Contents and arrangement of the tables

Hahn, Th.; Looijenga-Vos, A.

doi:10.1107/97809553602060000505

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. A. ch. 2.2, pp. 17-41
https://doi.org/10.1107/97809553602060000505

Chapter 2.2. Contents and arrangement of the tables

Th. Hahn^a ^* and A. Looijenga-Vos^b ^‡

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

This chapter and Chapter 2.1 form the main guide to understanding and using the plane-group and space-group tables in Parts 6 and 7 . Chapter 2.2 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, subgroups and supergroups, monoclinic space groups, and the two crystallographic space groups in one dimension (line groups).

Keywords: space groups; plane groups; origin; international space-group symbols; Hermann–Mauguin symbols; Schoenflies symbols; Patterson symmetry; space-group diagrams; asymmetric unit; symmetry operations; generators; Wyckoff positions; site symmetry; reflection conditions; maximal subgroups; minimal supergroups; monoclinic space groups; one-dimensional groups.

2.2.1. General layout

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The presentation of the plane-group and space-group data in Parts 6 and 7 follows the style of the previous editions of International Tables. The entries for a space group are printed on two facing pages as shown below; an example (Cmm2, No. 35) is provided inside the front and back covers . Deviations from this standard sequence (mainly for cubic space groups) are indicated on the relevant pages.

Left-hand page:
(1)	Headline
(2)	Diagrams for the symmetry elements and the general position (for graphical symbols of symmetry elements see Chapter 1.4 )
(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)
(10)	Maximal non-isomorphic subgroups
(11)	Maximal isomorphic subgroups of lowest index
(12)	Minimal non-isomorphic supergroups

Note: Symbols for Lattice complexes of the plane groups and space groups are given in Tables 14.2.3.1

and 14.2.3.2

. Normalizers of space groups are listed in Part 15

2.2.2. 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 2.2.7 )

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 2.2.16 .
(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). For both descriptions, the same space-group symbol is used. The relations between the cell parameters of the two cells are listed in Chapter 2.1 .

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), 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. Chapter 1.2 .

2.2.3. Headline

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The description of each plane group or space group starts with a headline on a left-hand page, 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 Chapter 12.2 , a brief summary in Section 2.2.4 .

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. $[P2_{1}/c]$ for No. 14 or 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 2.2.16 ).
(2) The Schoenflies symbol for the space group.

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. Chapter 12.1 ).
(4) The name of the crystal system (cf. Table 2.1.2.1 ).

Second line

(5) The sequential number of the plane or space group, as introduced in IT (1952).
(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 2.2.5 ).

Third line

This line is used, where appropriate, to indicate origin choices, settings, cell choices and coordinate axes (see Section 2.2.2 ). For five orthorhombic space groups, an entry `Former space-group symbol' is given; cf. Chapter 1.3 , Note (x) .

2.2.4. International (Hermann–Mauguin) symbols for plane groups and space groups (cf. Chapter 12.2 )

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2.2.4.1. Present symbols

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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 Chapter 1.2 . 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). Symmetry directions occur either as singular directions (as in the monoclinic and orthorhombic crystal systems) or as sets of symmetrically equivalent symmetry directions (as in the higher-symmetrical 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 2.2.4.1 . According to their position in this sequence, the symmetry directions are referred to as `primary', `secondary' and `tertiary' directions.

Table 2.2.4.1| 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.

Lattice	Symmetry direction (position in Hermann–Mauguin symbol)
Lattice	Primary	Secondary	Tertiary
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 Section 2.2.4.1

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. Chapter 1.3 ) 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').

For the different crystal lattices, the Hermann–Mauguin space-group symbols have the following form:

(i) Triclinic lattices have no symmetry direction because they have, in addition to translations, only centres of symmetry, $[\bar{1}]$ . Thus, only two triclinic space groups, P1 (1) and $[P\bar{1}]$ (2), exist.
(ii) Monoclinic lattices have one symmetry direction. Thus, for monoclinic space groups, only one position after the centring letter is needed. This is used in the short Hermann–Mauguin symbols, as in $[P2_{1}]$ . Conventionally, the symmetry direction is labelled either b (`unique axis b') or c (`unique axis c').

In order to distinguish between the different settings, the full Hermann–Mauguin symbol contains two extra entries `1'. They indicate those two axial directions that are not symmetry directions of the lattice. Thus, the symbols P121, P112 and P211 show that the b axis, c axis and a axis, respectively, is the unique axis. Similar considerations apply to the three rectangular plane groups pm, pg and cm (e.g. plane group No. 5: short symbol cm, full symbol c1m1 or c11m).
(iii) Rhombohedral lattices have two kinds of symmetry directions. Thus, the symbols of the seven rhombohedral space groups contain only two entries after the letter R, as in R3m or R3c.
(iv) Orthorhombic , tetragonal , hexagonal and cubic lattices have three kinds of symmetry directions. Hence, the corresponding space-group symbols have three entries after the centring letter, as in Pmna, P3m1, P6cc or $[Ia\bar{3}d]$ .

Lattice symmetry directions that carry no symmetry elements for the space group under consideration are represented by the symbol `1', as in P3m1 and P31m. If no misinterpretation is possible, entries `1' at the end of a space-group symbol are omitted, as in P6 (instead of P611), $[R\bar{3}]$ (instead of $[R\bar{3}1]$ ), $[I4_{1}]$ (instead of $[I4_{1}11]$ ), F23 (instead of F231); similarly for the plane groups.

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 two symbols are $[I4_{1}/a\;\bar{3}\;2/d]$ and $[Ia\bar{3}d]$ .

Many space groups contain more kinds of symmetry elements than are indicated in the full symbol (`additional symmetry elements', cf. Chapter 4.1 ). A complete listing of the symmetry elements is given in Tables 4.2.1.1 and 4.3.2.1 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.

2.2.4.2. Changes in Hermann–Mauguin space-group symbols as compared with the 1952 and 1935 editions of International Tables

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Extensive changes in the space-group symbols were applied in IT (1952) as compared with the original Hermann–Mauguin symbols of IT (1935), 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). In the present edition, the symbols of the 1952 edition are retained, except for the following four cases (cf. Chapter 12.4 ).

(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 listed plane groups in Table 2.2.4.2 .

Table 2.2.4.2| 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: $[\quad \hbox{No. 2: Change from }p211 \hbox{ to }p2.\hfill]$

With these changes, the symbols of the two-dimensional groups follow the rules that were introduced in IT (1952) 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 4.3.2.1 ; further details are given in Section 2.2.16 .

For standard short monoclinic space-group symbols see Sections 2.2.3 and 2.2.16 .
(iii) Cubic 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 recent introduction of the `double glide plane' e into five space-group symbols, see Chapter 1.3 , Note (x) .

2.2.5. Patterson symmetry

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The entry Patterson symmetry in the headline gives the space group of the Patterson function P(x, y, z). With neglect of anomalous dispersion, this function is defined by the formula $[P(x,y,z) = {1 \over V} \sum_{h} \sum_{k} \sum_{l} |F(hkl)|^{2} \cos 2\pi (hx + ky + lz).]$ The Patterson function represents the convolution of a structure with its inverse or the pair-correlation function of a structure. A detailed discussion of its use for structure determination is given by Buerger (1959). The space group of the Patterson function is identical to that of the `vector set' of the structure, and is thus always centrosymmetric and symmorphic .¹

The symbol for the Patterson space group of a crystal structure can be deduced from that of its space group in two steps:

(i) Glide planes and screw axes have to be replaced by the corresponding mirror planes and rotation axes, resulting in a symmorphic space group.
(ii) If this symmorphic space group is not centrosymmetric, inversions have to be added.

There are 7 different Patterson symmetries in two dimensions and 24 in three dimensions. They are listed in Table 2.2.5.1 . Account is taken of the fact that the Laue class $[\bar{3}m]$ combines in two ways with the hexagonal translation lattice, namely as $[\bar{3}m1]$ and as $[\bar{3}1m]$ .

Table 2.2.5.1| top | pdf |
Patterson symmetries for two and three dimensions

Laue class	Lattice type					Patterson symmetry (with space-group number)
Two dimensions
2	p					p2 (2)
2mm	p	c				p2mm (6)	c2mm (9)
4	p					p4 (10)
4mm	p					p4mm (11)
6	p					p6 (16)
6mm	p					p6mm (17)
Three dimensions
$[\bar{1}]$	P					$[P\bar{1}\ (2)]$
	P	C				$[P2/m\ (10)]$	$[C2/m\ (12)]$
mmm	P	C	I	F		Pmmm (47)	Cmmm (65)	Immm (71)	Fmmm (69)
	P		I			$[P4/m\ (83)]$		$[I4/m\ (87)]$
	P		I			$[P4/mmm\ (123)]$		$[I4/mmm\ (139)]$
$[\bar{3}]$	P				R	$[P\bar{3}\ (147)]$				$[R\bar{3}\ (148)]$
$[\!\left\{\matrix{\bar{3}m1\hfill\cr \bar{3}1m\hfill\cr}\right.]$	P				R	$[P\bar{3}m1\ (164)]$				$[R\bar{3}m\ (166)]$
	P					$[P\bar{3}1m\ (162)]$
	P					$[P6/m\ (175)]$
	P					$[P6/mmm\ (191)]$
$[m\bar{3}]$	P		I	F		$[Pm\bar{3}\ (200)]$		$[Im\bar{3}\ (204)]$	$[Fm\bar{3}\ (202)]$
$[m\bar{3}m]$	P		I	F		$[Pm\bar{3}m\ (221)]$		$[Im\bar{3}m\ (229)]$	$[Fm\bar{3}m\ (225)]$

Note: For the four orthorhombic space groups with A cells (Nos. 38–41), the standard symbol for their Patterson symmetry, Cmmm, is added (between parentheses) after the actual symbol Ammm in the space-group tables.

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 ). In the absence of anomalous dispersion, the Laue class of a crystal expresses the point symmetry of its diffraction record, i.e. the symmetry of the reciprocal lattice weighted with I(hkl).

2.2.6. 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 symmetrically equivalent points of the general position.

All 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 2.2.6.6 ), 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 $[c_{p}]$ . This applies to one or two axes for triclinic and monoclinic space groups (cf. Figs. 2.2.6.1 to 2.2.6.3 ), as well as to the three rhombohedral axes in Fig. 2.2.6.9 .

