [I\ 2_{1}/b\;2_{1}/c\;2_{1}/a\ (73)]
1 P222 (16)   [P\ 2/m\;2/m\;2/m\ (47)]
(10 space groups) [P2_{1}2_{1}2_{1}\ (19)]   [P\ 2/n\;2/n\;2/n\ (48)]
  F222 (22)   [F\ 2/m\;2/m\;2/m\ (69)]
  I222 (23)   [F\ 2/d\;2/d\;2/d\ (70)]
  [I2_{1}2_{1}2_{1}\ (24)]   [I\ 2/m\;2/m\;2/m\ (71)]
Total: 59 9 22 28

In the earlier (1935 and 1952) editions of International Tables, only one setting was illustrated, in a projection along c, so that it was usual to consider it as the `standard setting' and to accept its cell edges as crystal axes and its space-group symbol as the `standard Hermann–Mauguin symbol'. In the present edition, following IT A (2002)[link], however, all six orthorhombic settings are illustrated, as explained below.

The three projections of the symmetry elements can be interpreted in two ways. First, in the sense indicated above, that is, as different projections of a single (standard) setting of the space group, with the projected basis vectors a, b, c labelled as in Fig. 2.1.3.5[link]. Second, each one of the three diagrams can be considered as the projection along c′ of either one of two different settings: one setting in which b′ is horizontal and one in which b′ is vertical (a′, b′, c′ refer to the setting under consideration). This second interpretation is used to illustrate in the same figure the space-group symbols corresponding to these two settings. In order to view these projections in conventional orientation (b′ horizontal, a′ vertical, origin in the upper left corner, projection down the positive c′ axis), the setting with b′ horizontal can be inspected directly with the figure upright; hence, the corresponding space-group symbol is printed above the projection. The other setting with b′ vertical and a′ horizontal, however, requires turning the figure by 90°, or looking at it from the side; thus, the space-group symbol is printed at the left, and it runs upwards.

The `setting symbols' for the six settings are attached to the three diagrams of Fig. 2.1.3.6[link], which correspond to those of Fig. 2.1.3.5[link]. In the orientation of the diagram where the setting symbol is read in the usual way, a′ is vertical pointing downwards, b′ is horizontal pointing to the right, and c′ is pointing upwards from the page. Each setting symbol is printed in the position that in the space-group tables is actually occupied by the corresponding full Hermann–Mauguin symbol. The changes in the space-group symbol that are associated with a particular setting symbol can easily be deduced by comparing Fig. 2.1.3.6[link] with the diagrams for the space group under consideration.

[Figure 2.1.3.6]

Figure 2.1.3.6 | top | pdf |

Orthorhombic space groups. The three projections of the symmetry elements with the six setting symbols (see text). For setting symbols printed vertically, the page has to be turned clockwise by 90° or viewed from the side. Note that in the actual space-group tables instead of the setting symbols the corresponding full Hermann–Mauguin space-group symbols are printed.

Not all of the 59 orthorhombic space groups have all six projections distinct, i.e. have different Hermann–Mauguin symbols for the six settings. This aspect is treated in Table 2.1.3.4[link]. Only 22 space groups have six, 25 have three, 2 have two different symbols, while 10 have all symbols the same. This information can be of help in the early stages of a crystal-structure analysis.

The six setting symbols, i.e. the six permutations of the labels of the axes, form the column headings of the ortho­rhombic entries in Table 1.5.4.4[link] , which contains the extended Hermann–Mauguin symbols for the six settings of each orthorhombic space group. Note that some of these setting symbols exhibit different sign changes compared with those in Fig. 2.1.3.6[link].

2.1.3.6.5. Tetragonal, trigonal P and hexagonal P space groups

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The pairs of diagrams for these space groups are similar to those in the previous editions of IT. Each pair consists of a general-position diagram (right) and a diagram of the symmetry elements (left), both projected along c, as illustrated in Figs. 2.1.3.7[link] and 2.1.3.8[link].

[Figure 2.1.3.7]

Figure 2.1.3.7 | top | pdf |

Tetragonal space groups ([\hbox{\sf G}] = general-position diagram).

[Figure 2.1.3.8]

Figure 2.1.3.8 | top | pdf |

Trigonal P and hexagonal P space groups ([\hbox{\sf G}] = general-position diagram).

2.1.3.6.6. Trigonal R (rhombohedral) space groups

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The seven rhombohedral space groups are treated in two versions, the first based on `hexagonal axes' (obverse setting), the second on `rhombohedral axes' (cf. Sections 2.1.1.2[link] and 2.1.3.2[link]). The pairs of diagrams are similar to those in IT (1952)[link] and IT A (2002)[link]; the left or top one displays the symmetry elements, the right or bottom one the general position. This is illustrated in Fig. 2.1.3.9[link], which gives the axes a and b of the triple hexagonal cell and the projections of the axes of the primitive rhombohedral cell, labelled arh, brh and crh. For convenience, all `heights' in the space-group diagrams are fractions of the hexagonal c axis. For `hexagonal axes', the projection direction is [001], for `rhombohedral axes' it is [111]. In the general-position diagrams, the circles drawn in heavier lines represent atoms that lie within the primitive rhombohedral cell (provided the symbol `−' is read as [1 - z] rather than as [- z]).

[Figure 2.1.3.9]

Figure 2.1.3.9 | top | pdf |

Rhombohedral space groups. Obverse triple hexagonal cell with `hexagonal axes' a, b and primitive rhombohedral cell with projections of `rhombohedral axes' arh, brh, crh. Note: In the actual space-group diagrams the edges of the primitive rhombohedral cell (dashed lines) are only indicated in the general-position diagram of the rhombohedral-axes description ([\hbox{\sf G}] = general-position diagram).

The symmetry-element diagrams for the hexagonal and the rhombohedral descriptions of a space group are the same. The edges of the primitive rhombohedral cell (cf. Fig. 2.1.3.9[link]) are only indicated in the general-position diagram of the rhombohedral description.

2.1.3.6.7. Cubic space groups

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For each cubic space group, one projection of the symmetry elements along [001] is given, Fig. 2.1.3.10[link]; for details of the diagrams, see Section 2.1.2[link] and Buerger (1956)[link]. For face-centred lattices F, only a quarter of the unit cell is shown; this is sufficient since the projected arrangement of the symmetry elements is translation-equivalent in the four quarters of an F cell. It is important to note that symmetry axes inclined to the projection plane are indicated where they intersect the plane of projection. Symmetry planes inclined to the projection plane that occur in classes [\bar{4}3m] and [m\bar{3}m] are shown as `inserts' around the high-symmetry points, such as [0,0,0]; [\textstyle{{1}\over{2}},0,0]; etc.

[Figure 2.1.3.10]

Figure 2.1.3.10 | top | pdf |

Cubic space groups. [\hbox{\sf G}] = general-position diagram, in which the equivalent positions are shown as the vertices of polyhedra.

The cubic diagrams given in IT (1935)[link] are different from the ones used here. No drawings for cubic space groups were provided in IT (1952)[link].

2.1.3.6.8. Diagrams of the general position (by K. Momma and M. I. Aroyo)

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Non-cubic space groups. In these diagrams, the `heights' of the points are z coordinates, except for monoclinic space groups with unique axis b where they are y coordinates. For rhombohedral space groups, the heights are always fractions of the hexagonal c axis. The symbols [+] and − stand for [+z] and [-z] (or [+y] and [-y]) in which z or y can assume any value. For points with symbols [+] or − preceded by a fraction, e.g. [{1 \over 2}+] or [{1 \over 3}-], the relative z or y coordinate is [{1 \over 2}] etc. higher than that of the point with symbol [+] or −.

Where a mirror plane exists parallel to the plane of projection, the two positions superimposed in projection are indicated by the use of a ring divided through the centre. The information given on each side refers to one of the two positions related by the mirror plane, as in [-\def\circhyp{\mathop{\bigcirc\hskip-.95em{{\raise0.15em\hbox{$\scriptscriptstyle{{\hbox to 1.5pt{}}\displaystyle{,}}$}}\hskip-0.05em \lower0.2em\hbox{\vrule height 0.9em}}}\;\;}\circhyp +].

Diagrams for cubic space groups (Fig. 2.1.3.10)[link]. Following the approach of IT (1935), for each cubic space group a diagram showing the points of the general position as the vertices of polyhedra is given. In these diagrams, the polyhedra are transparent, but the spheres at the vertices are opaque. For most of the space groups, `starting points' with the same coordinate values, x = 0.048, y = 0.12, z = 0.089, have been used. The origins of the polyhedra are chosen at special points of highest site symmetry, which for most space groups coincide with the origin (and its equivalent points in the unit cell). Polyhedra with origins at sites [({\textstyle{1\over 8}}, {\textstyle{1\over 8}}, {\textstyle{1\over 8}})] have been chosen for the space groups [P4_332] (212) and [I4_132] (214), and [({\textstyle{3\over 8}}, {\textstyle{3\over 8}}, {\textstyle{3\over 8}})] for [P4_132] (213). The two diagrams shown for the space groups [I\bar{4}3d] (220) and [Ia\bar{3}d] (230) correspond to polyhedra with origins chosen at two different special sites with site-symmetry groups of equal (32 versus [\bar{3}] in [Ia\bar{3}d]) or nearly equal order (3 versus [\bar{4}] in [I\bar{4}3d]). The height h of the centre of each polyhedron is given on the diagram, if different from zero. For space-group Nos. 198, 199 and 220, h refers to the special point to which the polyhedron (triangle) is connected. Polyhedra with height 1 are omitted in all the diagrams. A grid of four squares is drawn to represent the four quarters of the basal plane of the cell. For space groups [F\bar{4}3c] (219), [Fm\bar{3}c] (226) and [Fd\bar{3}c] (228), where the number of points is too large for one diagram, two diagrams are provided, one for the upper half and one for the lower half of the cell.

Notes:

  • (i) For space group [P4_132] (213), the coordinates [\bar{x}, \bar{y}, \bar{z}] have been chosen for the `starting point' to show the enantiomorphism with [P4_332] (212).

  • (ii) For the description of a space group with `origin choice 2', the coordinates x, y, z of all points have been shifted with the origin to retain the same polyhedra for both origin choices.

An additional general-position diagram is shown on the fourth page for each of the ten space groups of the [m\bar{3}m] crystal class. To provide a clearer three-dimensional-style overview of the arrangements of the polyhedra, these general-position diagrams are shown in tilted projection (in contrast to the orthogonal-projection diagrams described above).

The general-position diagrams of the cubic groups in both orthogonal and tilted projections were generated using the program VESTA (Momma & Izumi, 2011[link]).

Readers who wish to compare other approaches to space-group diagrams and their history are referred to IT (1935), IT (1952), the fifth edition of IT A (2002)[link] (where general-position stereodiagrams of the cubic space groups are shown) and the following publications: Astbury & Yardley (1924)[link], Belov et al. (1980)[link], Buerger (1956)[link], Fedorov (1895[link]; English translation, 1971), Friedel (1926)[link], Hilton (1903)[link], Niggli (1919)[link] and Schiebold (1929)[link].

2.1.3.7. 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] and IT A (2002)[link], the origin of the unit cell has been chosen according to the following conventions (cf. Sections 2.1.1[link] and 2.1.3.2[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. As an example, study the origin choice 1 and origin choice 2 descriptions of [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, see for example space groups [Pca2_1] (29) and [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 2.1.3.6[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,{\textstyle{1 \over 3}}]; [0,x,{\textstyle{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 2.1.3.12[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.

Origin statement. 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 Ccce (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 2.1.3.5[link], which should be studied together with the appropriate space-group diagrams.

Table 2.1.3.5| top | pdf |
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 2.1.3.1[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 2.1.3.1[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 2.1.3.1[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 2.1.3.9[link] and 1.4.2.1[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.

2.1.3.8. Asymmetric unit

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An asymmetric unit of a space group is a (simply connected) smallest closed part of space from which, by application of all symmetry operations of the space group, the whole of space is filled. This implies that mirror planes and rotation axes must form boundary planes and boundary edges of the asymmetric unit. A twofold rotation axis may bisect a boundary plane. Centres of inversion must either form vertices of the asymmetric unit or be located at the midpoints of boundary planes or boundary edges. For glide planes and screw axes, these simple restrictions do not hold. An asymmetric unit contains all the information necessary for the complete description of the crystal structure. In mathematics, an asymmetric unit is called `fundamental region' or `fundamental domain'.

