International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 2.2, pp. 3840
Section 2.2.16. Monoclinic space groups^{a}Institut für Kristallographie, RheinischWestfälische Technische Hochschule, Aachen, Germany, and ^{b}Laboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands 
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 spacegroup symbol, depends upon two choices:
One edge of the cell, i.e. one crystal axis, is always chosen along the monoclinic symmetry direction. The other two edges are located in the plane perpendicular to this direction and coincide with translation vectors in this `monoclinic plane'. It is sensible and common practice (see below) to choose these two basis vectors from the shortest three translation vectors in that plane. They are shown in Fig. 2.2.16.1 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 , whereas for the other two primitive meshes larger angles are possible.
Other choices of the basis vectors in the monoclinic plane are possible, provided they span a primitive mesh. It turns out, however, that the spacegroup symbol for any of these (nonreduced) 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 .
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 or 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 nonacute, i.e. , and the coordinate system righthanded. 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 baxis, caxis and aaxis `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 spacegroup symbols for all monoclinic settings and cell choices. Again, the corresponding transformation matrices are listed in Table 5.1.3.1 .

In the spacegroup tables, only the settings with b and c unique are treated and for these only the lefthand members of the double entries in Table 2.2.16.1. This implies, for instance, that the caxis setting is obtained from the baxis setting by cyclic permutation of the labels, i.e. by the transformation 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 aaxis setting is obtained from the caxis setting also by cyclic permutation of the labels and from the baxis setting by the reverse cyclic permutation: .
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 uniqueaxis vector have been selected. Six of the nine resulting possibilities are illustrated in Fig. 2.2.6.4.
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: 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: 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 secondshortest or the thirdshortest 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 spacegroup 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 . Cell choices 2 and 3 follow from the anticlockwise order 1–2–3 in Fig. 2.2.6.4 and their spacegroup symbols can be obtained from Table 2.2.16.2. The caxis and the aaxis settings then are derived from the baxis setting by cyclic permutations of the axial labels, as described in Section 2.2.16.2.

In the two space groups Cc (9) and , 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), and , which contain only one type of glide plane, the lefthand member of each pair of glide planes in Table 2.2.16.2 applies.
In the spacegroup tables of this volume, the following treatments of monoclinic space groups are given:
All settings and cell choices are identified by the appropriate full Hermann–Mauguin symbols (cf. Section 2.2.4), e.g. or . For the two space groups Cc (9) and 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 glideplane 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 Application of the priority rule would have resulted in the following symbols Here, the transformation properties are obscured.
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 caxis setting (first setting) was newly introduced, in addition to the baxis 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 baxis 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. and ) and of two different standard short symbols (e.g. and ) 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 caxis setting (primed labels) was obtained from the baxis setting (unprimed labels) by the following transformation 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 spacegroup symbols that indicate the setting unambiguously; thus the lattice letters were chosen as C (baxis setting) and B (caxis 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 caxis settings in IT (1952) and in this volume, some spacegroup symbols have changed. This is apparent from a comparison of pairs such as and in IT (1952) with the corresponding pairs in this volume, and . The symbols with Bcentred cells appear now for cell choice 2, as can be seen from Table 2.2.16.2.
In practice, the selection of the (righthanded) unit cell of a monoclinic crystal can be approached in three ways, whereby the axes refer to the bunique setting; for c unique similar considerations apply:

References
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]International Tables for Xray 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).]