The graphical symbols for symmetry elements, as used in the drawings, are displayed in Chapter 1.4 .

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 cubic diagrams). In the space-group diagrams, only the symmetry element at height h is indicated (cf. Chapter 1.4 ).

Schematic representations of the diagrams, displaying the origin, the labels of the axes, and the projection direction [uvw], are given in Figs. 2.2.6.1 to 2.2.6.10 (except Fig. 2.2.6.6). The general-position diagrams are indicated by the letter $[\hbox{\sf G}]$ .

2.2.6.1. 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.

2.2.6.2. 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. 2.2.6.1 .

Figure 2.2.6.1| 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, reference is made to Section 9.2.2 .

2.2.6.3. Monoclinic space groups (cf. Sections 2.2.2 and 2.2.16 )

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The `complete treatment' of each of the two settings contains four diagrams (Figs. 2.2.6.2 and 2.2.6.3 ). 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 2.2.6.2| top | pdf |

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

Figure 2.2.6.3| top | pdf |

Monoclinic space groups, setting with unique axis c.

The `synoptic descriptions' of the three cell choices (for each setting) are headed by a pair of diagrams, as illustrated in Fig. 2.2.6.4 . 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. 2.2.6.4 . 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 2.2.6.4| 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 2.2.16 ). The unique axis points upwards from the page.

2.2.6.4. 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. 2.2.6.5 . 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 2.2.6.5| 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 2.2.16 : $[{\bf abc} \quad {\bf ba}\overline{{\bf c}} \quad {\bf cab} \quad \overline{{\bf c}}{\bf ba} \quad {\bf bca} \quad {\bf a}\overline{{\bf c}}{\bf b.}]$ The symbol for each setting, here called `setting symbol', is a short-hand notation for the transformation of the basis vectors of the standard setting, a, b, c, into those of the setting considered. 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 $[({\bf a}'{\bf b}'{\bf c}') = ({\bf abc}) \pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr} = ({\bf cab}),]$ 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 $[({\bf a}'{\bf b}'{\bf c}') = ({\bf abc}) \pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr} = ({\bf ba}\overline{{\bf c}}).]$ The six orthorhombic settings correspond to six Hermann–Mauguin symbols which, however, need not all be different; cf. Table 2.2.6.1 .²

Table 2.2.6.1| top | pdf |
Numbers of distinct projections and different Hermann–Mauguin symbols for the orthorhombic space groups (space-group number placed between parentheses), listed according to point group as indicated in the headline

Number of distinct projections	222	mm2
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)	$[C\ 2/m\;2/c\;2_{1}/m\ (63)]$
		Abm2 (39)	$[C\ 2/m\;2/c\;2_{1}/a\ (64)]$
		Ama2 (40)	$[I\ 2_{1}/m\;2_{1}/m\;2_{1}/a\ (74)]$
		Aba2 (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)	$[C\ 2/m\;2/m\;2/a\ (67)]$
		Iba2 (45)	$[C\ 2/c\;2/c\;2/a\ (68)]$
			$[I\ 2/b\;2/a\;2/m\ (72)]$
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 `standard Hermann–Mauguin symbol'. In the present edition, 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. 2.2.6.5 . 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 over 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. 2.2.6.6 , which correspond to those of Fig. 2.2.6.5 . 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. 2.2.6.6 with the diagrams for the space group under consideration.

Figure 2.2.6.6| 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 2.2.6.1 . 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 listed in the second paragraph of this section form the column headings of the orthorhombic entries in Table 4.3.2.1 , 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. 2.2.6.6 .

2.2.6.5. 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 IT (1935) and IT (1952). 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. 2.2.6.7 and 2.2.6.8 .

Figure 2.2.6.7| top | pdf |

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

Figure 2.2.6.8| top | pdf |

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

2.2.6.6. Rhombohedral (trigonal R) space groups

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The seven rhombohedral R space groups are treated in two versions, the first based on `hexagonal axes' (obverse setting ), the second on `rhombohedral axes' (cf. Sections 2.1.3 and 2.2.2 ). The pairs of diagrams are similar to those in IT (1952); the left or top one displays the symmetry elements, the right or bottom one the general position. This is illustrated in Fig. 2.2.6.9 , which gives the axes a and b of the triple hexagonal cell and the projections of the axes of the primitive rhombohedral cell, labelled $[a_{p},\ b_{p}]$ and $[c_{p}]$ . 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 2.2.6.9| top | pdf |

Rhombohedral R space groups. Obverse triple hexagonal cell with `hexagonal axes' a, b and primitive rhombohedral cell with projections of `rhombohedral axes' $[a_{p}, b_{p}, c_{p}]$ . Note: In the actual space-group diagrams only the upper edges (full lines), not the lower edges (dashed lines) of the primitive rhombohedral cell are shown ( $[\hbox{\sf G}]$ = general-position diagram).

The pairs of drawings for the hexagonal and the rhombohedral descriptions of a space group are the same. In the rhombohedral descriptions of space groups Nos. 166 and 167, $[R\bar{3}m]$ and $[R\bar{3}c]$ , the diagrams are omitted for reasons of space, and the reader is referred to the drawings in the hexagonal descriptions.

2.2.6.7. Cubic space groups

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For each cubic space group, one projection of the symmetry elements along [001] is given, Fig. 2.2.6.10 ; for details of the diagrams, see Chapter 1.4 and Buerger (1956). 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. The three stereoscopic general-position diagrams in the lower part of the page are explained below.

Figure 2.2.6.10| top | pdf |

Cubic space groups ( $[\hbox{\sf G}]$ = general-position stereodiagrams).

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

2.2.6.8. Diagrams of the general position

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(i) 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 and (or and ) 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 −.

Points represented by $[\bigcirc]$ and $[\def\circcom{\mathop{\bigcirc\hskip -6.5pt{\raise2pt\hbox{,}} \ }} \circcom]$ are related by inversion, rotoinversion or mirror symmetry and are thus enantiomorphs of each other. If $[\bigcirc]$ were to be occupied by the centre of a right-handed molecule, the molecule at $[\def\circcom{\mathop{\bigcirc\hskip -6.5pt{\raise2pt\hbox{,}} \ }} \circcom]$ would be left-handed.

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{}},}$}}\hskip-.05em \lower0.2em\hbox{\vrule height 0.9em}}}\;\;}\circhyp +]$ .

(ii) Stereodiagrams for cubic space groups (Fig. 2.2.6.10 )

For each cubic space group, three diagrams are given with the points of the general position as vertices of transparent polyhedra. (The spheres at the vertices are depicted as opaque, however.) For the `starting point', the same coordinate values, x = 0.048, y = 0.12, z = 0.08, as in the cubic diagrams of IT (1935) have been used. The diagram on the left corresponds to that published in IT (1935); in this figure, the height h of the centre of each polyhedron is given, if different from zero. For space groups Nos. 198, 199 and 220, h refers to the special point to which the polyhedron (triangle) is connected by dotted lines. In all diagrams, polyhedra with height 1 are omitted. A grid of four squares is drawn to represent the four quarters of the basal plane of the cell.

Of the three diagrams, the image on the left and the central one form a stereopair, as well as the central image and that on the right (Langlet, 1972 ). The stereoscopic effect is obtained by a 6° tilt between each view. The separation of neighbouring images has the standard value of 55 mm. The presence of the two stereopairs has the advantage that difficulties in seeing the polyhedra due to overlap in one pair do not occur in the other. For space groups Nos. 219, 226 and 228, where the number of points was too large for one set, two sets of three drawings are provided, one for the upper and one for the lower half of the cell.

Notes:

(i) For space group $[P4_{1}32\ (213)]$ , the coordinates $[\bar{x},\bar{y},\bar{z}]$ have been chosen for the `starting point' to bring out the enantiomorphism with $[P4_{3}32\ (212)]$ .
(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.

Readers who wish to compare other approaches to space-group diagrams and their history are referred to IT (1935), IT (1952) and the following publications: Astbury & Yardley (1924); Belov et al. (1980); Buerger (1956); Fedorov (1895 ; English translation, 1971); Friedel (1926); Hilton (1903); Niggli (1919); Schiebold (1929).

2.2.7. Origin

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

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

Example: $[I4_{1}/amd]$ (141).
(ii) For noncentrosymmetric space groups, the origin is at a point of highest site symmetry, as in $[P{\bar 6}m2]$ (187). If no site symmetry is higher than 1, except for the cases listed below under (iii), the origin is placed on a screw axis, or a glide plane, or at the intersection of several such symmetry elements.

Examples: $[Pca2_{1}]$ (29); $[P6_{1}]$ (169).
(iii) In space group $[P2_{1}2_{1}2_{1}]$ (19), the origin is chosen in such a way that it is surrounded symmetrically by three pairs of $[2_{1}]$ axes. This principle is maintained in the following noncentrosymmetric cubic space groups of classes 23 and 432, which contain $[P2_{1}2_{1}2_{1}]$ as subgroup: $[P2_{1}3]$ (198), $[I2_{1}3]$ (199), $[F4_{1}32]$ (210). It has been extended to other noncentrosymmetric orthorhombic and cubic space groups with $[P2_{1}2_{1}2_{1}]$ as subgroup, even though in these cases points of higher site symmetry are available: $[I2_{1}2_{1}2_{1}]$ (24), $[P4_{3}32]$ (212), $[P4_{1}32]$ (213), $[I4_{1}32]$ (214).

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

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

2.2.7.1. Origin statement

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

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

Table 2.2.7.1| top | pdf |
Examples of origin statements

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

These examples illustrate the following points:

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

The presence of a symmetry centre at the origin is always stated explicitly, either by giving the symbol $[\bar{1}]$ (E1 and E4) or by the words `at centre', followed by the full site symmetry between parentheses (E2 and E5). This completes the origin line, if no further glide planes or screw axes are present at the origin.
(ii) If glide planes or screw axes are present, as in examples E4–E11, they are given in the order of the symmetry directions listed in Table 2.2.4.1 . Such a set of symmetry elements is described here in the form of a `point-group-like' symbol (although it does not describe a group). With the help of the orthorhombic symmetry directions, the symbols in E4–E6 can be interpreted easily. The shortened notation of E6 and E7 is used for space groups of crystal classes mm2, 4mm, $[\bar{4}2m]$ , 3m, 6mm and $[\bar{6}2m]$ if the site symmetry at the origin can be easily recognized from the shortened symbol.
(iii) For the tetragonal, trigonal and hexagonal space groups, the situation is more complicated than for the orthorhombic groups. The tetragonal space groups have one primary, two secondary and two tertiary symmetry directions. For hexagonal groups, these numbers are one, three and three (Table 2.2.4.1 ). If the symmetry elements passing through the origin are the same for the two (three) secondary or the two (three) tertiary directions, only one entry is given at the relevant position of the origin statement [example E7: `on 41g' instead of `on 41(g, g)']. An exception occurs for the site-symmetry group 2mm (example E8), which is always written in full rather than as 2m1.