Example

The boundary planes of the asymmetric unit in space group Pmmm (47) are fixed by the six mirror planes x, y, 0; [x, y, {1 \over 2}]; x, 0, z; [x, {1 \over 2}, z]; 0, y, z; and [{1 \over 2},y,z]. For space group [P2_{1}2_{1}2_{1}] (19), on the other hand, a large number of connected regions, each with a volume of [{1 \over 4}V](cell), may be chosen as asymmetric unit.

In cases where the asymmetric unit is not uniquely determined by symmetry, its choice may depend on the purpose of its application. For the description of the structures of molecular crystals, for instance, it is advantageous to select asymmetric units that contain one or more complete molecules. In the space-group tables of this volume, following IT A (2002)[link], the asymmetric units are chosen in such a way that Fourier summations can be performed conveniently.

For all triclinic, monoclinic and orthorhombic space groups, the asymmetric unit is chosen as a parallelepiped with one vertex at the origin of the cell and with boundary planes parallel to the faces of the cell. It is given by the notation [0 \leq x_{i} \leq \hbox{upper limit of } x_{i},]where [x_{i}] stands for x, y or z.

For space groups with higher symmetry, cases occur where the origin does not coincide with a vertex of the asymmetric unit or where not all boundary planes of the asymmetric unit are parallel to those of the cell. In all these cases, parallelepipeds [\hbox{lower limit of } x_{i} \leq x_{i} \leq \hbox{upper limit of } x_{i}]are given that are equal to or larger than the asymmetric unit. Where necessary, the boundary planes lying within these parallelepipeds are given by additional inequalities, such as [x \leq y], [y \leq {1 \over 2} - x] etc.

In the trigonal, hexagonal and especially the cubic crystal systems, the asymmetric units have complicated shapes. For this reason, they are also specified by the coordinates of their vertices. Drawings of asymmetric units for cubic space groups have been published by Koch & Fischer (1974)[link]. Fig. 2.1.3.11[link] shows the boundary planes occurring in the tetragonal, trigonal and hexagonal systems, together with their algebraic equations.

[Figure 2.1.3.11]

Figure 2.1.3.11 | top | pdf |

Boundary planes of asymmetric units occurring in the space-group tables. (a) Tetragonal system. (b) Trigonal and hexagonal systems. The point coordinates refer to the vertices in the plane [z = 0].

Examples

  • (1) In space group P4mm (99), the boundary plane [y = x] occurs in addition to planes parallel to the unit-cell faces; the asymmetric unit is given by [0 \leq x \leq {\textstyle{1 \over 2}}{\hbox{;}} \quad 0 \leq y \leq {\textstyle{1 \over 2}}{\hbox{;}} \quad 0 \leq z \leq 1{\hbox{;}} \quad x \leq y.]

  • (2) In P4bm (100), one of the boundary planes is [y = {1 \over 2} - x]. The asymmetric unit is given by [0 \leq x \leq {\textstyle{1 \over 2}}{\hbox{;}} \quad 0 \leq y \leq {\textstyle{1 \over 2}}{\hbox{;}} \quad 0 \leq z \leq 1{\hbox{;}} \quad y \leq {\textstyle{1 \over 2}} - x.]

  • (3) In space group R32 [(]155; hexagonal axes[)], the boundary planes are, among others, [x = (1 + y)/2], [y = 1 - x], [y = (1 + x)/2]. The asymmetric unit is defined by[\displaylines{\qquad0 \leq x \leq {\textstyle{2 \over 3}}\semi \quad 0 \leq y \leq {\textstyle{2 \over 3}}\semi \quad 0 \leq z \leq {\textstyle{1 \over 6}}\semi\hfill\cr \qquad x \leq (1 + y)/2\semi \quad y \leq \min(1 - x, (1 + x)/2)\hfill\cr}][\let\normalbaselines\relax\openup2pt\matrix {{\rm Vertices{:}}\hfill& 0, 0, 0\hfill &\textstyle{1 \over 2}, 0, 0 \hfill&\textstyle{2 \over 3}, \textstyle{1 \over 3}, 0\hfill &\textstyle{1 \over 3}, \textstyle{2 \over 3}, 0\hfill &0, \textstyle{1 \over 2}, 0\hfill\cr &0, 0, \textstyle{1 \over 6}\hfill &\textstyle{1 \over 2}, 0, \textstyle{1 \over 6} \hfill&\textstyle{2 \over 3}, \textstyle{1 \over 3}, \textstyle{1 \over 6}\hfill &\textstyle{1 \over 3}, \textstyle{2 \over 3}, \textstyle{1 \over 6} \hfill&0, \textstyle{1 \over 2}, \textstyle{1 \over 6}.\cr}]It is obvious that the indication of the vertices is of great help in drawing the asymmetric unit.

Fourier syntheses. For complicated space groups, the easiest way to calculate Fourier syntheses is to consider the parallelepiped listed, without taking into account the additional boundary planes of the asymmetric unit. These planes should be drawn afterwards in the Fourier synthesis. For the computation of integrated properties from Fourier syntheses, such as the number of electrons for parts of the structure, the values at the boundaries of the asymmetric unit must be applied with a reduced weight if the property is to be obtained as the product of the content of the asymmetric unit and the multiplicity.

Example

In the parallelepiped of space group Pmmm (47), the weights for boundary planes, edges and vertices are [{1 \over 2}], [{1 \over 4}] and [{1 \over 8}], respectively.

Asymmetric units of the plane groups have been discussed by Buerger (1949[link], 1960[link]) in connection with Fourier summations.

2.1.3.9. Symmetry operations

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As explained in Sections 1.3.3.2[link] and 1.4.2.3[link] , the coordinate triplets of the General position of a space group may be interpreted as a shorthand description of the symmetry operations in matrix notation. The geometric description of the symmetry operations is found in the space-group tables under the heading Symmetry operations.

Numbering scheme. The numbering [(1) \ldots (\,p) \ldots] of the entries in the blocks Symmetry operations and General position (first block below Positions) is the same. Each listed coordinate triplet of the general position is preceded by a number between parentheses (p). The same number (p) precedes the corresponding symmetry operation. For space groups with primitive cells, the two lists contain the same number of entries.

For space groups with centred cells, several (2, 3 or 4) blocks of Symmetry operations correspond to the one General position block. The numbering scheme of the general position is applied to each one of these blocks. The number of blocks equals the multiplicity of the centred cell, i.e. the number of centring translations below the subheading Coordinates, such as [(0,0,0)\!\, +, (\,{2 \over 3}, {1 \over 3}, {1 \over 3}\,)\!\, +, (\,{1 \over 3}, {2 \over 3}, {2 \over 3}\,) \,\!+].

Whereas for the Positions the reader is expected to add these centring translations to each printed coordinate triplet themselves (in order to obtain the complete general position), for the Symmetry operations the corresponding data are listed explicitly. The different blocks have the subheadings `For (0, 0, 0)+ set', `For [(\,{1 \over 2},{1 \over 2},{1 \over 2}\,) +] set', etc. Thus, an obvious one-to-one correspondence exists between the analytical description of a symmetry operation in the form of its general-position coordinate triplet and the geometrical description under Symmetry operations. Note that the coordinates are reduced modulo 1, where applicable, as shown in the example below.

Example: Ibca (73)

The centring translation is [t(\,{1 \over 2},{1 \over 2},{1 \over 2}\,)]. Accordingly, above the general position one finds [({0}, {0}, {0})+] and [(\,{1 \over 2},{1 \over 2},{1 \over 2}\,) +]. In the block Symmetry operations, under the subheading `For [(0, 0, 0)+] set', entry (2) refers to the coordinate triplet [\bar{x} + {1 \over 2},\;\bar{y},\;z + {1 \over 2}]. Under the subheading `For [(\,{1 \over 2},{1 \over 2},{1 \over 2}\,) +] set', however, entry (2) refers to [\bar{x},\;\bar{y} + {1 \over 2},\;z]. The triplet [\bar{x},\;\bar{y} + {1 \over 2},\;z] is selected rather than [\bar{x} + 1,\; \bar{y} + {1 \over 2},\;z + 1], because the coordinates are reduced modulo 1.

The coordinate triplets of the general position represent the symmetry operations chosen as coset representatives of the decomposition of the space group with respect to its translation subgroup (cf. Section 1.4.2[link] for a detailed discussion). In space groups with two origins the origin shift may lead to the choice of symmetry operations of different types as coset representatives of the same coset (e.g. mirror versus glide plane, rotation versus screw axis, see Tables 1.4.2.2[link] and 1.4.2.3[link] ) and designated by the same number (p) in the general-position blocks of the two descriptions. Thus, in [P4/nmm] (129), (p) = (7) represents a 2 and a 21 axis, both in [x, x, 0], whereas (p) = (16) represents a g and an m plane, both in [x, x, z].

Designation of symmetry operations. An entry in the block Symmetry operations is characterized as follows.

  • (i) A symbol denoting the type of the symmetry operation (cf. Section 2.1.2[link]), including its glide or screw part, if present. In most cases, the glide or screw part is given explicitly by fractional coordinates between parentheses. The sense of a rotation is indicated by the superscript [+] or −. Abbreviated notations are used for the glide reflections [a(\,{1 \over 2},0,0) \equiv a]; [b(0,{1 \over 2},0) \equiv b]; [c(0,0,{1 \over 2}\,) \equiv c]. Glide reflections with complicated and unconventional glide parts are designated by the letter g, followed by the glide part between parentheses.

  • (ii) A coordinate triplet indicating the location and orientation of the symmetry element which corresponds to the symmetry operation. For rotoinversions, the location of the inversion point is also given.

Examples

  • (1) [g(\,{1 \over 4},{1 \over 4},{1 \over 2}\,)\ \ \ x,x,z]

    Glide reflection with glide component [(\,{1 \over 4},{1 \over 4},{1 \over 2}\,)] through the plane x, x, z, i.e. the plane parallel to [(1\bar{1}0)] containing the point 0, 0, 0.

  • (2) [g(\,{1 \over 3},{1 \over 6},{1 \over 6}\,)\ \ \ 2x - {1 \over 2}, x,z] (hexagonal axes)

    Glide reflection with glide component [(\,{1 \over 3},{1 \over 6},{1 \over 6}\,)] through the plane [2x-{1 \over 2}, x,z], i.e. the plane parallel to [(1\bar{2}10)], which intersects the a axis at [-{1 \over 2}] and the b axis at [{1 \over 4}]; this operation occurs in [R\bar{3}c] (167, hexagonal axes).

  • (3) Symmetry operations in Ibca (73)

    Under the subheading `For (0, 0, 0)[+] set', the operation generating the coordinate triplet (2) [\bar{x} + {1 \over 2},\bar{y},z + {1 \over 2}] from (1) x, y, z is symbolized by [2 (0,0, {1 \over 2}\,)\ \ {1 \over 4},0,z]. This indicates a twofold screw rotation with screw part [(0,0,{1 \over 2}\,)] for which the corresponding screw axis coincides with the line [{1 \over 4},0,z], i.e. runs parallel to [001] through the point [{1 \over 4},0,0]. Under the subheading `For [(\,{1 \over 2},{1 \over 2},{1 \over 2}\,) +] set', the operation generating the coordinate triplet (2) [\bar{x},\bar{y} + {1 \over 2}, z] from (1) x, y, z is symbolized by [2\ \ 0, {1 \over 4}, z]. It is thus a twofold rotation (without screw part) around the line [0,{1 \over 4},z].

Details on the symbolism and further illustrative examples are presented in Section 1.4.2.1[link] .

2.1.3.10. Generators

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The line Generators selected states the symmetry operations and their sequence, selected to generate all symmetry-equivalent points of the General position from a point with coordinates x, y, z. Generating translations are listed as t(1, 0, 0), t(0, 1, 0), t(0, 0, 1); likewise for additional centring translations. The other symmetry operations are given as numbers (p) that refer to the corresponding coordinate triplets of the general position and the corresponding entries under Symmetry operations, as explained in Section 2.1.3.9[link] [for centred space groups the first block `For (0, 0, 0)+ set' must be used].

For all space groups, the identity operation given by (1) is selected as the first generator. It is followed by the generators t(1, 0, 0), t(0, 1, 0), t(0, 0, 1) of the integral lattice translations and, if necessary, by those of the centring translations, e.g. [t(\,{1 \over 2},{1 \over 2},0)] for a C-centred lattice. In this way, point x, y, z and all its translationally equivalent points are generated. (The remark `and its translationally equivalent points' will hereafter be omitted.) The sequence chosen for the generators following the translations depends on the crystal class of the space group and is set out in Table 1.4.3.1[link] .

