International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 6871
Section 1.4.5.2. Sections
B. Souvignier^{d}

For a space group and a point X in the threedimensional point space , the sitesymmetry group of X is the subgroup of operations of that fix X. Analogously, one can also look at the subgroup of operations fixing a onedimensional line or a twodimensional plane. If the line is along a rational direction, it will be fixed at least by the translations of along that direction. However, it may also be fixed by a symmetry operation that reverses the direction of the line. The resulting subgroup of that fixes the line is a rod group.
Similarly, a plane having a normal vector along a rational direction is fixed by translations of corresponding to a twodimensional lattice. Again, the plane may also be fixed by additional symmetry operations, e.g. by a twofold rotation around an axis lying in the plane, by a rotation around an axis normal to the plane or by a reflection in the plane.
Definition
A rational planar section of a crystal pattern is the intersection of the crystal pattern with a plane containing two linearly independent translation vectors of the crystal pattern. The intersecting plane is called the section plane.
A rational linear section of a crystal pattern is the intersection of the crystal pattern with a line containing a translation vector of the crystal pattern. The intersecting line is called the penetration line.
A planar section is determined by a vector d which is perpendicular to the section plane and a continuous parameter s, called the height, which gives the position of the plane on the line along d.
A linear section is specified by a vector d parallel to the penetration line and a point in a plane perpendicular to d giving the intersection of the line with that plane.
Definition
From now on we will only consider rational sections and omit this attribute. Moreover, we will concentrate on the case of planar sections, since this is by far the most relevant case for crystallographic applications. The treatment of onedimensional sections is analogous, but in general much easier.
Let d be a vector perpendicular to the section plane. In most cases, d is chosen as the shortest lattice vector perpendicular to the section plane. However, in the triclinic and monoclinic crystal family this may not be possible, since the translations of the crystal pattern may not contain a vector perpendicular to the section plane. In that case, we assume that d captures the periodicity of the crystal pattern perpendicular to the section plane. This is achieved by choosing d as the shortest nonzero projection of a lattice vector to the line through the origin which is perpendicular to the section plane. Because of the periodicity of the crystal pattern along d, it is enough to consider heights s with , since for an integer m the sectional layer groups at heights s and s + m are conjugate subgroups of . This is a consequence of the orbit–stabilizer theorem in Section 1.1.7 , applied to the group acting on the planes in . The layer at height s is mapped to the layer at height s + m by the translation through md. Thus, the two layers lie in the same orbit under . According to the orbit–stabilizer theorem, the corresponding stabilizers, being just the layer groups at heights s and s + m, are then conjugate by the translation through md.
Since we assume a rational section, the sectional layer group will always contain translations along two independent directions , which, we assume, form a crystallographic basis for the lattice of translations fixing the section plane. The points in the section plane at height s are then given by . In order to determine whether the sectional layer group contains additional symmetry operations which are not translations, the following simple remark is crucial:
Let be an operation of a sectional layer group. Then the rotational part of maps d either to +d or to −d. In the former case, is sidepreserving, in the latter case it is sidereversing. Moreover, since the section plane remains fixed under , the vectors and are mapped to linear combinations of and by the rotational part of . Therefore, with respect to the (usually nonconventional) basis , , d of threedimensional space and some choice of origin, the operation has an augmented matrix of the formHere, . Moreover, if , i.e. is sidepreserving, then is necessarily zero, since otherwise the plane is shifted along d. On the other hand, if , i.e. is sidereversing, then a plane situated at height s along d is only fixed if .
From these considerations it is straightforward to determine the conditions under which a spacegroup operation belongs to a certain sectional layer group (excluding translations):
The sidepreserving operations will belong to the sectional layer groups for all planes perpendicular to d, independent of the height s:
Sidereversing operations will only occur in the sectional layer groups for planes at special heights along d:
Note that, because of the periodicity along d, a sidereversing operation that occurs at height s gives rise to a sidereversing operation of the same type occurring at height : if is a sidereversing symmetry operation fixing a layer at height s, then maps a point in the layer at height with coordinates (with respect to the layeradapted basis ) to a point with coordinates and hence the composition of with the translation by d maps to , i.e. it fixes the layer at height . This shows that the composition with the translation by d provides a onetoone correspondence between the sidereversing symmetry operations in the layer group at height s with those at height .
If a section allows any sidereversing symmetry at all, then the sidepreserving symmetries of the section form a subgroup of index 2 in the sectional layer group. Since the sidepreserving symmetries exist independently of the height parameter s, the full sectional layer group is always generated by the sidepreserving subgroup and either none or a single sidereversing symmetry.
Summarizing, one can conclude that for a given space group the interesting sections are those for which the perpendicular vector d is parallel or perpendicular to a symmetry direction of the group, e.g. an axis of a rotation or rotoinversion or the normal vector of a reflection or glide reflection.
Example
Consider the space group of type (31). In its standard setting, the cosets of relative to the translation subgroup are represented by the operations given in Table 1.4.5.1.

Since this is an orthorhombic group, it is natural to consider sections along the coordinate axes. The spacegroup diagrams displayed in Fig. 1.4.5.2, which show the orthogonal projections of the symmetry elements along these directions, are very helpful.