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

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

2.2.8. 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, 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). Fig. 2.2.8.1 shows the boundary planes occurring in the tetragonal, trigonal and hexagonal systems, together with their algebraic equations.

Figure 2.2.8.1| 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 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, , , . The asymmetric unit is defined by $[\eqalign{&\qquad 0 \leq x \leq {\textstyle{2 \over 3}}{\hbox{;}} \quad 0 \leq y \leq {\textstyle{2 \over 3}}{\hbox{;}} \quad 0 \leq z \leq {\textstyle{1 \over 6}}{\hbox{;}}\cr &\qquad x \leq (1 + y)/2{\hbox{;}} \quad y \leq \min(1 - x, (1 + x)/2)\cr &\matrix{\hbox{Vertices:} &0, 0, 0 &{1 \over 2}, 0, 0 &{2 \over 3}, {1 \over 3}, 0 &{1 \over 3}, {2 \over 3}, 0 &0, {1 \over 2}, 0\hfill \cr &0, 0, {1 \over 6} &{1 \over 2}, 0, {1 \over 6} &{2 \over 3}, {1 \over 3}, {1 \over 6} &{1 \over 3}, {2 \over 3}, {1 \over 6} &0, {1 \over 2}, {1 \over 6}.\cr}}]$ 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 , 1960 ) in connection with Fourier summations.

2.2.9. Symmetry operations

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As explained in Sections 8.1.6 and 11.1.1 , 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.

2.2.9.1. Numbering scheme

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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, both lists contain the same number of entries.

For space groups with centred cells, to the one block General position several (2, 3 or 4) blocks Symmetry operations correspond. 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 himself (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.

In space groups with two origins where a `symmetry element' and an `additional symmetry element' are of different type (e.g. mirror versus glide plane, rotation versus screw axis, Tables 4.1.2.2 and 4.1.2.3 ), the origin shift may interchange the two different types in the same location (referred to the appropriate origin) under the same number (p). Thus, in [P4/nmm] (129), (p) = (7) represents a $[\bar 2]$ and a 2₁ axis, both in [x, x, 0] , whereas (p) = (16) represents a g and an m plane, both in [x, x, z] .

2.2.9.2. Designation of symmetry operations

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

(i) A symbol denoting the type of the symmetry operation (cf. Chapter 1.3 ), 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 given in addition.

Details of this symbolism are presented in Section 11.1.2 .

Examples

(1) $[a\ \ \ x,y,{1 \over 4}]$

Glide reflection with glide component $[({1 \over 2},0,0)]$ through the plane $[x,y,{1 \over 4}]$ , i.e. the plane parallel to (001) at $[z = {1 \over 4}]$ .
(2) $[\bar{4}^{+}\ \ \ {1 \over 4},{1 \over 4},z\semi \ \ {1 \over 4},{1 \over 4},{1 \over 4}]$

Fourfold rotoinversion, consisting of a counter clockwise rotation by 90° around the line $[{1 \over 4},{1 \over 4}, z]$ , followed by an inversion through the point $[{1 \over 4},{1 \over 4},{1 \over 4}]$ .
(3) $[g({1 \over 4},{1 \over 4},{1 \over 2})\ \ \ x,x,z]$

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.
(4) $[g({1 \over 3},{1 \over 6},{1 \over 6})\ \ \ 2x - {1 \over 2}, x,z]$ (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).
(5) 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 $[0,{1 \over 4},z]$ .

2.2.10. Generators

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The line Generators selected states the symmetry operations and their sequence, selected to generate all symmetrically 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 2.2.9 [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 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 8.3.5.1 of Section 8.3.5 .

Example: P12₁/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 symmetrically 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 8.3.5 .

For different descriptions of the same space group (settings, cell choices, origin choices), the generating operations are the same. Thus, the transformation relating the two coordinate systems transforms also the generators of one description into those of the other.

From the Fifth Edition onwards, this applies also to the description of the seven rhombohedral (R) space groups by means of `hexagonal' and `rhombohedral' axes. In previous editions, there was a difference in the sequence (not the data) of the `coordinate triplets' and the `symmetry operations' in both descriptions (cf. Section 2.10 in the First to Fourth Editions).

2.2.11. Positions

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The entries under Positions³ (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 M of the Wyckoff position. This is the number of equivalent points per unit cell. For primitive cells, the multiplicity M of the general position is equal to the order of the point group of the space group; for centred cells, M 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.
(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 (the theoretical background on Wyckoff positions is given in Section 8.3.2 ).
(iii) Site symmetry. This is explained in Section 2.2.12 .
(iv) Coordinates. The sequence of the coordinate triplets is based on the Generators (cf. Section 2.2.10 ). 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 M coordinate triplets of a position represent the coordinates of the M 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 2.2.6 .
(v) Reflection conditions. These are described in Section 2.2.13 .

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

(i) General position

A set of symmetrically equivalent points, i.e. a `crystallographic orbit ', is said to be in `general position' if each of its points 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 short-hand form of the matrix representation of the symmetry operations of the space group; this viewpoint is further described in Sections 8.1.6 and 11.1.1 .
(ii) Special position(s)

A set of symmetrically equivalent points is said to be in `special position' if each of its points 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. $[x = {1 \over 4},y = 0]$ , leading to the triplet $[{1 \over 4},0,z]$ ; or $[y = x + {1 \over 2}]$ , leading to the triplet $[x, x + {1 \over 2},z]$ . The number of special positions in a space group [up to 26 in Pmmm (No. 47)] depends on the number and types of symmetry operations that map a point onto itself.

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 2.2.12 . 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. 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 consequent 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'. The 13 space groups of this kind are listed in Section 8.3.2 .

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 entries `Symmetry operations', the locations of the four twofold axes can be deduced as $[0,y,{1 \over 4};\ 0,y,{3 \over 4};\ {1 \over 2},y,{1 \over 4};]$ $[{1 \over 2},y,{3 \over 4}]$ .

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) or of this edition are taken. This problem is analysed by Parthé et al. (1988).

2.2.12. Oriented site-symmetry symbols

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The third column of each Wyckoff position gives the Site symmetry⁴ 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 (symmetrically equivalent) subgroups of the space group. For this reason, all points of one special position are described by the same site-symmetry symbol.

Oriented site-symmetry symbols (cf. Fischer et al., 1973 ) 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 2.2.4.1 ). 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).

2.2.13. Reflection conditions

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The Reflection conditions⁵ 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.

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

(1) General conditions. They apply to all Wyckoff positions of a space group, i.e. they 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 occur always 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.

2.2.13.1. General reflection conditions

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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 2.2.13.1 . 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 2.2.13.1| top | pdf |
Integral reflection conditions for centred cells (lattices)

Reflection condition	Centring type of cell	Centring symbol
None	Primitive	$[\left\{\matrix{P\cr R\cr}\right.]$ ^† $[{\lower6pt\hbox{(rhombohedral axes)}}]$
	C-face centred	C
	A-face centred	A
	B-face centred	B
	Body centred	I
and	All-face centred	F
or:
all odd or all even (`unmixed')
	Rhombohedrally centred, obverse setting (standard)	$[\!\!\left.{\matrix{{}\cr{}\cr{}\cr{}\cr{}\cr{}}}\right\}{R}]$ ^† (hexagonal axes)
	Rhombohedrally centred, reverse setting
	Hexagonally centred	H^‡

^†For further explanations see Chapters 1.2

and 2.1

.
^‡For the use of the unconventional H cell, see Chapter 1.2

(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 , 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.

Reflection conditions of types (ii) and (iii) are listed in Table 2.2.13.2 . 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 2.2.14 ) 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 2.2.13.2| top | pdf |
Zonal and serial reflection conditions for glide planes and screw axes (cf. Chapter 1.3 )

(a) Glide planes

Type of reflections	Reflection condition	Glide plane			Crystallographic coordinate system to which condition applies
Type of reflections	Reflection condition	Orientation of plane	Glide vector	Symbol
0kl		(100)	$[{\bf b}/2]$	b	$[\!\left.\matrix{\cr\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (a\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}]$	$[\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]$
			$[{\bf c}/2]$	c
			$[{\bf b}/2 + {\bf c}/2]$	n
	$[\!\matrix{k + l = 4n\hfill\cr \quad (k,l = 2n)\hfill\cr}]$ ^†		$[{\bf b}/4 \pm {\bf c}/4]$	d
h0l		(010)	$[{\bf c}/2]$	c	$[\!\left.\matrix{\cr\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (b\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}]$	$[\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]$
			$[{\bf a}/2]$	a
			$[{\bf c}/2 + {\bf a}/2]$	n
	$[\!\matrix{l + h = 4n\hfill\cr \quad (l,h = 2n)\hfill\cr}]$ ^†		$[{\bf c}/4 \pm {\bf a}/4]$	d
hk0		(001)	$[{\bf a}/2]$	a	$[\!\left.\matrix{\cr\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic (}c\ \hbox{unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}]$	$[\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]$
			$[{\bf b}/2]$	b
			$[{\bf a}/2 + {\bf b}/2]$	n
	$[\!\matrix{h + k = 4n\hfill\cr \quad (h,k = 2n)\hfill\cr}]$ ^†		$[{\bf a}/4 \pm {\bf b}/4]$	d
$[\matrix{h\bar{h}0l\hfill\cr 0k\bar{k}l\hfill\cr \bar{h}0hl\hfill\cr}]$		$[\left.\!\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}\right\} \{11\bar{2}0\}]$	$[{\bf c}/2]$	c	$[\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]$
$[\matrix{hh.\overline{2h}.l\hfill\cr \overline{2h}.hhl\hfill\cr h.\overline{2h}.hl\hfill\cr}]$		$[\left.\!\matrix{(1\bar{1}00)\hfill\cr (01\bar{1}0)\hfill\cr (\bar{1}010)\hfill\cr}\right\} \{1\bar{1}00\}]$	$[{\bf c}/2]$	c	$[\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]$
$[\matrix{hhl\hfill\cr hkk\hfill\cr hkh\hfill\cr}]$	$[\matrix{l = 2n\hfill\cr h = 2n\hfill\cr k = 2n\hfill\cr}]$	$[\left.\!\matrix{(1\bar{1}0)\hfill\cr (01\bar{1})\hfill\cr (\bar{1}01)\hfill\cr}\right\} \{1\bar{1}0\}]$	$[\matrix{{\bf c}/2\hfill\cr {\bf a}/2\hfill\cr {\bf b}/2\hfill\cr}]$	$[\matrix{c,n\hfill\cr a,n\hfill\cr b,n\hfill\cr}]$	$[\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Rhombohedral}]$ ^‡
$[hhl, h\bar{h}l]$		$[(1\bar{1}0), (110)]$	$[{\bf c}/2]$	c, n	$[\!\left.\matrix{\cr\cr\cr}\right\}\hbox{Tetragonal}]$ ^§	$[\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr\cr\cr\cr}\right\}\hbox{Cubic}]$ ^¶
$[hhl, h\bar{h}l]$		$[(1\bar{1}0), (110)]$	$[{\bf a}/4 \pm {\bf b}/4 \pm {\bf c}/4]$	d
$[hkk, hk\bar{k}]$		$[(01\bar{1}), (011)]$	$[{\bf a}/2]$	a, n
$[hkk, hk\bar{k}]$		$[(01\bar{1}), (011)]$	$[\pm {\bf a}/4 + {\bf b}/4 \pm {\bf c}/4]$	d
$[hkh, \bar{h}kh]$		$[(\bar{1}01), (101)]$	$[{\bf b}/2]$	b, n
$[hkh, \bar{h}kh]$		$[(\bar{1}01), (101)]$	$[\pm {\bf a}/4 \pm {\bf b}/4 + {\bf c}/4]$	d