Example: [P12_1/c1] (14, unique axis b, cell choice 1)

After the generation of (1) x, y, z, the operation (2) which stands for a twofold screw rotation around the axis 0, y, [{1 \over 4}] generates point (2) of the general position with coordinate triplet [\bar{x},y + {1 \over 2},\bar{z} + {1 \over 2}]. Finally, the inversion (3) generates point (3) [\bar{x},\bar{y},\bar{z}] from point (1), and point (4′) [x,\bar{y} - {1 \over 2},z - {1 \over 2}] from point (2). Instead of (4′), however, the coordinate triplet (4) [x, \bar{y} + {1 \over 2},z + {1 \over 2}] is listed, because the coordinates are reduced modulo 1.

The example shows that for the space group [P12_{1}/c1] two operations, apart from the identity and the generating translations, are sufficient to generate all symmetry-equivalent points. Alternatively, the inversion (3) plus the glide reflection (4), or the glide reflection (4) plus the twofold screw rotation (2), might have been chosen as generators. The process of generation and the selection of the generators for the space-group tables, as well as the resulting sequence of the symmetry operations, are discussed in Section 1.4.3[link] .

The generating operations for different descriptions of the same space group (settings, cell choices, origin choices) are chosen in such a way that the transformation relating the two coordinate systems also transforms the generators of one description into those of the other (cf. Section 1.5.3[link] ).

2.1.3.11. Positions

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The entries under Positions2 (more explicitly called Wyckoff positions) consist of the one General position (upper block) and the Special positions (blocks below). The columns in each block, from left to right, contain the following information for each Wyckoff position.

  • (i) Multiplicity of the Wyckoff position. This is the number of equivalent points per unit cell. For primitive cells, the multiplicity of the general position is equal to the order of the point group of the space group; for centred cells, it is the product of the order of the point group and the number (2, 3 or 4) of lattice points per cell. The multiplicity of a special position is always a divisor of the multiplicity of the general position and the quotient of the two is equal to the order of the site-symmetry group.

  • (ii) Wyckoff letter. This letter is merely a coding scheme for the Wyckoff positions, starting with a at the bottom position and continuing upwards in alphabetical order.

  • (iii) Site symmetry. This is explained in Section 2.1.3.12[link].

  • (iv) Coordinates. The sequence of coordinate triplets is produced in the same order as the symmetry operations, generated by the chosen set of generators, omitting duplicates (cf. Sections 1.4.3[link] and 2.1.3.10[link]). For centred space groups, the centring translations, for instance [(0,0,0)\! +\ (\,{1 \over 2},{1 \over 2},{1 \over 2}\,) +], are listed above the coordinate triplets. The symbol `[+]' indicates that, in order to obtain a complete Wyckoff position, the components of these centring translations have to be added to the listed coordinate triplets. Note that not all points of a position always lie within the unit cell; some may be outside since the coordinates are formulated modulo 1; thus, for example, [\bar{x},\bar{y},\bar{z}] is written rather than [\bar{x} + 1,\bar{y} + 1,\bar{z} + 1].

    The coordinate triplets of a position represent the coordinates of the equivalent points (atoms) in the unit cell. A graphic representation of the points of the general position is provided by the general-position diagram; cf. Section 2.1.3.6[link].

  • (v) Reflection conditions. These are described in Section 2.1.3.13[link].

Detailed treatment of general and special Wyckoff positions, including definitions, theoretical background and examples, is given in Section 1.4.4[link] .

The two types of positions, general and special, are characterized as follows:

  • (i) General position

    A point is said to be in general position if it is left invariant only by the identity operation but by no other symmetry operation of the space group. Each space group has only one general position.

    The coordinate triplets of a general position (which always start with x, y, z) can also be interpreted as a shorthand form of the matrix representation of the symmetry operations of the space group; this viewpoint is described further in Sections 1.3.3.2[link] and 1.4.2.3[link] .

  • (ii) Special position(s)

    A point is said to be in `special position' if it is mapped onto itself by the identity and at least one further symmetry operation of the space group. This implies that specific constraints are imposed on the coordinates of each point of a special position; e.g. [x = {\textstyle{1 \over 4}}], [y = 0], leading to the triplet [{\textstyle{1 \over 4}},0,z]; or [y = x + {\textstyle{1 \over 2}}], leading to the triplet [x, x + {\textstyle{1 \over 2}},z]. The number of special positions of a space group depends on the space-group type and can vary from 1 to 26 (in Pmmm, No. 47).

The set of all symmetry operations that map a point onto itself forms a group, known as the `site-symmetry group' of that point. It is given in the third column by the `oriented site-symmetry symbol' which is explained in Section 2.1.3.12[link]. General positions always have site symmetry 1, whereas special positions have higher site symmetries, which can differ from one special position to another.

If in a crystal structure the centres of finite objects, such as molecules, are placed at the points of a special position, each such object must display a point symmetry that is at least as high as the site symmetry of the special position. Geometrically, this means that the centres of these objects are located on symmetry elements without translations (centre of symmetry, mirror plane, rotation axis, rotoinversion axis) or at the intersection of several symmetry elements of this kind (cf. the space-group diagrams).

Note that the location of an object on a screw axis or on a glide plane does not lead to an increase in the site symmetry and to a corresponding reduction of the multiplicity for that object. Accordingly, a space group that contains only symmetry elements with translation components does not have any special position. Such a space group is called `fixed-point-free' (for further discussion, see Section 1.4.4.2[link] ).

Example: Space group C12/c1 (15, unique axis b, cell choice 1)

The general position 8f of this space group contains eight equivalent points per cell, each with site symmetry 1. The coordinate triplets of four points, (1) to (4), are given explicitly, the coordinates of the other four points are obtained by adding the components [{1 \over 2},{1 \over 2},0] of the C-centring translation to the coordinate triplets (1) to (4).

The space group has five special positions with Wyckoff letters a to e. The positions 4a to 4d require inversion symmetry, [\bar{1}], whereas Wyckoff position 4e requires twofold rotation symmetry, 2, for any object in such a position. For position 4e, for instance, the four equivalent points have the coordinates [0,y,{1 \over 4};\ 0,\bar{y},{3 \over 4};\ {1 \over 2},y + {1 \over 2},{1 \over 4};\ {1 \over 2},\bar{y} + {1 \over 2},{3 \over 4}]. The values of x and z are specified, whereas y may take any value. Since each point of position 4e is mapped onto itself by a twofold rotation, the multiplicity of the position is reduced from 8 to 4, whereas the order of the site-symmetry group is increased from 1 to 2.

From the symmetry-element diagram of C2/c, the locations of the four twofold axes can be deduced as [0,y,{\textstyle{1 \over 4}}]; [0,y,{\textstyle{3 \over 4}}]; [{\textstyle{1 \over 2}},y,{\textstyle{1 \over 4}}]; [{\textstyle{1 \over 2}},y,{\textstyle{3 \over 4}}].

From this example, the general rule is apparent that the product of the position multiplicity and the order of the corresponding site-symmetry group is constant for all Wyckoff positions of a given space group; it is the multiplicity of the general position.

Attention is drawn to ambiguities in the description of crystal structures in a few space groups, depending on whether the coordinate triplets of IT (1952)[link] or of this edition are taken. This problem is analysed by Parthé et al. (1988)[link].

2.1.3.12. Oriented site-symmetry symbols

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The third column of each Wyckoff position gives the Site symmetry3 of that position. The site-symmetry group is iso­morphic to a (proper or improper) subgroup of the point group to which the space group under consideration belongs. The site-symmetry groups of the different points of the same special position are conjugate (symmetry-equivalent) subgroups of the space group. For this reason, all points of one special position are described by the same site-symmetry symbol. (See Section 1.4.4[link] for a detailed discussion of site-symmetry groups.)

Oriented site-symmetry symbols (cf. Fischer et al., 1973[link]) are employed to show how the symmetry elements at a site are related to the symmetry elements of the crystal lattice. The site-symmetry symbols display the same sequence of symmetry directions as the space-group symbol (cf. Table 2.1.3.1[link]). Sets of equivalent symmetry directions that do not contribute any element to the site-symmetry group are represented by a dot. In this way, the orientation of the symmetry elements at the site is emphasized, as illustrated by the following examples.

Examples

  • (1) In the tetragonal space group [P4_{2}2_{1}2\ (94)], Wyckoff position 4f has site symmetry ..2 and position 2b has site symmetry 2.22. The easiest way to interpret the symbols is to look at the dots first. For position 4f, the 2 is preceded by two dots and thus must belong to a tertiary symmetry direction. Only one tertiary direction is used. Consequently, the site symmetry is the monoclinic point group 2 with one of the two tetragonal tertiary directions as twofold axis.

    Position b has one dot, with one symmetry symbol before and two symmetry symbols after it. The dot corresponds, therefore, to the secondary symmetry directions. The first symbol 2 indicates a twofold axis along the primary symmetry direction (c axis). The final symbols 22 indicate two twofold axes along the two mutually perpendicular tertiary directions [[1\bar{1}0]] and [110]. The site symmetry is thus orthorhombic, 222.

  • (2) In the cubic space group I23 (197), position 6b has 222.. as its oriented site-symmetry symbol. The orthorhombic group 222 is completely related to the primary set of cubic symmetry directions, with the three twofold axes parallel to the three equivalent primary directions [100], [010], [001].

  • (3) In the cubic space group [Pn\bar{3}n\ (222)], position 6b has 42.2 as its site-symmetry symbol. This `cubic' site-symmetry symbol displays a tetragonal site symmetry. The position of the dot indicates that there is no symmetry along the four secondary cubic directions. The fourfold axis is connected with one of the three primary cubic symmetry directions and two equivalent twofold axes occur along the remaining two primary directions. Moreover, the group contains two mutually perpendicular (equivalent) twofold axes along those two of the six tertiary cubic directions [\langle 110 \rangle] that are normal to the fourfold axis. Each pair of equivalent twofold axes is given by just one symbol 2. (Note that at the six sites of position 6b the fourfold axes are twice oriented along a, twice along b and twice along c.)

  • (4) In the tetragonal space group [P4_{2}/nnm\ (134)], position 2a has site symmetry [\bar{4}2m]. The site has symmetry for all symmetry directions. Because of the presence of the primary [\bar{4}] axis, only one of the twofold axes along the two secondary directions need be given explicitly and similarly for the mirror planes m perpendicular to the two tertiary directions.

The above examples show:

  • (i) The oriented site-symmetry symbols become identical to Hermann–Mauguin point-group symbols if the dots are omitted.

  • (ii) Sets of symmetry directions having more than one equivalent direction may require more than one character if the site-symmetry group belongs to a lower crystal system than the space group under consideration.

To show, for the same type of site symmetry, how the oriented site-symmetry symbol depends on the space group under discussion, the site-symmetry group mm2 will be considered in orthorhombic and tetragonal space groups. Relevant crystal classes are mm2, mmm, 4mm, [\bar{4}2m] and [4/mmm]. The site symmetry mm2 contains two mutually perpendicular mirror planes intersecting in a twofold axis.

For space groups of crystal class mm2, the twofold axis at the site must be parallel to the one direction of the rotation axes of the space group. The site-symmetry group mm2, therefore, occurs only in the orientation mm2. For space groups of class mmm (full symbol [2/m\ 2/m\ 2/m]), the twofold axis at the site may be parallel to a, b or c and the possible orientations of the site symmetry are 2mm, m2m and mm2. For space groups of the tetragonal crystal class 4mm, the twofold axis of the site-symmetry group mm2 must be parallel to the fourfold axis of the crystal. The two mirror planes must belong either to the two secondary or to the two tertiary tetragonal directions so that 2mm. and 2.mm are possible site-symmetry symbols. Similar considerations apply to class [\bar{4}2m], which can occur in two settings, [\bar{4}2m] and [\bar{4}m2]. Finally, for class [4/mmm] (full symbol [4/m\ 2/m\ 2/m]), the twofold axis of 2mm may belong to any of the three kinds of symmetry directions and possible oriented site symmetries are 2mm., 2.mm, m2m. and m.2m. In the first two symbols, the twofold axis extends along the single primary direction and the mirror planes occupy either both secondary or both tertiary directions; in the last two cases, one mirror plane belongs to the primary direction and the second to either one secondary or one tertiary direction (the other equivalent direction in each case being occupied by the twofold axis).

2.1.3.13. Reflection conditions

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The Reflection conditions4 are listed in the right-hand column of each Wyckoff position.

These conditions are formulated here, in accordance with general practice, as `conditions of occurrence' (structure factor not systematically zero) and not as `extinctions' or `systematic absences' (structure factor zero). Reflection conditions are listed for all those three-, two- and one-dimensional sets of reflections for which extinctions exist; hence, for those nets or rows that are not listed, no reflection conditions apply. The theoretical background of reflection conditions and their derivation are discussed in detail in Section 1.6.3[link] .

There are two types of systematic reflection conditions for diffraction of radiation by crystals:

  • (1) General conditions. They are associated with systematic absences caused by the presence of lattice centrings, screw axes and glide planes. The general conditions are always obeyed, irrespective of which Wyckoff positions are occupied by atoms in a particular crystal structure.

  • (2) Special conditions (`extra' conditions). They apply only to special Wyckoff positions and always occur in addition to the general conditions of the space group. Note that each extra condition is valid only for the scattering contribution of those atoms that are located in the relevant special Wyckoff position. If the special position is occupied by atoms whose scattering power is high in comparison with the other atoms in the structure, reflections violating the extra condition will be weak. One should note that the special conditions apply only to isotropic and spherical atoms (cf. Section 1.6.3[link] ).

General reflection conditions. These are due to one of three effects:

  • (i) Centred cells. The resulting conditions apply to the whole three-dimensional set of reflections hkl. Accordingly, they are called integral reflection conditions. They are given in Table 2.1.3.6[link]. These conditions result from the centring vectors of centred cells. They disappear if a primitive cell is chosen instead of a centred cell. Note that the centring symbol and the corresponding integral reflection condition may change with a change of the basis vectors (e.g. monoclinic: [C \rightarrow A \rightarrow I]).

    Table 2.1.3.6| top | pdf |
    Integral reflection conditions for centred cells (lattices)

    Reflection conditionCentring type of cellCentring symbol
    None Primitive [\bigg\{] P
        R (rhombohedral axes)
    [h + k = 2n] C-face centred   C
    [k + l = 2n] A-face centred   A
    [h + l = 2n] B-face centred   B
    [h + k + l = 2n] Body centred   I
    [h + k, h + l] and All-face centred   F
    [k + l = 2n] or:      
    [h, k, l] all odd or all even (`unmixed')      
    [{-h + k + l = 3n}] Rhombohedrally centred, obverse setting (standard) [\!\!\left.{\matrix{{}\cr{}\cr{}\cr{}\cr{}\cr{}}}\right\}] R (hexagonal axes)
    [h - k + l = 3n] Rhombohedrally centred, reverse setting  
    [h - k = 3n] Hexagonally centred   H
    For further explanations see Section 2.1.1[link] and Table 2.1.1.2[link].
    For the use of the unconventional H cell, see Section 1.5.4[link] and Table 2.1.1.2[link].
  • (ii) Glide planes. The resulting conditions apply only to two-dimensional sets of reflections, i.e. to reciprocal-lattice nets containing the origin (such as hk0, h0l, 0kl, hhl). For this reason, they are called zonal reflection conditions. The indices hkl of these `zonal reflections' obey the relation [hu + kv + lw = 0], where [uvw], the direction of the zone axis, is normal to the reciprocal-lattice net. Note that the symbol of a glide plane and the corresponding zonal reflection condition may change with a change of the basis vectors (e.g. monoclinic: [c \rightarrow n \rightarrow a]).

  • (iii) Screw axes. The resulting conditions apply only to one-dimensional sets of reflections, i.e. reciprocal-lattice rows containing the origin (such as h00, 0k0, 00l). They are called serial reflection conditions. It is interesting to note that some diagonal screw axes do not give rise to systematic absences (cf. Section 1.6.3[link] for more details).

Reflection conditions of types (ii)[link] and (iii)[link] are listed in Table 2.1.3.7[link]. They can be understood as follows: Zonal and serial reflections form two- or one-dimensional sections through the origin of reciprocal space. In direct space, they correspond to projections of a crystal structure onto a plane or onto a line. Glide planes or screw axes may reduce the translation periods in these projections (cf. Section 2.1.3.14[link]) and thus decrease the size of the projected cell. As a consequence, the cells in the corresponding reciprocal-lattice sections are increased, which means that systematic absences of reflections occur.

Table 2.1.3.7| top | pdf |
Zonal and serial reflection conditions for glide planes and screw axes (cf. Table 2.1.2.1[link])

(a) Glide planes

Type of reflectionsReflection conditionGlide planeCrystallographic coordinate system to which condition applies
Orientation of planeGlide vectorSymbol
0kl [k = 2n] (100) [{\bf b}/2] b [\!\left.\matrix{\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (a\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}] [\!\left.\matrix{\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]
[l = 2n] [{\bf c}/2] c
[k + l = 2n] [{\bf b}/2 + {\bf c}/2] n
[\!\matrix{k + l = 4n\hfill\cr \quad (k,l = 2n)\hfill\cr}] [{\bf b}/4 \pm {\bf c}/4] d
h0l [l = 2n] (010) [{\bf c}/2] c [\!\left.\matrix{\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (b\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}] [\!\left.\matrix{\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]
[h = 2n] [{\bf a}/2] a
[l + h = 2n] [{\bf c}/2 + {\bf a}/2] n
[\!\matrix{l + h = 4n\hfill\cr \quad (l,h = 2n)\hfill\cr}] [{\bf c}/4 \pm {\bf a}/4] d
hk0 [h = 2n] (001) [{\bf a}/2] a [\!\left.\matrix{\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic (}c\ \hbox{unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}] [\!\left.\matrix{\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]
[k = 2n] [{\bf b}/2] b
[h + k = 2n] [{\bf a}/2 + {\bf b}/2] n
[\!\matrix{h + k = 4n\hfill\cr \quad (h,k = 2n)\hfill\cr}] [{\bf a}/4 \pm {\bf b}/4] d
[\matrix{h\bar{h}0l\hfill\cr 0k\bar{k}l\hfill\cr \bar{h}0hl\hfill\cr}] [l = 2n] [\left.\!\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}\right\} \{11\bar{2}0\}] [{\bf c}/2] c [\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]
[\matrix{hh.\overline{2h}.l\hfill\cr \overline{2h}.hhl\hfill\cr h.\overline{2h}.hl\hfill\cr}] [l = 2n] [\left.\!\matrix{(1\bar{1}00)\hfill\cr (01\bar{1}0)\hfill\cr (\bar{1}010)\hfill\cr}\right\} \{1\bar{1}00\}] [{\bf c}/2] c [\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]
[\matrix{hhl\hfill\cr hkk\hfill\cr hkh\hfill\cr}] [\matrix{l = 2n\hfill\cr h = 2n\hfill\cr k = 2n\hfill\cr}] [\left.\!\matrix{(1\bar{1}0)\hfill\cr (01\bar{1})\hfill\cr (\bar{1}01)\hfill\cr}\right\} \{1\bar{1}0\}] [\matrix{{\bf c}/2\hfill\cr {\bf a}/2\hfill\cr {\bf b}/2\hfill\cr}] [\matrix{c,n\hfill\cr a,n\hfill\cr b,n\hfill\cr}] [\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Rhombohedral}]
[hhl, h\bar{h}l] [l = 2n] [(1\bar{1}0), (110)] [{\bf c}/2] c, n [\!\left.\let\normalbaselines\relax\openup4pt\matrix{\cr\cr}\right\}\hbox{Tetragonal}]§ [\!\left.\let\normalbaselines\relax\openup1pt\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\hbox{Cubic}]
[2h + l = 4n] [{\bf a}/4 \pm {\bf b}/4 \pm {\bf c}/4] d
[hkk, hk\bar{k}] [h = 2n] [(01\bar{1}), (011)] [{\bf a}/2] a, n
[2k + h = 4n] [\pm {\bf a}/4 + {\bf b}/4 \pm {\bf c}/4] d
[hkh, \bar{h}kh] [k = 2n] [(\bar{1}01), (101)] [{\bf b}/2] b, n
[2h + k = 4n] [\pm {\bf a}/4 \pm {\bf b}/4 + {\bf c}/4] d

(b) Screw axes

Type of reflectionsReflection conditionsScrew axisCrystallographic coordinate system to which condition applies
Direction of axisScrew vectorSymbol
h00 [h = 2n] [100] [{\bf a}/2] [2_{1}] [\left\{\matrix{\hbox{Monoclinic }(a\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.] [\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]
[4_{2}]
[h = 4n] [{\bf a}/4] [4_{1},4_{3}]
0k0 [k = 2n] [010] [{\bf b}/2] [2_{1}] [\left\{\matrix{\hbox{Monoclinic }(b\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.] [\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]
[4_{2}]
[k = 4n] [{\bf b}/4] [4_{1},4_{3}]
00l [l = 2n] [001] [{\bf c}/2] [2_{1}] [\matrix{\left\{\matrix{\hbox{Monoclinic } (c\hbox{ unique}),\hfill\cr\quad\hbox{ Orthorhombic}\hfill\cr}\right.\cr\left.\matrix{\noalign{\vskip31pt}\cr}\right\}\hbox{Tetragonal}\hfill}] [\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]
[4_{2}]
[l = 4n] [{\bf c}/4] [4_{1},4_{3}]
000l [l = 2n] [001] [{\bf c}/2] [6_{3}] [\left.{\hbox to 2pt{}}\matrix{\noalign{\vskip44pt}}\right\}\hbox{Hexagonal}]
[l = 3n] [{\bf c}/3] [3_{1},3_{2},6_{2},6_{4}]
[l = 6n] [{\bf c}/6] [6_{1},6_{5}]
Glide planes d with orientations (100), (010) and (001) occur only in orthorhombic and cubic F space groups. Combination of the integral reflection condition (hkl: all odd or all even) with the zonal conditions for the d glide planes leads to the further conditions given between parentheses.
For rhombohedral space groups described with `rhombohedral axes', the three reflection conditions [(l = 2n, h = 2n, k = 2n)] imply interleaving of c and n glides, a and n glides, and b and n glides, respectively. In the Hermann–Mauguin space-group symbols, c is always used, as in R3c (161) and [R\bar{3}c\ (167)], because c glides also occur in the hexagonal description of these space groups.
§For tetragonal P space groups, the two reflection conditions (hhl and [h\bar{h}l] with [l = 2n]) imply interleaving of c and n glides. In the Hermann–Mauguin space-group symbols, c is always used, irrespective of which glide planes contain the origin: cf. P4cc (103), [P\bar{4}2c\ (112)] and [P4/nnc\ (126)].
For cubic space groups, the three reflection conditions [(l = 2n, h = 2n, k = 2n)] imply interleaving of c and n glides, a and n glides, and b and n glides, respectively. In the Hermann–Mauguin space-group symbols, either c or n is used, depending upon which glide plane contains the origin, cf. [P\bar{4}3n\ (218)], [Pn\bar{3}n\ (222)], [Pm\bar{3}n\ (223)] versus [F\bar{4}3c\ (219)], [Fm\bar{3}c\ (226)], [Fd\bar{3}c\ (228)].

For the two-dimensional groups, the reasoning is analogous. The reflection conditions for the plane groups are assembled in Table 2.1.3.8[link].

Table 2.1.3.8| top | pdf |
Reflection conditions for the plane groups

Type of reflectionsReflection conditionCentring type of plane cell; or glide line with glide vectorCoordinate system to which condition applies
hk None Primitive p All systems
[h + k = 2n] Centred c Rectangular
[h - k = 3n] Hexagonally centred h Hexagonal
h0 [h = 2n] Glide line g normal to b axis; glide vector [{1 \over 2}{\bf a}] [\left.\matrix{\noalign{\vskip 50pt}}\right\}\matrix{\hbox{Rectangular, }\hfill\cr\quad\hbox{Square}\hfill}]
0k [k = 2n] Glide line g normal to a axis; glide vector [{1 \over 2}{\bf b}]
For the use of the unconventional h cell see Table 2.1.1.2[link].

For the interpretation of observed reflections, the general reflection conditions must be studied in the order (i)[link][link] to (iii)[link], as conditions of type (ii)[link] may be included in those of type (i)[link], while conditions of type (iii)[link] may be included in those of types (i)[link] or (ii)[link]. This is shown in the example below.