(b) Screw axes

Type of reflections	Reflection conditions	Screw axis			Crystallographic coordinate system to which condition applies
Type of reflections	Reflection conditions	Direction of axis	Screw vector	Symbol
h00		[100]	$[{\bf a}/2]$	$[2_{1}]$	$[\left\{\matrix{\hbox{Monoclinic }(a\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.]$	$[\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]$
			$[{\bf a}/2]$	$[4_{2}]$
			$[{\bf a}/4]$	$[4_{1},4_{3}]$
0k0		[010]	$[{\bf b}/2]$	$[2_{1}]$	$[\left\{\matrix{\hbox{Monoclinic }(b\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.]$	$[\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]$
			$[{\bf b}/2]$	$[4_{2}]$
			$[{\bf b}/4]$	$[4_{1},4_{3}]$
00l		[001]	$[{\bf c}/2]$	$[2_{1}]$	$[\matrix{\left\{\matrix{\hbox{Monoclinic } (c\hbox{ unique}),\hfill\cr\quad\hbox{ Orthorhombic}\hfill\cr}\right.\cr\left.\matrix{\noalign{\vskip31pt}\cr}\right\}\hbox{Tetragonal}\hfill}]$	$[\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]$
			$[{\bf c}/2]$	$[4_{2}]$
			$[{\bf c}/4]$	$[4_{1},4_{3}]$
000l		[001]	$[{\bf c}/2]$	$[6_{3}]$	$[\left.{\hbox to 6pt{}}\matrix{\noalign{\vskip54pt}}\right\}\hbox{Hexagonal}]$
			$[{\bf c}/3]$	$[3_{1},3_{2},6_{2},6_{4}]$
			$[{\bf c}/6]$	$[6_{1},6_{5}]$

^†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, 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 occur also 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. $[P\bar{4}3n\ (218)]$ , $[Pn\bar{3}n\ (222)]$ , $[Pm\bar{3}n\ (223)]$ vs $[F\bar{4}3c\ (219)]$ , $[Fm\bar{3}c\ (226)]$ , $[Fd\bar{3}c\ (228)]$ .

For the two-dimensional groups , the reasoning is analogous. The reflection conditions for the plane groups are assembled in Table 2.2.13.3 .

Table 2.2.13.3| top | pdf |
Reflection conditions for the plane groups

Type of reflections	Reflection condition	Centring type of plane cell; or glide line with glide vector	Coordinate system to which condition applies
hk	None	Primitive p	All systems
		Centred c	Rectangular
		Hexagonally centred h^†	Hexagonal
h0		Glide line g normal to b axis; glide vector $[{1 \over 2}{\bf a}]$	$[\left.\matrix{\noalign{\vskip 54pt}}\right\}\matrix{\hbox{Rectangular, }\hfill\cr\quad\hbox{Square}\hfill}]$
0k		Glide line g normal to a axis; glide vector $[{1 \over 2}{\bf b}]$

^†For the use of the unconventional h cell see Chapter 1.2

For the interpretation of observed reflections, the general reflection conditions must be studied in the order (i) to (iii), as conditions of type (ii) may be included in those of type (i), while conditions of type (iii) may be included in those of types (i) or (ii). 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 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), 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 rather than and . The same applies to sums of indices. Thus, there are several different ways to express the integral conditions for an F-centred lattice: `' or ` and ' or ` and ' (cf. Table 2.2.13.1 );
(iv) conditions separated by `OR' are alternative conditions. For example, ` or ' means that hkl is `present' if either the condition or the alternative condition is fulfilled. Obviously, hkl is a `present' reflection also if both conditions are satisfied. Note that `or' conditions occur only for the special conditions described in Section 2.2.13.2 ;
(v) in crystal systems with two or more symmetrically equivalent nets or rows (tetragonal and higher), only one representative set (the first one in Table 2.2.13.2 ) 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 2.2.13.1 ). 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) $[hkl\!\! :h + k = 2n{\hbox{;}}]$
(2) $[0kl\!\! :k = 2n{\hbox{;}}\quad h0l\!\!:h,l = 2n{\hbox{;}}\quad hk0\!\!:h + k = 2n{\hbox{;}}]$
(3) $[h00\!\! :h = 2n{\hbox{;}}\quad 0k0\!\!:k = 2n{\hbox{;}}\quad 00l\!\!:l = 2n.]$

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 and 0k0 with follow from the integral condition (1), whereas 00l with 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). On the basis of diffraction symmetry and reflection conditions, no choice between the two space groups can be made (cf. Part 3 ).

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 Part 7 ).

2.2.13.2. Special or `extra' reflection conditions

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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 symmetrically 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, whence the special condition. Analogously for : .
(2) (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) (15, unique axis b, cell choice 3)

For the description of space group No. 15 with cell choice 3 (see Section 2.2.16 and 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.

2.2.13.3. Structural or non-space-group absences

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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 ' (Engel et al., 1984 )]. Non-space-group absences may also occur for special arrangements of atoms (`false symmetry') in a crystal structure (cf. Templeton, 1956 ; Sadanaga et al., 1978 ). Non-space-group absences may occur also for polytypic structures; this is briefly discussed by Durovič in Section 9.2.2.2.5 of International Tables for Crystallography (2004), 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 Part 3 .

2.2.14. Symmetry of special projections

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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.

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.

2.2.14.1. Data listed in the space-group tables

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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 2.2.4.1 ). If a lattice contains less 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: $[\openup 2pt\eqalign{&\left.\matrix{\hbox{Triclinic}\hfill\cr \hbox{Monoclinic}\hfill\cr \hbox{(both settings)}\hfill\cr \hbox{Orthorhombic}\hfill\cr}\right\}{\hskip 3.5pt{[001]}}{\hskip 12.5pt{[100]}} {\hskip 12pt{[010]}} \cr &\matrix{\hbox{Tetragonal}\hfill &\ [001] &\ [100] &\ [110]\cr \hbox{Hexagonal}\hfill &\ [001] &\ [100] &\ [210]\cr \hbox{Rhombohedral}\hfill &\ [111] &\ [1\bar{1}0] &\ [2\bar{1}\bar{1}]\cr \hbox{Cubic}\hfill &\ [001] &\ [111] &\ [110]\cr}}]$
(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.

(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 Parts 6 and 7 . 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 2.2.14.1 . The additional relations for centred cells can be derived easily from the table.

Table 2.2.14.1| 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 Part 7

Projection direction	Triclinic	Monoclinic		Orthorhombic	Projection direction	Tetragonal
Projection direction	Triclinic	Unique axis b	Unique axis c	Orthorhombic	Projection direction	Tetragonal
[001]	$[a' = a \sin \beta]$	$[a' = a \sin \beta]$			[001]
	$[b' = b \sin \alpha]$
	$[\gamma' = 180^{\circ} - \gamma^{*}]$ ^†	$[\gamma' = 90^{\circ}]$	$[\gamma' = \gamma]$	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$
[100]	$[a' = b \sin \gamma]$		$[a' = b \sin \gamma]$		[100]
	$[b' = c \sin \beta]$	$[b' = c \sin \beta]$
	$[\gamma' = 180^{\circ} - \alpha^{*}]$ ^†	$[\gamma' = 90^{\circ}]$	$[\gamma' = 90^{\circ}]$	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$
[010]	$[a' = c \sin \alpha]$				[110]	$[a' = (a/2) \sqrt{2}]$
	$[b' = \alpha \sin \gamma]$		$[b' = a \sin \gamma]$
	$[\gamma' = 180^{\circ} - \beta^{*}]$ ^†	$[\gamma' = \beta]$	$[\gamma' = 90^{\circ}]$	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$

Projection direction	Hexagonal	Projection direction	Rhombohedral^‡	Projection direction	Cubic
[001]		[111]	$[a' = {\displaystyle{2 \over \sqrt{3}}}\; a \sin (\alpha/2)]$	[001]
			$[b' = {\displaystyle{2 \over \sqrt{3}}} \;a \sin (\alpha/2)]$
	$[\gamma' = 120^{\circ}]$		$[\gamma' = 120^{\circ}]$		$[\gamma' = 90^{\circ}]$
[100]	$[a' = (a/2) \sqrt{3}]$	$[[1\bar{1}0]]$	$[a' = a \cos (\alpha/2)]$	[111]	$[a' = a\sqrt{2/3}]$
					$[b' = a\sqrt{2/3}]$
	$[\gamma' = 90^{\circ}]$		$[\gamma' = \delta]$ ^§		$[\gamma' = 120^{\circ}]$
[210]		$[[\bar{2}11]]$	$[a' = {\displaystyle{1 \over \sqrt{3}}}\; a\sqrt{1 + 2 \cos \alpha}]$	[110]	$[a' = (a/2)\sqrt{2}]$
			$[b' = a \sin (\alpha/2)]$
	$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$		$[\gamma' = 90^{\circ}]$

^† $[\cos \alpha^{*} = {\displaystyle{\cos \beta \cos \gamma - \cos \alpha \over \sin \beta \sin \gamma}}; \;\cos \beta^{*} = {\displaystyle{\cos \gamma \cos \alpha - \cos \beta \over \sin \gamma \sin \alpha}};\; \cos \gamma^{*} = {\displaystyle{\cos \alpha \cos \beta - \cos \gamma \over \sin \alpha \sin \beta}}.]$
^‡The entry `Rhombohedral' refers to the primitive rhombohedral cell with $[a = b = c, \alpha = \beta = \gamma]$ (cf. Table 2.1.2.1

).
^§ $[\cos \delta = {\displaystyle{\cos \alpha \over \cos \alpha/2\;}}]$ .

(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 2.2.9 ).

Example

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

2.2.14.2. Projections of centred cells (lattices)

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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 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 [], [ $[\bar{1}11]$ ], [ $[1\bar{1}1]$ ] 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 [ $[\bar{1}11]$ ], [] or [ $[\bar{1}\bar{2}1]$ ] for the obverse setting. For the reverse setting, the corresponding directions are [ $[1\bar{1}1]$ ], [ $[\bar{2}\bar{1}1]$ ], []; cf. Chapter 1.2 .

(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 [] 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 [] of an orthorhombic I-centred cell leads to a rectangular c-centred plane cell, which is conventional.
(3) Projection along [] 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. Chapter 1.2 ). A simple cell transformation leads to the conventional hexagonal p cell.

2.2.14.3. Projections of symmetry elements

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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 2.2.14.2 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 2.2.14.2| top | pdf |
Projections of crystallographic symmetry elements

Symmetry element in three dimensions	Symmetry element in projection
Arbitrary orientation
$[\left.\!\matrix{\hbox{Symmetry centre}\hfill &\bar{1}\hfill\cr \hbox{Rotoinversion axis} &\bar{3} \equiv 3 \times \bar{1}\hfill\cr}\right\}]$	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 has not 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.

Example: (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) and Biedl (1966).

2.2.15. Maximal subgroups and minimal supergroups

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The present section gives a brief summary, without theoretical explanations, of the sub- and supergroup data in the space-group tables. The theoretical background is provided in Section 8.3.3 and Part 13 . Detailed sub- and supergroup data are given in International Tables for Crystallography Volume A1 (2004).

2.2.15.1. Maximal non-isomorphic subgroups⁶

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The maximal non-isomorphic subgroups $[{\cal H}]$ of a space group $[{\cal G}]$ are divided into two types: $[\eqalign{&{\bf I}\quad translationengleiche\quad \hbox{or } t\ \hbox{subgroups}\cr &{\bf II}\;\ klassengleiche\phantom{leiche\quad } \hbox{or } k\ \hbox{subgroups}.}]$ For practical reasons, type II is subdivided again into two blocks:

IIa the conventional cells of $[{\cal G}]$ and $[{\cal H}]$ are the same

IIb the conventional cell of $[{\cal H}]$ is larger than that of $[{\cal G}]$ . ⁷

Block IIa has no entries for space groups $[{\cal G}]$ with a primitive cell. For space groups $[{\cal G}]$ with a centred cell, it contains those maximal subgroups $[{\cal H}]$ that have lost some or all centring translations of $[{\cal G}]$ but none of the integral translations (`decentring ' of a centred cell).

Within each block, the subgroups are listed in order of increasing index [i] and in order of decreasing space-group number for each value of i.

(i) Blocks I and IIa

In blocks I and IIa, every maximal subgroup $[{\cal H}]$ of a space group $[{\cal G}]$ is listed with the following information:

[i] HMS1 (HMS2, No.) Sequence of numbers.

The symbols have the following meaning:

[i]: index of $[{\cal H}]$ in $[{\cal G}]$ (cf. the footnote to Section 8.1.6);

HMS1: Hermann–Mauguin symbol of $[{\cal H}]$ , referred to the coordinate system and setting of $[{\cal G}]$ ; this symbol may be unconventional;

(HMS2, No.): conventional short Hermann–Mauguin symbol of $[{\cal H}]$ , given only if HMS1 is not in conventional short form, and the space-group number of $[{\cal H}]$ .

Sequence of numbers: coordinate triplets of $[{\cal G}]$ retained in $[{\cal H}]$ . The numbers refer to the numbering scheme of the coordinate triplets of the general position of $[{\cal G}]$ (cf. Section 2.2.9 ). The following abbreviations are used:

Block I (all translations retained):
Number	Coordinate triplet given by Number, plus those obtained by adding all centring translations of $[{\cal G}]$ .
(Numbers)	The same, but applied to all Numbers between parentheses.
Block IIa (not all translations retained):
$[Number + (t_{1},t_{2},t_{3})]$	Coordinate triplet obtained by adding the translation $[t_{1},t_{2},t_{3}]$ to the triplet given by Number.
$[(Numbers) + (t_{1},t_{2},t_{3})]$	The same, but applied to all Numbers between parentheses.

In blocks I and IIa, sets of conjugate subgroups are linked by left-hand braces. For an example, see space group $[R\bar{3}]$ (148) below.

Examples

(1) $[{\cal G}\!\!: C1m1\;(8)]$ $[\displaylines{\quad\;\;{\matrix{{\bf I}\hfill\cr {\bf IIa}\hfill\cr\cr}\quad\matrix{[2]\;C1\;(P1,1)\hfill\cr [2]\;P1a1\;(Pc,7)\hfill\cr[2]\;P1m1\;(Pm,6)\hfill\cr}\qquad\matrix{1+\hfill\cr 1{\hbox{;}}\;2+(1/2,1/2,0)\hfill\cr 1{\hbox {;}}\;2\hfill\cr}\hfill}\hfill}]$ where the numbers have the following meaning: $[\eqalign{\matrix{1 + \hfill &x, y, z{\hbox{;}} &x + 1/2, y + 1/2, z\cr 1{\hbox{;}}\;2 \hfill &x, y, z{\hbox{;}} &x, \bar{y}, z\hfill \cr 1{\hbox{;}}\;2 + (1/2, 1/2, 0) &x, y, z{\hbox{;}} &x + 1/2, \bar{y} + 1/2, z.\cr}}]$
(2) $[{\cal G}\!\!: Fdd2\;(43)]$ $[\;\;{\bf I}\quad [2]\;F112 \;(C2, 5)\qquad \qquad (1;2) +]$ where the numbers have the following meaning: $[\phantom{\;\;{\bf I}}\quad\eqalign{\matrix{(1{\hbox{; }}2) + &x, y, z{\hbox{;}}\; \quad x + 1/2, y + 1/2, z{\hbox {;}}\hfill \cr & x + 1/2, y, z + 1/2{\hbox{;}}\;\quad x, y + 1/2, z + 1/2{\hbox{;}}\hfill\cr& \bar{x}, \bar{y}, z{\hbox {;}}\;\quad \bar{x} + 1/2, \bar{y} + 1/2, z{\hbox{;}}\hfill\cr& \bar{x} + 1/2, \bar{y}, z + 1/2{\hbox{;}}\;\quad \bar{x}, \bar{y} + 1/2, z + 1/2.\hfill\cr}}\hfill]$
(3) $[{\cal G}\!\!: P4_{2}/nmc = P4_{2}/n 2_{1}/m 2/c\ (137)]$ $[\quad\;\;{\bf I}\quad [2]\; P2/n2_{1}/m1\ (Pmmn, 59)\ 1{\hbox{;}}\; 2{\hbox{;}}\; 5{\hbox{;}}\; 6{\hbox{;}}\; 9{\hbox{;}}\; 10{\hbox{;}}\; 13{\hbox{;}}\; 14.]$

Operations $[4_{2}, 2]$ and c, occurring in the Hermann–Mauguin symbol of $[{\cal G}]$ , are lacking in $[{\cal H}]$ . In the unconventional `tetragonal version' $[P2/n2_{1}/m 1]$ of the symbol of $[{\cal H}, 2_{1}/m]$ stands for two sets of $[2_{1}/m]$ (along the two orthogonal secondary symmetry directions), implying that $[{\cal H}]$ is orthorhombic. In the conventional `orthorhombic version', the full symbol of $[{\cal H}]$ reads $[P2_{1}/m 2_{1}/m 2/n]$ and the short symbol Pmmn.

(ii) Block IIb

Whereas in blocks I and IIa every maximal subgroup $[{\cal H}]$ of $[{\cal G}]$ is listed, this is no longer the case for the entries of block IIb. The information given in this block is:

[i] HMS1 (Vectors) (HMS2, No.)

The symbols have the following meaning:

[i]: index of $[{\cal H}]$ in $[{\cal G}]$ ;

HMS1: Hermann–Mauguin symbol of $[{\cal H}]$ , referred to the coordinate system and setting of $[{\cal G}]$ ; this symbol may be unconventional;⁸

(Vectors): basis vectors a′, b′, c′ of $[{\cal H}]$ in terms of the basis vectors a, b, c of $[{\cal G}]$ . No relations are given for unchanged axes, e.g. $[{\bf a}' = {\bf a}]$ is not stated;

(HMS2, No.): conventional short Hermann–Mauguin symbol, given only if HMS1 is not in conventional short form, and the space-group number of $[{\cal H}]$ .

In addition to the general rule of increasing index [i] and decreasing space-group number (No.), the sequence of the IIb subgroups also depends on the type of cell enlargement. Subgroups with the same index and the same kind of cell enlargement are listed together in decreasing order of space-group number (see example 1 below).

In contradistinction to blocks I and IIa, for block IIb the coordinate triplets retained in $[{\cal H}]$ are not given. This means that the entry is the same for all subgroups $[{\cal H}]$ that have the same Hermann–Mauguin symbol and the same basis-vector relations to $[{\cal G}]$ , but contain different sets of coordinate triplets. Thus, in block IIb, one entry may correspond to more than one subgroup,⁹ as illustrated by the following examples.