In the space-group tables, the reflection conditions are given according to the following rules:

  • (i) for a given space group, all reflection conditions [up to symmetry equivalence, cf. rule (v)[link]] are listed; hence for those nets or rows that are not listed no conditions apply. No distinction is made between `independent' and `included' conditions, as was done in IT (1952)[link], where `included' conditions were placed in parentheses;

  • (ii) the integral condition, if present, is always listed first, followed by the zonal and serial conditions;

  • (iii) conditions that have to be satisified simultaneously are separated by a comma or by `AND'. Thus, if two indices must be even, say h and l, the condition is written [h, l = 2n] rather than [h = 2n] and [l = 2n]. The same applies to sums of indices. Thus, there are several different ways to express the integral conditions for an F-centred lattice: `[h + k, h + l, k + l = 2n]' or `[h + k, h + l = 2n] and [k + l = 2n]' or `[h + k = 2n] and [h + l, k + l = 2n]' (cf. Table 2.1.3.6[link]);

  • (iv) conditions separated by `OR' are alternative conditions. For example, `[hkl{:}\ h = 2n + 1] or [h + k + l = 4n]' means that hkl is `present' if either the condition [h = 2n + 1] or the alternative condition [h + k + l = 4n] is fulfilled. Obviously, hkl is also a `present' reflection if both conditions are satisfied. Note that `or' conditions occur only for the special conditions described below;

  • (v) in crystal systems with two or more symmetry-equivalent nets or rows (tetragonal and higher), only one representative set (the first one in Table 2.1.3.7[link]) is listed; e.g. tetragonal: only the first members of the equivalent sets 0kl and h0l or h00 and 0k0 are listed;

  • (vi) for cubic space groups, it is stated that the indices hkl are `cyclically permutable' or `permutable'. The cyclic permutability of h, k and l in all rhombohedral space groups, described with `rhombohedral axes', and of h and k in some tetragonal space groups are not stated;

  • (vii) in the `hexagonal-axes' descriptions of trigonal and hexagonal space groups, Bravais–Miller indices hkil are used. They obey two conditions:

    • (a) [h + k + i = 0,\ i.e.\ i = - (h + k)];

    • (b) the indices h, k, i are cyclically permutable; this is not stated. Further details can be found in textbooks of crystallography.

Note that the integral reflection conditions for a rhombohedral lattice, described with `hexagonal axes', permit the presence of only one member of the pair hkil and [\bar{h}\bar{k}\bar{i{\phantom l}}\!l] for [l \neq 3n] (cf. Table 2.1.3.6[link]). This applies also to the zonal reflections [h\bar{h}0l] and [\bar{h}h0l], which for the rhombohedral space groups must be considered separately.

Example

For a monoclinic crystal (b unique), the following reflection conditions have been observed:

  • (1) [hkl\!\! :h + k = 2n\semi]

  • (2) [0kl\!\! :k = 2n\semi\quad h0l\!\!:h,l = 2n\semi\quad hk0\!\!:h + k = 2n\semi]

  • (3) [h00\!\! :h = 2n\semi\quad 0k0\!\!:k = 2n\semi\quad 00l\!\!:l = 2n.]

Line (1) states that the cell used for the description of the space group is C centred. In line (2), the conditions 0kl with [k = 2n], h0l with [h = 2n] and hk0 with [h + k = 2n] are a consequence of the integral condition (1), leaving only h0l with [l = 2n] as a new condition. This indicates a glide plane c. Line (3) presents no new condition, since h00 with [h = 2n] and 0k0 with [k = 2n] follow from the integral condition (1), whereas 00l with [l = 2n] is a consequence of a zonal condition (2). Accordingly, there need not be a twofold screw axis along [010]. Space groups obeying the conditions are Cc (9, b unique, cell choice 1) and [C2/c] (15, b unique, cell choice 1). Under certain conditions, using methods based on resonant scattering, it is possible to determine whether the structure space group is centrosymmetric or not (cf. Section 1.6.5.1[link] ).

For a different choice of the basis vectors, the reflection conditions would appear in a different form owing to the transformation of the reflection indices (cf. cell choices 2 and 3 for space groups Cc and [C2/c] in Chapter 2.3[link] ). The transformations of reflection conditions under coordinate transformations are discussed and illustrated in Sections 1.5.2[link] and 1.5.3[link] .

Special or `extra' reflection conditions. These apply either to the integral reflections hkl or to particular sets of zonal or serial reflections. In the space-group tables, the minimal special conditions are listed that, on combination with the general conditions, are sufficient to generate the complete set of conditions. This will be apparent from the examples below.

Examples

  • (1) [P4_{2}22\ (93)]

    General position [8p{:}\ 00l{:}\ l = 2n], due to [4_{2}]; the projection on [001] of any crystal structure with this space group has periodicity [{1 \over 2}c].

    Special position [4i{:}\ \ hkl{:}\ \ h + k + l = 2n]; any set of symmetry-equivalent atoms in this position displays additional I centring.

    Special position [4n{:}\ \,0kl{:}\ \, l = 2n]; any set of equivalent atoms in this position displays a glide plane [c \perp [100]]. Projection of this set along [100] results in a halving of the original c axis, hence the special condition. Analogously for [h0l]: [l=2n].

  • (2) [C12/c1] (15, unique axis b, cell choice 1)

    General position [8f{:}\ hkl{:}\ h + k = 2n], due to the C-centred cell.

    Special position [4d{:}\ hkl{:}\ k + l = 2n], due to additional A and B centring for atoms in this position. Combination with the general condition results in [hkl{:}\ h + k, h + l], [k + l = 2n] or hkl all odd or all even; this corresponds to an F-centred arrangement of atoms in this position.

    Special position [4b{:}\ hkl{:}\ l = 2n], due to additional halving of the c axis for atoms in this position. Combination with the general condition results in [hkl{:}\ h + k,l = 2n]; this corresponds to a C-centred arrangement in a cell with half the original c axis. No further condition results from the combination.

  • (3) [I12/a1] (15, unique axis b, cell choice 3)

    For the description of space group No. 15 with cell choice 3 (see Section 2.1.3.15[link] and the space-group tables), the reflection conditions appear as follows:

    General position [8f{:}\ \, hkl{:}\ \,h + k + l = 2n], due to the I-centred cell.

    Special position [4b{:}\ hkl{:}\ h = 2n], due to additional halving of the a axis. Combination gives [hkl{:}\ h,k + l = 2n], i.e. an A-centred arrangement of atoms in a cell with half the original a axis.

    An analogous result is obtained for position 4d.

  • (4) Fmm2 (42)

    General position [16e{:}\ hkl{:}\ h + k,h + l,k + l = 2n], due to the F-centred cell.

    Special position [8b{:}\ hkl{:}\h = 2n], due to additional halving of the a axis. Combination results in [hkl{:}\ h,k,l = 2n], i.e. all indices even; the atoms in this position are arranged in a primitive lattice with axes [{1 \over 2}a,\; {1 \over 2}b] and [{1 \over 2}c].

For the cases where the special reflection conditions are described by means of combinations of `OR' and `AND' instructions, the `AND' condition always has to be evaluated with priority, as shown by the following example.

Example: [P\bar{4}3n] (218)

Special position [6d{:}\ hkl{:}\ h+k+l = 2n] or [h = 2n+1], [k = 4n] and [l = 4n+2].

This expression contains the following two conditions:

(a) [hkl{:}\ h+k+l=2n];

(b) [h = 2n+1] and [k = 4n] and [l = 4n+2].

A reflection is `present' (occurring) if either condition (a) is satisfied or if a permutation of the three conditions in (b) are simultaneously fulfilled.

Structural or non-space-group absences. Note that in addition non-space-group absences may occur that are not due to the symmetry of the space group (i.e. centred cells, glide planes or screw axes). Atoms in general or special positions may cause additional systematic absences if their coordinates assume special values [e.g. `noncharacteristic orbits'; cf. Section 1.4.4.4[link] and Engel et al. (1984[link])]. Non-space-group absences may also occur for special arrangements of atoms (`false symmetry') in a crystal structure (cf. Templeton, 1956[link]; Sadanaga et al., 1978[link]). Non-space-group absences may occur also for polytypic structures; this is briefly discussed by Durovič in Section 9.2.2.2.5[link] of International Tables for Crystallography (2004)[link], Vol. C. Even though all these `structural absences' are fortuitous and due to the special arrangements of atoms in a particular crystal structure, they have the appearance of space-group absences. Occurrence of structural absences thus may lead to an incorrect assignment of the space group. Accordingly, the reflection conditions in the space-group tables must be considered as a minimal set of conditions.

The use of reflection conditions and of the symmetry of reflection intensities for space-group determination is described in Chapter 1.6[link] .

2.1.3.14. Symmetry of special projections

| top | pdf |

Projections of crystal structures are used by crystallographers in special cases. Use of so-called `two-dimensional data' (zero-layer intensities) results in the projection of a crystal structure along the normal to the reciprocal-lattice net. A detailed treatment of projections of space groups, including basic definitions and illustrative examples, is given in Section 1.4.5.3[link] .

Even though the projection of a finite object along any direction may be useful, the projection of a periodic object such as a crystal structure is only sensible along a rational lattice direction (lattice row). Projection along a nonrational direction results in a constant density in at least one direction.

Data listed in the space-group tables. Under the heading Symmetry of special projections, the following data are listed for three projections of each space group; no projection data are given for the plane groups.

  • (i) The projection direction. All projections are orthogonal, i.e. the projection is made onto a plane normal to the projection direction. This ensures that spherical atoms appear as circles in the projection. For each space group, three projections are listed. If a lattice has three kinds of symmetry directions, the three projection directions correspond to the primary, secondary and tertiary symmetry directions of the lattice (cf. Table 2.1.3.1[link]). If a lattice contains fewer than three kinds of symmetry directions, as in the triclinic, monoclinic and rhombohedral cases, the additional projection direction(s) are taken along coordinate axes, i.e. lattice rows lacking symmetry.

    The directions for which projection data are listed are as follows: [\openup 1pt\eqalign{&\left.\matrix{\hbox{Triclinic}\hfill\cr \hbox{Monoclinic}\hfill\cr \hbox{(both settings)}\hfill\cr \hbox{Orthorhombic}\hfill\cr}\right\}{\hskip 3.5pt{[001]}}{\hskip 12.5pt{[100]}} {\hskip 12pt{[010]}} \cr &\matrix{\hbox{Tetragonal}\hfill &\ [001] &\ [100] &\ [110]\cr \hbox{Hexagonal}\hfill &\ [001] &\ [100] &\ [210]\cr \hbox{Rhombohedral}\hfill &\ [111] &\ [1\bar{1}0] &\ [2\bar{1}\bar{1}]\cr \hbox{Cubic}\hfill &\ [001] &\ [111] &\ [110]\cr}}]

  • (ii) The Hermann–Mauguin symbol of the plane group resulting from the projection of the space group. If necessary, the symbols are given in oriented form; for example, plane group pm is expressed either as p1m1 or as p11m (cf. Section 1.4.1.5[link] for explanations of Hermann–Mauguin symbols of plane groups).

  • (iii) Relations between the basis vectors a′, b′ of the plane group and the basis vectors a, b, c of the space group. Each set of basis vectors refers to the conventional coordinate system of the plane group or space group, as employed in Chapters 2.2[link] and 2.3[link] . The basis vectors of the two-dimensional cell are always called a′ and b′ irrespective of which two of the basis vectors a, b, c of the three-dimensional cell are projected to form the plane cell. All relations between the basis vectors of the two cells are expressed as vector equations, i.e. a′ and b′ are given as linear combinations of a, b and c. For the triclinic or monoclinic space groups, basis vectors a, b or c inclined to the plane of projection are replaced by the projected vectors [{\bf a}_{p},{\bf b}_{p},{\bf c}_{p}].

    For primitive three-dimensional cells, the metrical relations between the lattice parameters of the space group and the plane group are collected in Table 2.1.3.9[link]. The additional relations for centred cells can be derived easily from the table.