Examples

(1) $[{\cal G}\!\!: Pmm2\ (25)]$ $[\displaylines{\quad\;\; \eqalign{{\bf IIb}\;\; &\ldots [2]\; Pbm2\; ({\bf b}' = 2{\bf b})\; (Pma2, 28); [2]\;Pcc2\;({\bf c}' = 2{\bf c})\; (27);\cr &\ldots [2]\;Cmm2\;({\bf a}' = 2{\bf a}, {\bf b}' = 2{\bf b})\; (35);\;\ldots\cr}\hfill}]$

Each of the subgroups is referred to its own distinct basis a′, b′, c′, which is different in each case. Apart from the translations of the enlarged cell, the generators of the subgroups, referred to a′, b′, c′, are as follows: $[{\hbox to -4pt{}}\matrix{Pbm2 \hfill &x,y,z{\hbox{;}}\; \hfill &\bar{x},\bar{y},z{\hbox{;}}\; \hfill &x,\bar{y} + 1/2,z &\hbox{or} \hfill &\cr &x,y,z{\hbox{;}}\; &\bar{x},\bar{y} + 1/2,z{\hbox{;}}\; \hfill &{\hbox to 2.5pt{}}x,\bar{y},z \hfill & &\cr Pcc2 \hfill &x,y,z{\hbox{;}}\; \hfill &\bar{x},\bar{y},z{\hbox{;}}\; \hfill &x,\bar{y},z + 1/2 & &\cr Cmm2 &x,y,z{\hbox{;}}\; &x + 1/2,y + 1/2,z{\hbox{;}}\; &{\hbox to 2.5pt{}}\bar{x},\bar{y},z{\hbox{;}}\; \hfill &x,\bar{y},z \hfill &\hbox{or}\cr &x,y,z{\hbox{;}}\; &x + 1/2,y + 1/2,z{\hbox{;}}\; &{\hbox to 2.5pt{}}\bar{x},\bar{y},z{\hbox{;}}\; \hfill &x,\bar{y} + 1/2,z &\hbox{or}\cr &x,y,z{\hbox{;}}\; &x + 1/2,y + 1/2,z{\hbox{;}}\; &{\hbox to 2.5pt{}}\bar{x},\bar{y} + 1/2,z{\hbox{;}}\; &x,\bar{y},z \hfill &\hbox{or}\cr &x,y,z{\hbox{;}}\; &x + 1/2,y + 1/2,z{\hbox{;}}\; &{\hbox to 2.5pt{}}\bar{x},\bar{y} + 1/2,z{\hbox{;}}\; &x,\bar{y} + 1/2,z. &\cr}]$

There are thus 2, 1 or 4 actual subgroups that obey the same basis-vector relations. The difference between the several subgroups represented by one entry is due to the different sets of symmetry operations of $[{\cal G}]$ that are retained in $[{\cal H}]$ . This can also be expressed as different conventional origins of $[{\cal H}]$ with respect to $[{\cal G}]$ .

(2) $[{\cal G}: P3m1\ (156)]$ $[{\bf IIb}\ \ldots [3] \;H3m1\; ({\bf a}' = 3{\bf a}, {\bf b}' = 3{\bf b})\; (P31m, 157)]$

The nine subgroups of type P31m may be described in two ways:

(i) By partial `decentring' of ninetuple cells ( $[{\bf a}' = 3{\bf a}]$ , $[{\bf b}' = 3{\bf b}]$ , $[{\bf c}' = {\bf c}]$ ) with the same orientations as the cell of the group $[{\cal G}({\bf a},{\bf b},{\bf c})]$ in such a way that the centring points 0, 0, 0; ; (referred to a′, b′, c′) are retained. The conventional space-group symbol P31m of these nine subgroups is referred to the same basis vectors $[{\bf a}'' = {\bf a} - {\bf b}]$ , $[{\bf b}'' = {\bf a} + 2{\bf b}]$ , $[{\bf c}'' = {\bf c}]$ , but to different origins; cf. Section 2.2.15.5 . This kind of description is used in the space-group tables of this volume.
(ii) Alternatively, one can describe the group $[{\cal G}]$ with an unconventional H-centred cell ( $[{\bf a}' = {\bf a} - {\bf b}]$ , $[{\bf b}' = {\bf a} + 2{\bf b}]$ , $[{\bf c}' = {\bf c}]$ ) referred to which the space-group symbol is H31m. `Decentring ' of this cell results in the conventional space-group symbol P31m for the subgroups, referred to the basis vectors a′, b′, c′. This description is used in Section 4.3.5 .

(iii) Subdivision of k subgroups into blocks IIa and IIb

The subdivision of k subgroups into blocks IIa and IIb has no group-theoretical background and depends on the coordinate system chosen. The conventional coordinate system of the space group $[{\cal G}]$ (cf. Section 2.1.3 ) is taken as the basis for the subdivision. This results in a uniquely defined subdivision, except for the seven rhombohedral space groups for which in the space-group tables both `rhombohedral axes' (primitive cell) and `hexagonal axes' (triple cell) are given (cf. Section 2.2.2 ). Thus, some k subgroups of a rhombohedral space group are found under IIa (klassengleich, centring translations lost) in the hexagonal description, and under IIb (klassengleich, conventional cell enlarged) in the rhombohedral description.

Example: $[{\cal G}\!\!:R\bar{3}\ (148)\qquad {\cal H}\!\!:P\bar{3}\ (147)]$

Hexagonal axes

$[\matrix{{\bf I}\hfill &\; \; \>\;[2]\;R3\;(146)\hfill&(1; 2; 3)+\hfill\cr&\; \; \>\;[3]\;R\bar{1}\;(P\bar{1},2)\hfill&(1;4)+\hfill\cr{\bf IIa}\hfill\cr \noalign{\vskip-9pt} &\cases{[3]\; P\bar{3}\; (147) \cr [3]\; P\bar{3}\; (147) \cr [3]\; P\bar{3}\; (147) \cr}\hfill\cr\noalign{\vskip-35pt} &&1 ; 2 ; 3 ; 4 ; 5 ; 6\hfill\cr&&1 ; 2 ; 3 ; (4 ; 5 ; 6) + ({1\over 3},{2\over 3},{2\over 3})\hfill\cr&&1 ; 2 ; 3 ; (4 ; 5 ; 6) + ({2\over 3},{1\over 3},{1\over 3})\hfill\cr\noalign{\vskip2pt}{\bf IIb}&\; \; \>\;\hbox{none}\hfill&\cr}]$

Rhombohedral axes

$[\matrix{{\bf I}\hfill&[2]\;R3\;(146)\quad\ \ 1;2;3\hfill\cr&[3]\;R\bar{1}\;(P\bar{1},2)\quad1;4\hfill\cr{\bf IIa}\hfill&\hbox{none}\hfill&\cr{\bf IIb}\hfill&[3]\;P\bar{3}\;({\bf a}'={\bf a}-{\bf b}, {\bf b}'={\bf b}-{\bf c},{\bf c}'={\bf a}+{\bf b}+{\bf c})\;(147).\hfill\cr}]$

Apart from the change from IIa to IIb, the above example demonstrates again the restricted character of the IIb listing, discussed above. The three conjugate subgroups $[P\bar{3}]$ of index [3] are listed under IIb by one entry only, because for all three subgroups the basis-vector relations between $[{\cal G}]$ and $[{\cal H}]$ are the same. Note the brace for the IIa subgroups, which unites conjugate subgroups into classes.

2.2.15.2. Maximal isomorphic subgroups of lowest index (cf. Part 13 )

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Another set of klassengleiche subgroups are the isomorphic subgroups listed under IIc, i.e. the subgroups $[{\cal H}]$ which are of the same or of the enantiomorphic space-group type as $[{\cal G}]$ . The kind of listing is the same as for block IIb. Again, one entry may correspond to more than one isomorphic subgroup.

As the number of maximal isomorphic subgroups of a space group is always infinite, the data in block IIc are restricted to the subgroups of lowest index. Different kinds of cell enlargements are presented. For monoclinic, tetragonal, trigonal and hexagonal space groups, cell enlargements both parallel and perpendicular to the main rotation axis are listed; for orthorhombic space groups, this is the case for all three directions, a, b and c. Two isomorphic subgroups $[{\cal H}_{1}]$ and $[{\cal H}_{2}]$ of equal index but with cell enlargements in different directions may, nevertheless, play an analogous role with respect to $[{\cal G}]$ . In terms of group theory, $[{\cal H}_{1}]$ and $[{\cal H}_{2}]$ then are conjugate subgroups in the affine normalizer of $[{\cal G}]$ , i.e. they are mapped onto each other by automorphisms of $[{\cal G}]$ .¹⁰ Such subgroups are collected into one entry, with the different vector relationships separated by `or' and placed within one pair of parentheses; cf. example (4).

Examples

(1) $[{\cal G}\!\!: P\bar{3}1c\ (163)]$ $[{\bf IIc}\ [3] \;P\bar{3}1c \;({\bf c}' = 3{\bf c})\; (163); [4] \;P\bar{3}1c\; ({\bf a}' = 2{\bf a}, {\bf b}' = 2{\bf b})\; (163).]$