    Table 2.1.3.9| top | pdf |
    Cell parameters a′, b′, γ′ of the two-dimensional cell in terms of cell parameters a, b, c, α, β, γ of the three-dimensional cell for the projections listed in the space-group tables of Chapter 2.3[link]

    Projection directionTriclinicMonoclinicOrthorhombic
    Unique axis bUnique axis c
    [001] [a' = a \sin \beta] [a' = a \sin \beta] [a' = a] [a' = a]
    [b' = b \sin \alpha] [b' = b] [b' = b] [b' = b]
    [\gamma' = 180^{\circ} - \gamma^{*}] [\gamma' = 90^{\circ}] [\gamma' = \gamma] [\gamma' = 90^{\circ}]
    [100] [a' = b \sin \gamma] [a' = b] [a' = b \sin \gamma] [a' = b]
    [b' = c \sin \beta] [b' = c \sin \beta] [b' = c] [b' = c]
    [\gamma' = 180^{\circ} - \alpha^{*}] [\gamma' = 90^{\circ}] [\gamma' = 90^{\circ}] [\gamma' = 90^{\circ}]
    [010] [a' = c \sin \alpha] [a' = c] [a' = c] [a' = c]
    [b' = \alpha \sin \gamma] [b' = a] [b' = a \sin \gamma] [b' = a]
    [\gamma' = 180^{\circ} - \beta^{*}] [\gamma' = \beta] [\gamma' = 90^{\circ}] [\gamma' = 90^{\circ}]

    Projection directionTetragonal
    [001] [a' = a]
    [b' = a]
    [\gamma' = 90^{\circ}]
    [100] [a' = a]
    [b' = c]
    [\gamma' = 90^{\circ}]
    [110] [a' = (a/2) \sqrt{2}]
    [b' = c]
    [\gamma' = 90^{\circ}]

    Projection directionHexagonal
    [001] [a' = a]
    [b' = a]
    [\gamma' = 120^{\circ}]
    [100] [a' = (a/2) \sqrt{3}]
    [b' = c]
    [\gamma' = 90^{\circ}]
    [210] [a' = a/2]
    [b' = c]
    [\gamma' = 90^{\circ}]

    Projection directionRhombohedral
    [111] [a' = {\displaystyle{2 \over \sqrt{3}}}\; a \sin (\alpha/2)]
    [b' = {\displaystyle{2 \over \sqrt{3}}} \;a \sin (\alpha/2)]
    [\gamma' = 120^{\circ}]
    [[1\bar{1}0]] [a' = a \cos (\alpha/2)]
    [b' = a]
    [\gamma' = \delta]§
    [[\bar{2}11]] [a' = {\displaystyle{1 \over \sqrt{3}}}\; a\sqrt{1 + 2 \cos \alpha}]
    [b' = a \sin (\alpha/2)]
    [\gamma' = 90^{\circ}]

    Projection directionCubic
    [001] [a' = a]
    [b' = a]
    [\gamma' = 90^{\circ}]
    [111] [a' = a\sqrt{2/3}]
    [b' = a\sqrt{2/3}]
    [\gamma' = 120^{\circ}]
    [110] [a' = (a/2)\sqrt{2}]
    [b' = a]
    [\gamma' = 90^{\circ}]
    [\cos \alpha^{*} = {\displaystyle{\cos \beta \cos \gamma - \cos \alpha \over \sin \beta \sin \gamma}}; \;\cos \beta^{*} = {\displaystyle{\cos \gamma \cos \alpha - \cos \beta \over \sin \gamma \sin \alpha}};\; \cos \gamma^{*} = {\displaystyle{\cos \alpha \cos \beta - \cos \gamma \over \sin \alpha \sin \beta}}.]
    The entry `Rhombohedral' refers to the primitive rhombohedral cell with [a = b = c], [\alpha = \beta = \gamma] (cf. Table 2.1.1.1[link]).
    §[\cos \delta = {\displaystyle{\cos \alpha \over \cos \alpha/2\;}}].
  • (iv) Location of the origin of the plane group with respect to the unit cell of the space group. The same description is used as for the location of symmetry elements (cf. Section 2.1.3.9[link]).

Example

`Origin at x, 0, 0' or `Origin at [{1 \over 4},{1 \over 4},z]'.

Projections of centred cells (lattices). For centred lattices, two different cases may occur:

  • (i) The projection direction is parallel to a lattice-centring vector. In this case, the projected plane cell is primitive for the centring types A, B, C, I and R. For F-centred lattices, the multiplicity is reduced from 4 to 2 because c-centred plane cells result from projections along face diagonals of three-dimensional F cells.

    Examples

    • (1) A body-centred lattice with centring vector [{1 \over 2}({\bf a} + {\bf b} + {\bf c})] gives a primitive net if projected along [[111]], [[\bar{1}11]], [[1\bar{1}1]] or [[11\bar{1}]].

    • (2) A C-centred lattice projects to a primitive net along the directions [110] and [[1\bar{1}0]].

    • (3) An R-centred lattice described with `hexagonal axes' (triple cell) results in a primitive net if projected along [[\bar{1}11]], [[211]] or [[\bar{1}\bar{2}1]] for the obverse setting. For the reverse setting, the corresponding directions are [[1\bar{1}1]], [[\bar{2}\bar{1}1]], [[121]]; cf. Table 2.1.1.2[link].

  • (ii) The projection direction is not parallel to a lattice-centring vector (general projection direction). In this case, the plane cell has the same multiplicity as the three-dimensional cell. Usually, however, this centred plane cell is unconventional and a transformation is required to obtain the conventional plane cell. This transformation has been carried out for the projection data in this volume.

    Examples

    • (1) Projection along [[010]] of a cubic I-centred cell leads to an unconventional quadratic c-centred plane cell. A simple cell transformation leads to the conventional quadratic p cell.

    • (2) Projection along [[010]] of an orthorhombic I-centred cell leads to a rectangular c-centred plane cell, which is conventional.

    • (3) Projection along [[001]] of an R-centred cell (both in obverse and reverse setting) results in a triple hexagonal plane cell h (the two-dimensional analogue of the H cell, cf. Table 2.1.1.2[link]). A simple cell transformation leads to the conventional hexagonal p cell.

Projections of symmetry elements. A symmetry element of a space group does not project as a symmetry element unless its orientation bears a special relation to the projection direction; all translation components of a symmetry operation along the projection direction vanish, whereas those perpendicular to the projection direction (i.e. parallel to the plane of projection) may be retained. This is summarized in Table 2.1.3.10[link] for the various crystallographic symmetry elements. From this table the following conclusions can be drawn:

  • (i) n-fold rotation axes and n-fold screw axes, as well as rotoinversion axes [\bar{4}], parallel to the projection direction project as n-fold rotation points; a [\bar{3}] axis projects as a sixfold, a [\bar{6}] axis as a threefold rotation point. For the latter, a doubling of the projected electron density occurs owing to the mirror plane normal to the projection direction [(\bar{6} \equiv 3/m)].

    Table 2.1.3.10| top | pdf |
    Projections of crystallographic symmetry elements

    Symmetry element in three dimensionsSymmetry element in projection
    Arbitrary orientation
    [\!\!\!\!\!\!\!\matrix{{\rm Symmetry\ centre}\!\!\!\!\hfill &\bar{1}\hfill\cr {\rm Rotoinversion\ axis} &\bar{3} \equiv 3 \times \bar{1}\hfill}\bigg\}] Rotation point 2 (at projection of centre)
    Parallel to projection direction
    Rotation axis 2; 3; 4; 6 Rotation point 2; 3; 4; 6
    [\!\matrix{\hbox{Screw axis}\!\!\!\!\hfill& 2_{1}\hfill\cr & 3_{1},3_{2}\hfill\cr & 4_{1},4_{2},4_{3}\hfill\cr & 6_{1},6_{2},6_{3},6_{4},6_{5}\hfill\cr}] [\!\matrix{\hbox{Rotation point}\hfill &\!\!\!2\hfill\cr & \!\!\!3\hfill\cr & \!\!\!4\hfill\cr & \!\!\!6\hfill\cr}]
    [\!\matrix{\hbox{Rotoinversion axis}\hfill &\!\!\!\bar{4}\hfill\cr & \!\!\!\bar{6} \equiv 3/m\hfill\cr\cr & \!\!\!\bar{3} \equiv 3 \times \bar{1}\hfill\cr}] [\!\matrix{\hbox{Rotation point}\hfill &\!\!\!4\hfill\cr &\!\!\!3, \hbox{with overlap}\hfill\cr & \quad\!\!\! \hbox{of atoms}\hfill\cr &\!\!\!6\hfill\cr}]
    Reflection plane m Reflection line m
    Glide plane with [\perp] component Glide line g
    Glide plane without [\perp] component Reflection line m
    Normal to projection direction
    [\!\matrix{\hbox{Rotation axis}\hfill &\!\!\!2\semi\ \!\!\!4\semi \ \!\!\!6\hfill\cr & \!\!\!3\hfill\cr}] [\!\matrix{\hbox{Reflection line } m\hfill\cr \hbox{None}\hfill\cr}]
    [\!\matrix{\hbox{Screw axis}\hfill & \!\!\!4_{2}\semi\ 6_{2},6_{4}\hfill\cr & \!\!\!2_{1}\semi\ 4_{1},4_{3}\semi\ 6_{1},6_{3},6_{5}\hfill\cr & \!\!\!3_{1},3_{2}\hfill\cr}] [\!\matrix{\hbox{Reflection line } m\hfill\cr \hbox{Glide line }g\hfill\cr \hbox{None}\hfill\cr}]
    [\!\matrix{\hbox{Rotoinversion axis}\hfill &\!\!\!\bar{4}\hfill\cr & \!\!\!\bar{6} \equiv 3/m\hfill\cr\cr\cr &\!\!\!\bar{3} \equiv 3 \times \bar{1}\hfill}] [\!\matrix{\hbox{Reflection line }m \hbox{ parallel to axis}\hfill\cr \hbox{Reflection line }m \hbox{ perpendicular}\hfill\cr\quad\hbox{to axis (through projection of}\hfill\cr\quad\hbox{inversion point)}\hfill\cr \hbox{Rotation point 2 (at projection}\hfill\cr\quad\hbox{of centre)}\hfill\cr}]
       
    Reflection plane m None, but overlap of atoms
    Glide plane with glide vector t Translation with translation vector t
    The term `with [\perp] component' refers to the component of the glide vector normal to the projection direction.
  • (ii) n-fold rotation axes and n-fold screw axes normal to the projection direction (i.e. parallel to the plane of projection) do not project as symmetry elements if n is odd. If n is even, all rotation and rotoinversion axes project as mirror lines: the same applies to the screw axes [4_{2}, 6_{2}] and [6_{4}] because they contain an axis 2. Screw axes [2_{1}], [4_{1}], [4_{3}], [6_{1}], [6_{3}] and [6_{5}] project as glide lines because they contain [2_{1}].

  • (iii) Reflection planes normal to the projection direction do not project as symmetry elements but lead to a doubling of the projected electron density owing to overlap of atoms. Projection of a glide plane results in an additional translation; the new translation vector is equal to the glide vector of the glide plane. Thus, a reduction of the translation period in that particular direction takes place.

  • (iv) Reflection planes parallel to the projection direction project as reflection lines. Glide planes project as glide lines or as reflection lines, depending upon whether the glide vector has or does not have a component parallel to the projection plane.

  • (v) Centres of symmetry, as well as [\bar{3}] axes in arbitrary orientation, project as twofold rotation points.

A detailed discussion of the correspondence between the symmetry elements and their projections is given in Section 1.4.5.3[link] .

Example: [C12/c1] (15, b unique, cell choice 1)

The C-centred cell has lattice points at 0, 0, 0 and [{1 \over 2},{1 \over 2},0]. In all projections, the centre [\bar{1}] projects as a twofold rotation point.

Projection along [001]: The plane cell is centred; [2 \ \| \ [010]] projects as m; the glide component [(0,0,{1 \over 2}\,)] of glide plane c vanishes and thus c projects as m.

Result: Plane group c2mm (9), [{\bf a}' = {\bf a}_{p}, {\bf b}' = {\bf b}].

Projection along [100]: The periodicity along b is halved because of the C centring; [2 \ \| \ [010]] projects as m; the glide component [(0,0,{1 \over 2}\,)] of glide plane c is retained and thus c projects as g.

Result: Plane group p2gm (7), [{\bf a}' = {\bf b}/2,\ {\bf b}' = {\bf c}_{p}].

Projection along [010]: The periodicity along a is halved because of the C centring; that along c is halved owing to the glide component [(0,0,{1 \over 2}\,)] of glide plane c; [2 \ \| \ [010]] projects as 2.

Result: Plane group p2 (2), [{\bf a}' = {\bf c}/2, {\bf b}' = {\bf a}/2].