The first subgroup of index [3] entails an enlargement of the c axis, the second one of index [4] an enlargement of the mesh size in the a,b plane.
(2) $[{\cal G}\!\!: P23\ (195)]$ $[{\bf IIc}\ [27] \;P23\; ({\bf a}' = 3{\bf a},{\bf b}' = 3{\bf b},{\bf c}' = 3{\bf c})\; (195).]$ It seems surprising that [27] is the lowest index listed, even though another isomorphic subgroup of index [8] exists. The latter subgroup, however, is not maximal, as chains of maximal non-isomorphic subgroups can be constructed as follows: $[\eqalign{&P23 \rightarrow [4] \;I23 \;({\bf a}' = 2{\bf a},{\bf b}' = 2{\bf b},{\bf c}' = 2{\bf c}) \rightarrow [2] \;P23 \;({\bf a}',{\bf b}',{\bf c}')\cr \hbox{ or}&\cr &P23 \rightarrow [2]\; F23\; ({\bf a}' = 2{\bf a},{\bf b}' = 2{\bf b},{\bf c}' = 2{\bf c}) \rightarrow [4] \;P23\; ({\bf a}',{\bf b}',{\bf c}').}]$
(3) $[{\cal G}\!\!: P3_{1} 12\ (151)]$ $[\eqalign{{\bf IIc}\ &[2]\; P3_{2} 12\; ({\bf c}' = 2{\bf c})\; (153){\hbox {;}}\; [4]\; P3_{1} 12\; ({\bf a}' = 2{\bf a},{\bf b}' = 2{\bf b})\; (151){\hbox {;}}\; \cr &[7] \;P3_{1} 12\;({\bf c}' = 7{\bf c})\; (151).}]$ Note that the isomorphic subgroup of index [4] with $[{\bf c}' = 4{\bf c}]$ is not listed, because it is not maximal. This is apparent from the chain $[P3_{1} 12 \rightarrow [2]\; P3_{2} 12 \;({\bf c}' = 2{\bf c}) \rightarrow [2]\; P3_{1} 12\; ({\bf c}'' = 2{\bf c}' = 4{\bf c}).]$
(4) $[{\cal G}_{1}\!\!: Pnnm\ (58)]$ $[\openup1pt\eqalign{&{\bf IIc}\ [3]\; Pnnm\; ({\bf a}' = 3{\bf a} \hbox{ or } {\bf b}' = 3{\bf b})\ (58){\hbox {;}}\; [3] \;Pnnm \;({\bf c}' = 3{\bf c})\; (58){\hbox {;}}\; \cr &\hbox {but}\ {\cal G}_{2}\!\!: Pnna\; (52)\cr &{\bf IIc}\ [3] \;Pnna\; ({\bf a}' = 3{\bf a})\; (52){\hbox{;}}\; [3]\; Pnna\; ({\bf b}' = 3{\bf b})\; (52){\hbox{;}}\; \cr&\phantom{{\bf IIc}\ } [3] \;Pnna\; ({\bf c}' = 3{\bf c})\; (52).\cr\cr}]$ For $[{\cal G}_{1} = Pnnm]$ , the x and y directions are analogous, i.e. they may be interchanged by automorphisms of $[{\cal G}_{1}]$ . Such an automorphism does not exist for $[{\cal G}_{2} = Pnna]$ because this space group contains glide reflections a but not b.

2.2.15.3. Minimal non-isomorphic supergroups

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If $[{\cal G}]$ is a maximal subgroup of a group $[{\cal S}]$ , then $[{\cal S}]$ is called a minimal supergroup of $[{\cal G}]$ . Minimal non-isomorphic supergroups are again subdivided into two types, the translationengleiche or t supergroups I and the klassengleiche or k supergroups II. For the minimal t supergroups I of $[{\cal G}]$ , the listing contains the index [i] of $[{\cal G}]$ in $[{\cal S}]$ , the conventional Hermann–Mauguin symbol of $[{\cal S}]$ and its space-group number in parentheses.

There are two types of minimal k supergroups II: supergroups with additional centring translations (which would correspond to the IIa type) and supergroups with smaller conventional unit cells than that of $[{\cal G}]$ (type IIb). Although the subdivision between IIa and IIb supergroups is not indicated in the tables, the list of minimal supergroups with additional centring translations (IIa) always precedes the list of IIb supergroups. The information given is similar to that for the non-isomorphic subgroups IIb, i.e., where applicable, the relations between the basis vectors of group and supergroup are given, in addition to the Hermann–Mauguin symbols of $[{\cal S}]$ and its space-group number. The supergroups are listed in order of increasing index and increasing space-group number.

The block of supergroups contains only the types of the non-isomorphic minimal supergroups $[{\cal S}]$ of $[{\cal G}]$ , i.e. each entry may correspond to more than one supergroup $[{\cal S}]$ . In fact, the list of minimal supergroups $[{\cal S}]$ of $[{\cal G}]$ should be considered as a backwards reference to those space groups $[{\cal S}]$ for which $[{\cal G}]$ appears as a maximal subgroup. Thus, the relation between $[{\cal S}]$ and $[{\cal G}]$ can be found in the subgroup entries of $[{\cal S}]$ .

Example: $[{\cal G}\!\!: Pna2_{1}\ (33)]$

Minimal non-isomorphic supergroups $[\;\;\eqalign{{\bf I}\phantom{I\ldots}&[2]\; Pnna\; (52); [2]\; Pccn\; (56); [2]\; Pbcn\; (60); [2] \;Pnma\; (62).\cr \ {\bf II} \ldots &[2]\; Pnm2_{1}\; ({\bf a}' = {\textstyle{1 \over 2}}{\bf a})\;(Pmn2_{1}, 31); \ldots.\cr}]$ Block I lists, among others, the entry [2] Pnma (62). Looking up the subgroup data of Pnma (62), one finds in block I the entry [2] $[Pn2_{1}a\ (Pna2_{1})]$ . This shows that the setting of Pnma does not correspond to that of $[Pna2_{1}]$ but rather to that of $[Pn2_{1}a]$ . To obtain the supergroup $[{\cal S}]$ referred to the basis of $[Pna2_{1}]$ , the basis vectors b and c must be interchanged. This changes Pnma to Pnam, which is the correct symbol of the supergroup of $[Pna2_{1}]$ .

Note on R supergroups of trigonal P space groups: The trigonal P space groups Nos. 143–145, 147, 150, 152, 154, 156, 158, 164 and 165 each have two rhombohedral supergroups of type II. They are distinguished by different additional centring translations which correspond to the `obverse ' and `reverse ' settings of a triple hexagonal R cell; cf. Chapter 1.2 . In the supergroup tables of Part 7 , these cases are described as [3] R3 (obverse) (146); [3] R3 (reverse) (146) etc.

2.2.15.4. Minimal isomorphic supergroups of lowest index

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No data are listed for isomorphic supergroups IIc because they can be derived directly from the corresponding data of subgroups IIc (cf. Part 13 ).

2.2.15.5. Note on basis vectors

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In the subgroup data, a′, b′, c′ are the basis vectors of the subgroup $[{\cal H}]$ of the space group $[{\cal G}]$ . The latter has the basis vectors a, b, c. In the supergroup data, a′, b′, c′ are the basis vectors of the supergroup $[{\cal S}]$ and a, b, c are again the basis vectors of $[{\cal G}]$ . Thus, a, b, c and a′, b′, c′ exchange their roles if one considers the same group–subgroup relation in the subgroup and the supergroup tables.

Examples

(1) $[{\cal G}\!\!: Pba2\ (32)]$

Listed under subgroups IIb, one finds, among other entries, $[[2]\;Pna2_{1}\;({\bf c}' = {\bf 2c})\ (33)]$ ; thus, $[{\bf c}(Pna2_{1}) = {\bf 2c}(Pba2)]$ .

Under supergroups II of $[Pna2_{1}\ (33)]$ , the corresponding entry reads $[[2]\; Pba2 \;({\bf c}' = {1 \over 2} {\bf c})\; (32)]$ ; thus $[{\bf c}(Pba2) = {1\over 2}{\bf c}(Pna2_{1})]$ .
(2) Tetragonal k space groups with P cells. For index [2], the relations between the conventional basis vectors of the group and the subgroup read (cf. Fig. 5.1.3.5 ) $[{\bf a}' = {\bf a} + {\bf b},{\hbox to 19pt{}} {\bf b}' = - {\bf a} + {\bf b}{\hbox to 21pt{}}({\bf a}', {\bf b}' \hbox{ for the subgroup}).\hfill]$ Thus, the basis vectors of the supergroup are $[{\bf a}' = {\textstyle{1 \over 2}} ({\bf a} - {\bf b}),\quad {\bf b}' = {\textstyle{1 \over 2}} ({\bf a} + {\bf b}){\hbox to 17pt{}} ({\bf a}', {\bf b}' \hbox{ for the supergroup}).\hfill]$ An alternative description is $[\displaylines{{\bf a}' = {\bf a} - {\bf b},{\hbox to 19pt{}}{\bf b}' = {\bf a} + {\bf b}\quad \quad\quad ({\bf a}', {\bf b}' \hbox{ for the subgroup})\hfill\cr {\bf a}' = {\textstyle{1 \over 2}} ({\bf a} + {\bf b}), {\hbox to 9pt{}}{\bf b}' = {\textstyle{1 \over 2}} (-{\bf a}+{\bf b})\quad ({\bf a}', {\bf b}' \hbox{ for the supergroup}).\hfill}]$
(3) Hexagonal k space groups. For index [3], the relations between the conventional basis vectors of the sub- and supergroup read (cf. Fig 5.1.3.8 ) $[{\bf a}' = {\bf a} - {\bf b},{\hbox to 19pt{}}{\bf b}' = {\bf a} + 2{\bf b}{\hbox to 23pt{}}({\bf a}', {\bf b}' \hbox{ for the subgroup}).\hfill]$ Thus, the basis vectors of the supergroup are $[{\bf a}' = {\textstyle{1 \over 3}} (2{\bf a} + {\bf b}),{\hbox to 4pt{}}{\bf b}' = {\textstyle{1 \over 3}}(-{\bf a} + {\bf b}){\hbox to 7pt{}}({\bf a}', {\bf b}' \hbox{ for the supergroup}).\hfill]$ An alternative description is $[\displaylines{{\bf a}' = 2{\bf a} + {\bf b},{\hbox to 14pt{}}{\bf b}' = - {\bf a} + {\bf b} \quad\quad\ ({\bf a}', {\bf b}' \hbox{ for the subgroup})\hfill\cr {\bf a}' = {\textstyle{1 \over 3}} ({\bf a} - {\bf b}),{\hbox to 8pt{}}{\bf b}' = {\textstyle{1 \over 3}} ({\bf a} + 2{\bf b}){\hbox to 14pt{}}({\bf a}', {\bf b}' \hbox{ for the supergroup}).\hfill}]$

2.2.16. 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 2.1.3 and 2.2.2 ). 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'.

2.2.16.1. Cell choices

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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. 2.2.16.1 and labelled e, f and g, in order of increasing length.¹¹ The two shorter vectors span the `reduced mesh', 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 2.2.16.1| 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. 2.2.16.1 ; 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. 2.2.6.4 . The transformation matrices for the three cell choices are listed in Table 5.1.3.1 .

2.2.16.2. Settings

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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 2.2.6.4 .)

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 $[({\bf a}' {\bf b}' {\bf c}') = ({\bf abc}) \pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr} = ({\bf bca}),]$ where a′, b′, c′ is the new set of basis vectors. (Note that the setting symbol bca does not mean that the old vector a changes its label to b, the old vector b changes to c, and the old c changes 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'.¹² 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. 2.2.6.4 .

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 2.2.16.1 lists the setting symbols for the six monoclinic settings in three equivalent forms, starting with the symbols a b c (first line), a b c (second line) and a b c (third line); the unique axis is underlined. These symbols are also found in the headline of the synoptic Table 4.3.2.1 , which lists the space-group symbols for all monoclinic settings and cell choices. Again, the corresponding transformation matrices are listed in Table 5.1.3.1 .

Table 2.2.16.1| top | pdf |
Monoclinic setting symbols (unique axis is underlined)

Unique axis b		Unique axis c		Unique axis a
	$[{\bf c}{\underline{\bar{\bf b}}}{\bf a}]$	$[{\bf c}{\bf a}\underline{\bf b}]$	$[{\bf a}{\bf c}\underline{\bar{\bf b}}]$	$[{\underline{\bf b}{\bf c}{\bf a}}]$	$[{\underline{\bar{\bf b}}{\bf a}{\bf c}}]$
$[{{\bf b}\underline{\bf c}{\bf a}}]$	$[{{\bf a}\underline{\bar{\bf c}}{\bf b}}]$		$[{{\bf b}{\bf a}\underline{\bar{\bf c}}}]$	$[{\underline{\bf c}{\bf a}{\bf b}}]$	$[{\underline{\bar{\bf c}}{\bf b}{\bf a}}]$
$[{{\bf c}\underline{\bf a}{\bf b}}]$	$[{{\bf b}\underline{\bar{\bf a}}{\bf c}}]$	$[{{\bf b}{\bf c}\underline{\bf a}}]$	$[{{\bf c}{\bf b}\underline{\bar{\bf a}}}]$		$[{\underline{\bar{\bf a}}{\bf c}{\bf b}}]$

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 2.2.16.1 . 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 $[({\bf a}'{\bf b}'\underline{{\bf c}}') = ({\bf a}\underline{{\bf b}}{\bf c}) \pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr} = ({\bf ca}\underline{{\bf b}}).]$ In the present discussion, also the setting with a unique is included, as this setting occurs in the subgroup entries of Part 7 and in Table 4.3.2.1 . The a-axis setting $[\underline{{\bf a}}'{\bf b}'{\bf c}' = \underline{{\bf c}} {\bf a b}]$ 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. 2.2.6.4 .

2.2.16.3. Cell choices and settings in the present tables

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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: $[P2\ (\hbox{No}.\ 3),\ P2_{1}\ (4),\ Pm\ (6),\ P2/m\ (10),\ P2_{1}/m\ (11).]$ 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 are always directed along one of the vectors e, f or g in Fig. 2.2.16.1 , i.e. are 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 2.2.4 ) for each setting of a space group.

Table 2.2.16.2 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 2.2.3 ) 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. 2.2.6.4 and their space-group symbols can be obtained from Table 2.2.16.2 . 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 Section 2.2.16.2 .

Table 2.2.16.2| top | pdf |
Symbols for centring types and glide planes of monoclinic space groups

Setting		Cell choice
Setting		1	2	3
Unique axis b	Centring type	C	A	I
Unique axis b	Glide planes	c, n	n, a	a, c
Unique axis c	Centring type	A	B	I
Unique axis c	Glide planes	a, n	n, b	b, a
Unique axis a	Centring type	B	C	I
Unique axis a	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 2.2.16.2 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).

(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) and to that of the b-axis setting in IT (1952), e.g. $[P2_{1}/c]$ or . 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 2.2.3 .
(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 2.2.6 .
(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 2.2.4 ), 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 `priority rule' (cf. Section 4.1.1 ) for glide planes (e before a before b before c before n) is not followed. 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

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.

2.2.16.4. Comparison with earlier editions of International Tables

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In IT (1935), 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), 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)]. 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)]. In the present volume, only one of these standard short symbols is retained (see above and Section 2.2.3 ).

The c-axis setting (primed labels) was obtained from the b-axis setting (unprimed labels) by the following transformation $[({\bf a}'{\bf b}'\underline{{\bf c}}') = ({\bf a}\underline{{\bf b}}{\bf c}) \pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr} = ({\bf ac}\bar{\underline{{\bf b}}}).]$ This corresponds to an interchange of two labels and not to the more logical cyclic permutation, as used in the present volume. 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); see also Table 2.2.16.2 ].

As a consequence of the different transformations between b- and c-axis settings in IT (1952) and in this volume, 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) 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 2.2.16.2 .

2.2.16.5. Selection of monoclinic cell

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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. 2.2.16.1 , 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) and IT (1952). 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.

2.2.17. 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. Chapter 1.2 .

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. Chapters 1.3 and 1.4 .

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. 2.2.17.1 . 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 2.2.17.1| 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 10.1.2.1 . The one-dimensional space groups occur as projection and section symmetries of crystal structures.

References

Astbury, W. T. & Yardley, K. (1924). Tabulated data for the examination of the 230 space groups by homogeneous X-rays. Philos. Trans. R. Soc. London Ser. A, 224, 221–257.Google Scholar

Belov, N. V., Zagal'skaja, Ju. G., Litvinskaja, G. P. & Egorov-Tismenko, Ju. K. (1980). Atlas of the space groups of the cubic system. Moscow: Nauka. (In Russian.)Google Scholar

Biedl, A. W. (1966). The projection of a crystal structure. Z. Kristallogr. 123, 21–26.Google Scholar

Buerger, M. J. (1949). Fourier summations for symmetrical crystals. Am. Mineral. 34, 771–788.Google Scholar

Buerger, M. J. (1956). Elementary crystallography. New York: Wiley.Google Scholar

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Buerger, M. J. (1960). Crystal-structure analysis, Ch. 17. New York: Wiley.Google Scholar

Buerger, M. J. (1965). The geometry of projections. Tschermaks Mineral. Petrogr. Mitt. 10, 595–607.Google Scholar

Engel, P., Matsumoto, T., Steinmann, G. & Wondratschek, H. (1984). The non-characteristic orbits of the space groups. Z. Kristallogr., Supplement Issue No. 1.Google Scholar

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Fischer, W., Burzlaff, H., Hellner, E. & Donnay, J. D. H. (1973). Space groups and lattice complexes. NBS Monograph No. 134. Washington, DC: National Bureau of Standards.Google Scholar

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Heesch, H. (1929). Zur systematischen Strukturtheorie. II. Z. Kristallogr. 72, 177–201.Google Scholar

Hilton, H. (1903). Mathematical crystallography. Oxford: Clarendon Press. [Reprint: New York: Dover (1963).]Google Scholar

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International Tables for Crystallography (2004). Vol. A1, edited by H. Wondratschek & U. Müller. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2004). Vol. C, 3rd ed., edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Google Scholar

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

Koch, E. & Fischer, W. (1974). Zur Bestimmung asymmetrischer Einheiten kubischer Raumgruppen mit Hilfe von Wirkungsbereichen. Acta Cryst. A30, 490–496.Google Scholar

Langlet, G. A. (1972). FIGATOM: a new graphic program for stereographic crystal structure illustrations. J. Appl. Cryst. 5, 66–71.Google Scholar

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International Tables for Crystallography (2006). Vol. A. ch. 2.2, pp. 17-41
https://doi.org/10.1107/97809553602060000505

Chapter 2.2. Contents and arrangement of the tables

2.2.1. General layout

2.2.2. Space groups with more than one description

Examples

2.2.3. Headline

2.2.4. International (Hermann–Mauguin) symbols for plane groups and space groups (cf. Chapter 12.2 )

2.2.4.1. Present symbols

2.2.4.2. Changes in Hermann–Mauguin space-group symbols as compared with the 1952 and 1935 editions of International Tables

Examples

2.2.5. Patterson symmetry

2.2.6. Space-group diagrams

2.2.6.1. Plane groups

2.2.6.2. Triclinic space groups

2.2.6.3. Monoclinic space groups (cf. Sections 2.2.2 and 2.2.16)

2.2.6.4. Orthorhombic space groups and orthorhombic settings

2.2.6.5. Tetragonal, trigonal P and hexagonal P space groups

2.2.6.6. Rhombohedral (trigonal R) space groups

2.2.6.7. Cubic space groups

2.2.6.8. Diagrams of the general position

2.2.7. Origin

2.2.7.1. Origin statement

2.2.8. Asymmetric unit

Example

Examples

Example

2.2.9. Symmetry operations

2.2.9.1. Numbering scheme

Example: Ibca (73)

2.2.9.2. Designation of symmetry operations

Examples

2.2.10. Generators

Example: P121/c1 (14, unique axis b, cell choice 1)

2.2.11. Positions

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

2.2.12. Oriented site-symmetry symbols

Examples

2.2.13. Reflection conditions

2.2.13.1. General reflection conditions

Example

2.2.13.2. Special or `extra' reflection conditions

Examples

Example: (218)

2.2.13.3. Structural or non-space-group absences

2.2.14. Symmetry of special projections

2.2.14.1. Data listed in the space-group tables

Example

2.2.14.2. Projections of centred cells (lattices)

Examples

Examples

2.2.14.3. Projections of symmetry elements

Example: (15, b unique, cell choice 1)

2.2.15. Maximal subgroups and minimal supergroups

2.2.15.1. Maximal non-isomorphic subgroups6

Examples

Examples

Example:

2.2.15.2. Maximal isomorphic subgroups of lowest index (cf. Part 13 )

Examples

2.2.15.3. Minimal non-isomorphic supergroups

Example:

2.2.15.4. Minimal isomorphic supergroups of lowest index

2.2.15.5. Note on basis vectors

Examples

2.2.16. Monoclinic space groups

2.2.16.1. Cell choices

2.2.16.2. Settings

2.2.16.3. Cell choices and settings in the present tables

Example: No. 15, standard short symbol

2.2.16.4. Comparison with earlier editions of International Tables

2.2.16.5. Selection of monoclinic cell

2.2.17. Crystallographic groups in one dimension

References

2.2.6.3. Monoclinic space groups (cf. Sections 2.2.2 and 2.2.16 )

Example: P12₁/c1 (14, unique axis b, cell choice 1)

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

2.2.15.1. Maximal non-isomorphic subgroups⁶

Example: $[{\cal G}\!\!:R\bar{3}\ (148)\qquad {\cal H}\!\!:P\bar{3}\ (147)]$

Example: $[{\cal G}\!\!: Pna2_{1}\ (33)]$