Further details about the geometry of projections can be found in publications by Buerger (1965)[link] and Biedl (1966)[link].

2.1.3.15. Monoclinic space groups

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In this volume, space groups are described by one (or at most two) conventional coordinate systems (cf. Sections 2.1.1.2[link] and 2.1.3.2[link]). Eight monoclinic space groups, however, are treated more extensively. In order to provide descriptions for frequently encountered cases, they are given in six versions.

The description of a monoclinic crystal structure in this volume, including its Hermann–Mauguin space-group symbol, depends upon two choices:

  • (i) the unit cell chosen, here called `cell choice';

  • (ii) the labelling of the edges of this cell, especially of the monoclinic symmetry direction (`unique axis'), here called `setting'.

Cell choices. One edge of the cell, i.e. one crystal axis, is always chosen along the monoclinic symmetry direction. The other two edges are located in the plane perpendicular to this direction and coincide with translation vectors in this `monoclinic plane'. It is sensible and common practice (see below) to choose these two basis vectors from the shortest three translation vectors in that plane. They are shown in Fig. 2.1.3.12[link] and labelled e, f and g, in order of increasing length.5 The two shorter vectors span the `reduced mesh' (where mesh means a two-dimensional unit cell), here e and f; for this mesh, the monoclinic angle is [\leq 120^{\circ}], whereas for the other two primitive meshes larger angles are possible.

[Figure 2.1.3.12]

Figure 2.1.3.12 | top | pdf |

The three primitive two-dimensional cells which are spanned by the shortest three translation vectors e, f, g in the monoclinic plane. For the present discussion, the glide vector is considered to be along e and the projection of the centring vector along f.

Other choices of the basis vectors in the monoclinic plane are possible, provided they span a primitive mesh. It turns out, however, that the space-group symbol for any of these (non-reduced) meshes already occurs among the symbols for the three meshes formed by e, f, g in Fig. 2.1.3.12[link]; hence only these cases need be considered. They are designated in this volume as `cell choice 1, 2 or 3' and are depicted in Fig. 2.1.3.4[link]. The transformation matrices for the three cell choices are listed in Table 1.5.1.1[link] .

Settings. The term setting of a cell or of a space group refers to the assignment of labels (a, b, c) and directions to the edges of a given unit cell, resulting in a set of basis vectors a, b, c. (For orthorhombic space groups, the six settings are described and illustrated in Section 2.1.3.6.4[link].)

The symbol for each setting is a shorthand notation for the transformation of a given starting set abc into the setting considered. It is called here `setting symbol'. For instance, the setting symbol bca stands for [{\bf a}' = {\bf b},\quad{\bf b}' = {\bf c},\quad{\bf c}' = {\bf a}]or [({\bf a}', {\bf b}', {\bf c}') = ({\bf a},{\bf b},{\bf c}) \pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr} = ({\bf b},{\bf c},{\bf a}),]where a′, b′, c′ is the new set of basis vectors. [Note that the setting symbol bca means that the `old' vector a changes its label to c′ (and not to b′), that the `old' vector b changes its label to a′ (and not to c′) and that the `old' vector c changes its label to b′ (and not to a′).] Transformation of one setting into another preserves the shape of the cell and its orientation relative to the lattice. The matrices of these transformations have one entry +1 or −1 in each row and column; all other entries are 0.

In monoclinic space groups, one axis, the monoclinic symmetry direction, is unique. Its label must be chosen first and, depending upon this choice, one speaks of `unique axis b', `unique axis c' or `unique axis a'.6 Conventionally, the positive directions of the two further (`oblique') axes are oriented so as to make the monoclinic angle non-acute, i.e. [\geq 90^{\circ}], and the coordinate system right-handed. For the three cell choices, settings obeying this condition and having the same label and direction of the unique axis are considered as one setting; this is illustrated in Fig. 2.1.3.4[link].

Note: These three cases of labelling the monoclinic axis are often called somewhat loosely b-axis, c-axis and a-axis `settings'. It must be realized, however, that the choice of the `unique axis' alone does not define a single setting but only a pair, as for each cell the labels of the two oblique axes can be interchanged.

Table 2.1.3.11[link] lists the setting symbols for the six monoclinic settings in three equivalent forms, starting with the symbols abc (first line), abc (second line) and abc (third line); the unique axis is underlined. These symbols are also found in the headline of the synoptic Table 1.5.4.4[link] , which lists the space-group symbols for all monoclinic settings and cell choices. Again, the corresponding transformation matrices are listed in Table 1.5.1.1[link] .

Table 2.1.3.11| top | pdf |
Monoclinic setting symbols

The settings are distributed between three sets (with two columns in each set) depending on the choice of unique axis. The unique axis is underlined. The setting symbol is a shorthand notation for the transformation of the corresponding starting setting (first row [{\bf a}\underline{\bf b}{\bf c}]; second row [{\bf ab}\underline{\bf c}]; third row [\underline{\bf a}{\bf bc}]). For example, the symbol [{\bf ca}\underline{\bf b}] represents a transformation of the starting setting [{\bf a}\underline{\bf b}{\bf c}] such that a′ = c, b′ = a, [\underline{\bf c}'] = [\underline{\bf b}].

Starting settingUnique axis bUnique axis cUnique axis a
[Scheme scheme82] [Scheme scheme79] [{\bf c}{\underline{\bar{\bf b}}}{\bf a}] [{\bf c}{\bf a}\underline{\bf b}] [{\bf a}{\bf c}\underline{\bar{\bf b}}] [{\underline{\bf b}{\bf c}{\bf a}}] [{\underline{\bar{\bf b}}{\bf a}{\bf c}}]
[Scheme scheme83] [{{\bf b}\underline{\bf c}{\bf a}}] [{{\bf a}\underline{\bar{\bf c}}{\bf b}}] [Scheme scheme80] [{{\bf b}{\bf a}\underline{\bar{\bf c}}}] [{\underline{\bf c}{\bf a}{\bf b}}] [{\underline{\bar{\bf c}}{\bf b}{\bf a}}]
[Scheme scheme84] [{{\bf c}\underline{\bf a}{\bf b}}] [{{\bf b}\underline{\bar{\bf a}}{\bf c}}] [{{\bf b}{\bf c}\underline{\bf a}}] [{{\bf c}{\bf b}\underline{\bar{\bf a}}}] [Scheme scheme81] [{\underline{\bar{\bf a}}{\bf c}{\bf b}}]

Note: An interchange of two axes involves a change of the handedness of the coordinate system. In order to keep the system right-handed, one sign reversal is necessary.

In the space-group tables, only the settings with b and c unique are treated and for these only the left-hand members of the double entries in Table 2.1.3.11[link]. This implies, for instance, that the c-axis setting is obtained from the b-axis setting by cyclic permutation of the labels, i.e. by the transformation [({\bf a}',{\bf b}',\underline{{\bf c}}') = ({\bf a},\underline{{\bf b}},{\bf c}) \pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr} = ({\bf c},{\bf a},\underline{{\bf b}}).]The setting with a unique is also included in the present discussion, as this setting occurs in Table 1.5.4.4[link] . The a-axis setting [\underline{{\bf a}}'{\bf b}'{\bf c}' = \underline{{\bf c}} {\bf a b}] (i.e. [\underline{\bf a}'=\underline{\bf c}], [{\bf b}'={\bf a}], [{\bf c}'={\bf b}]) is obtained from the c-axis setting also by cyclic permutation of the labels and from the b-axis setting by the reverse cyclic permutation: [\underline{{\bf a}}'{\bf b}'{\bf c}' = \underline{{\bf b}} {\bf c a}].

By the conventions described above, the setting of each of the cell choices 1, 2 and 3 is determined once the label and the direction of the unique-axis vector have been selected. Six of the nine resulting possibilities are illustrated in Fig. 2.1.3.4[link].

Cell choices and settings in the present tables. There are five monoclinic space groups for which the Hermann–Mauguin symbols are independent of the cell choice, viz those space groups that do not contain centred lattices or glide planes: [P2\ (3),\ P2_{1}\ (4),\ Pm\ (6),\ P2/m\ (10),\ P2_{1}/m\ (11).]In these cases, description of the space group by one cell choice is sufficient.

For the eight monoclinic space groups with centred lattices or glide planes, the Hermann–Mauguin symbol depends on the choice of the oblique axes with respect to the glide vector and/or the centring vector. These eight space groups are: [\eqalign{&C2\ (5),\ Pc\ (7),\ Cm\ (8),\ Cc\ (9),\ C2/m\ (12),\ P2/c\ (13),\cr &P2_{1}/c\ (14),\ C2/c\ (15).}]Here, the glide vector or the projection of the centring vector onto the monoclinic plane is always directed along one of the vectors e, f or g in Fig. 2.1.3.12[link], i.e. is parallel to the shortest, the second-shortest or the third-shortest translation vector in the monoclinic plane (note that a glide vector and the projection of a centring vector cannot be parallel). This results in three possible orientations of the glide vector or the centring vector with respect to these crystal axes, and thus in three different full Hermann–Mauguin symbols (cf. Section 2.1.3.4[link]) for each setting of a space group.

Table 2.1.3.12[link] lists the symbols for centring types and glide planes for the cell choices 1, 2, 3. The order of the three cell choices is defined as follows: The symbols occurring in the familiar `standard short monoclinic space-group symbols' (see Section 2.1.3.3[link]) define cell choice 1; for `unique axis b', this applies to the centring type C and the glide plane c, as in Cm (8) and [P2_{1}/c\ (14)]. Cell choices 2 and 3 follow from the anti­clockwise order 1–2–3 in Fig. 2.1.3.4[link] and their space-group symbols can be obtained from Table 2.1.3.12[link]. The c-axis and the a-axis settings then are derived from the b-axis setting by cyclic permutations of the axial labels, as described in this section.

Table 2.1.3.12| top | pdf |
Symbols for centring types and glide planes of monoclinic space groups

Setting Cell choice
123
Unique axis b Centring type C A I
Glide planes c, n n, a a, c
Unique axis c Centring type A B I
Glide planes a, n n, b b, a
Unique axis a Centring type B C I
Glide planes b, n n, c c, b

In the two space groups Cc (9) and [C2/c\ (15)], glide planes occur in pairs, i.e. each vector e, f, g is associated either with a glide vector or with the centring vector of the cell. For Pc (7), [P2/c\ (13)] and [P2_{1}/c\ (14)], which contain only one type of glide plane, the left-hand member of each pair of glide planes in Table 2.1.3.12[link] applies.

In the space-group tables of this volume, the following treatments of monoclinic space groups are given:

  • (1) Two complete descriptions for each of the five monoclinic space groups with primitive lattices and without glide planes, one for `unique axis b' and one for `unique axis c', similar to the treatment in IT (1952)[link].

  • (2) A total of six descriptions for each of the eight space groups with centred lattices or glide planes, as follows:

    • (a) One complete description for `unique axis b' and `cell choice' 1. This is considered the standard description of the space group, and its short Hermann–Mauguin symbol is used as the standard symbol of the space group.

      This standard short symbol corresponds to the one symbol of IT (1935)[link] and to that of the b-axis setting in IT (1952)[link], e.g. [P2_{1}/c] or [C2/c]. It serves only to identify the space-group type but carries no information about the setting or cell choice of a particular description. The standard short symbol is given in the headline of every description of a monoclinic space group; cf. Section 2.1.3.3[link].

    • (b) Three condensed (synoptic) descriptions for `unique axis b' and the three `cell choices' 1, 2, 3. Cell choice 1 is repeated to facilitate comparison with the other cell choices. Diagrams are provided to illustrate the three cell choices: cf. Section 2.1.3.6[link].

    • (c) One complete description for `unique axis c' and `cell choice' 1.

    • (d) Three condensed (synoptic) descriptions for `unique axis c' and the three `cell choices' 1, 2, 3. Again cell choice 1 is repeated and appropriate diagrams are provided.

All settings and cell choices are identified by the appropriate full Hermann–Mauguin symbols (cf. Section 2.1.3.4[link]), e.g. [C12/c1] or [I112/b]. For the two space groups Cc (9) and [C2/c\ (15)] with pairs of different glide planes, the `simplest operation rule' for reflections (m > a, b, c > n) is not followed (cf. Section 1.4.1[link] ). Instead, in order to bring out the relations between the various settings and cell choices, the glide-plane symbol always refers to that glide plane which intersects the conventional origin.

Example: No. 15, standard short symbol [C2/c]

The full symbols for the three cell choices (rows) and the three unique axes (columns) read [\matrix{C12/c1 &A12/n1 &I12/a1\cr A112/a &B112/n &I112/b\cr B2/b11 &C2/n11 &I2/c11.\cr}]Application of the priority rule would have resulted in the following symbols: [\matrix{C12/c1 &A12/a1 &I12/a1\cr A112/a &B112/b &I112/a\cr B2/b11 &C2/c11 &I2/b11.\cr}]Here, the transformation properties are obscured.

Comparison with earlier editions of International Tables. In IT (1935)[link], each monoclinic space group was presented in one description only, with b as the unique axis. Hence, only one short Hermann–Mauguin symbol was needed.

In IT (1952)[link], the c-axis setting (first setting) was newly introduced, in addition to the b-axis setting (second setting). This extension was based on a decision of the Stockholm General Assembly of the International Union of Crystallography in 1951 [cf. Acta Cryst. (1951), 4, 569 and Preface to IT (1952)[link]]. According to this decision, the b-axis setting should continue to be accepted as standard for morphological and structural studies. The two settings led to the introduction of full Hermann–Mauguin symbols for all 13 monoclinic space groups (e.g. [P12_{1}/c1] and [P112_{1}/b]) and of two different standard short symbols (e.g. [P2_{1}/c] and [P2_{1}/b]) for the eight space groups with centred lattices or glide planes [cf. p. 545 of IT (1952)[link]]. In the present volume (as in the previous editions of this series), only one of these standard short symbols is retained (see above and Section 2.1.3.3[link]).

The c-axis setting (primed labels) was obtained from the b-axis setting (unprimed labels) by the following transformation: [({\bf a}',{\bf b}',\underline{{\bf c}}') = ({\bf a},\underline{{\bf b}},{\bf c}) \pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr} = ({\bf a},{\bf c},\bar{\underline{{\bf b}}}).]This corresponds to an interchange of two labels and not to the more logical cyclic permutation, as used in all editions of this series. The reason for this particular transformation was to obtain short space-group symbols that indicate the setting unambiguously; thus the lattice letters were chosen as C (b-axis setting) and B (c-axis setting). The use of A in either case would not have distinguished between the two settings [cf. pp. 7, 55 and 543 of IT (1952)[link]; see also Table 2.1.3.12[link]].

As a consequence of the different transformations between b- and c-axis settings in IT (1952)[link] and in this volume (and all editions of this series), some space-group symbols have changed. This is apparent from a comparison of pairs such as [P12_{1}/c1] & [P112_{1}/b] and [C12/c1] & [B112/b] in IT (1952)[link] with the corresponding pairs in this volume, [P12_{1}/c1] & [P112_{1}/a] and [C12/c1] & [A112/a]. The symbols with B-centred cells appear now for cell choice 2, as can be seen from Table 2.1.3.12[link].

Selection of monoclinic cell. In practice, the selection of the (right-handed) unit cell of a monoclinic crystal can be approached in three ways, whereby the axes refer to the b-unique setting; for c unique similar considerations apply:

  • (i) Irrespective of their lengths, the basis vectors are chosen such that, in Fig. 2.1.3.12[link], one obtains [{\bf c} = {\bf e}], [{\bf a} = {\bf f}] and b normal to a and c pointing upwards. This corresponds to a selection of cell choice 1. It ensures that the crystal structure can always be referred directly to the description and the space-group symbol in IT (1935)[link] and IT (1952)[link]. However, this is at the expense of possibly using a non-reduced and, in many cases, even a very awkward cell.

  • (ii) Selection of the reduced mesh, i.e. the shortest two translation vectors in the monoclinic plane are taken as axes and labelled a and c, with either [a \,\lt\, c] or [c \,\lt \,a]. This results with equal probability in one of the three cell choices described in the present volume.

  • (iii) Selection of the cell on special grounds, e.g. to compare the structure under consideration with another related crystal structure. This may result again in a non-reduced cell and it may even necessitate use of the a-axis setting. In all these cases, the coordinate system chosen should be carefully explained in the description of the structure.

2.1.3.16. Crystallographic groups in one dimension

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In one dimension, only one crystal family, one crystal system and one Bravais lattice exist. No name or common symbol is required for any of them. All one-dimensional lattices are primitive, which is symbolized by the script letter [{\scr p}]; cf. Table 2.1.1.1[link].

There occur two types of one-dimensional point groups, 1 and [m \equiv \bar{1}]. The latter contains reflections through a point (reflection point or mirror point). This operation can also be described as inversion through a point, thus [m \equiv \bar{1}] for one dimension; cf. Section 2.1.2[link].

Two types of line groups (one-dimensional space groups) exist, with Hermann–Mauguin symbols [{\scr p}1] and [{\scr p}m \equiv {\scr p}\bar{1}], which are illustrated in Fig. 2.1.3.13[link]. Line group [{\scr p}1], which consists of one-dimensional translations only, has merely one (general) position with coordinate x. Line group [{\scr p}m] consists of one-dimensional translations and reflections through points. It has one general and two special positions. The coordinates of the general position are x and [\bar{x}]; the coordinate of one special position is 0, that of the other [{1 \over 2}]. The site symmetries of both special positions are [m \equiv \bar{1}]. For [{\scr p}1], the origin is arbitrary, for [{\scr p}m] it is at a reflection point.

[Figure 2.1.3.13]

Figure 2.1.3.13 | top | pdf |

The two line groups (one-dimensional space groups). Small circles are reflection points; large circles represent the general position; in line group [{\scr p}1], the vertical bars are the origins of the unit cells.

The one-dimensional point groups are of interest as `edge symmetries' of two-dimensional `edge forms'; they are listed in Table 3.2.3.1[link] . The one-dimensional space groups occur as projection and section symmetries of crystal structures.

2.1.4. Computer production of the space-group tables

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P. Konstantinovf and K. Mommae

The space-group tables for the first (1983) edition of Volume A were produced and typeset by a computer-aided process as described by Fokkema (1983)[link]. However, the computer programs used and the data files were then lost. All corrections in the subsequent three editions were done by photocopying and `cut-and-paste' work based on the printed version of the book.

Hence, in October 1997, a new project for the electronic production of the fifth edition of Volume A of International Tables for Crystallography was started. Part of this project concerned the computer generation of the plane- and space-group tables [Parts 6 and 7 of IT A (2002)[link]], excluding the space-group diagrams. The aim was to be able to produce PostScript and Portable Document Format (PDF) documents that could be used for printing and displaying the tables. The layout of the tables had to follow exactly that of the previous editions of Volume A. Having the space-group tables in electronic form opened the way for easy corrections and modifications of later editions and made possible the online edition of Volume A in 2006 (http://it.iucr.org/A/ ). Although the plane- and space-group data were encoded in a format designed for printing, they were later machine-read and transformed to other electronic formats, and were incorporated into the data files on which the Symmetry Database in the online version of International Tables for Crystallography (http://it.iucr.org/resources/symmetrydatabase/ ) and the online services offered by the Bilbao Crystallographic Server (http://www.cryst.ehu.es/ ) are based.

The [\LaTeX] document preparation system (Lamport, 1994[link]), which is based on the [\TeX] typesetting software, was used for the preparation of these tables. It was chosen because of its high versatility and general availability on almost any computer platform. It is also worth noting the longevity of the system. Even though the hardware and operating system software used almost 20 years ago, when the project began, are now obsolete, the software is still available on all major modern computer systems. All the material – [\LaTeX] code and data – created for the fifth edition of 2002 was re-used in the preparation of the current edition with only small changes concerning the presentation.

A separate file was created for each plane and space group and each setting. These data files contain the information listed in the plane- and space-group tables and are encoded using standard [\LaTeX] constructs. As is customary, these specially designed commands and environments are defined in a separate package file, which essentially contains routines, called macros, that control the typographical layout of the data. Thus, the main principle of [\LaTeX] – that of keeping content and presentation separate – was followed as closely as possible.

The final typesetting of all the plane- and space-group tables was done by running a single computer job. References in the tables from one page to another are automatically computed. The result is a PostScript file which can be fed to a laser printer or other printing or typesetting equipment. It can also be easily converted to a PDF file. It is also possible to generate the output for just one group, as accessed in the online edition, or a series of groups.

The different types of data in the [\LaTeX] files were either keyed by hand or computer generated, and were additionally checked by specially written programs. The preparation of the data files can be summarized as follows:

Headline, Origin, Asymmetric unit: hand keyed.

Symmetry operations: partly created by a computer program. The algorithm for the derivation of symmetry operations from their matrix representation is similar to that described in the literature (cf. Section 1.2.2[link] ; see also Hahn & Wondratschek, 1994[link]). The data were additionally checked by automatic comparison with the output of the computer program SPACER (Stróż, 1997[link]).

Generators: transferred automatically from the data for Volume A1 of International Tables for Crystallography (2010)[link], hereafter referred to as IT A1.

General positions: created by a program. The algorithm uses the well known generating process for space groups based on their solvability property (cf. Section 1.4.3[link] ).

Special positions: The first representatives of the Wyckoff positions were typed in by hand. The Wyckoff letters are assigned automatically by the [\TeX] macros according to the order of appearance of the special positions in the data file. The multiplicity of the position, the oriented site-symmetry symbol and the rest of the representatives of the Wyckoff position were generated by a program. Again, the data were compared with the results of the program SPACER.

Reflection conditions: hand keyed. A program for automatic checking of the special-position coordinates and the corresponding reflection conditions with h, k, l ranging from −20 to 20 was developed.

Symmetry of special projections: hand keyed.

Maximal subgroups and minimal supergroups: this information appeared in the fifth revised edition of Volume A, as in the previous editions. Most of the data were automatically transferred from the data files used for the production of IT A1. The macros for typesetting these data were re-implemented to obtain exactly the layout of Volume A. For the current edition these data have been omitted by redefining the macros to ignore the content, which is still present in the data files.

The symmetry-element diagrams were scanned and processed in the IUCr Editorial Office in Chester.

In the first edition of IT A published in 1983, the general-position diagrams of the cubic groups presented in the 1935[link] edition of Internationale Tabellen zur Bestimmung von Kristallstrukturen were replaced by small stereodiagrams. At that time, such stereodiagrams were probably the easiest way to allow three-dimensional visualization, and the same stereodiagrams were reproduced in the following editions. However, the sizes of the stereodiagrams were limited by the page size, so they were very small, and they also lacked any indication of general-position `enantiomorph' points. The situation has changed a lot since then and three-dimensional visualization of the general positions is easily achieved with structure-drawing programs. Therefore, in this sixth edition, new general-position diagrams of the cubic groups, which are similar to those of the non-cubic groups, were created with a focus on better two-dimensional representation in print.

The new diagrams were created by K. Momma using the computer program VESTA (Momma & Izumi, 2011[link]), which was extended for this purpose. The diagrams that were generated were carefully checked by comparing them with the original diagrams in the 1935 edition. The coordinates of general positions are slightly different from those used in the original diagrams and were chosen so that the general positions overlap as little as possible in the two-dimensional orthogonal projection of the diagrams. The coordinates of general positions used are:

  • (i) 0.0375, 0.1125, 0.098 for space-group Nos. 198, 199, 205, 206, 212, 213, 214, 220 and 230;

  • (ii) 0.065, 0.12, 0.048 for the second diagrams of space-group Nos. 212, 214 and 230;

  • (iii) 0.435, 0.38, 0.452 for the second diagram of space-group No. 213;

  • (iv) 0.31, 0.095, 0.21 for the second diagram of space-group No. 220; and

  • (v) 0.048x, 0.12y, 0.089z for the remaining space groups, where x, y, z is the first coordinate triplet of the highest symmetry Wyckoff position of the space group.

In addition, three-dimensional-style tilted general-position diagrams were created by VESTA for each of the ten space groups of the [m\bar3m] crystal class. These diagrams can be reproduced and visualized in three dimensions using VESTA. They were provided in the form of Portable Network Graphics (PNG) raster images and were included in the page layout of the space-group tables with some scaling and cropping.

The preparation of the plane- and space-group tables was carried out on various computer platforms in Sofia, Bilbao, Karlsruhe, Tsukuba and Chester. The development of the computer programs and the layout macros in the [\LaTeX] package file, and the preparation of the diagrams were done in parallel by different members of the team, which included Asen Kirov (Sofia), Eli Kroumova (Bilbao), Koichi Momma, Preslav Konstantinov and Mois Aroyo, and staff at the Editorial Office in Chester.

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