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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 68-71

Section 1.4.5.2. Sections

B. Souvignierd

1.4.5.2. Sections

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For a space group [{\cal G}] and a point X in the three-dimensional point space [{\bb E}^3], the site-symmetry group of X is the subgroup of operations of [{\cal G}] that fix X. Analogously, one can also look at the subgroup of operations fixing a one-dimensional line or a two-dimensional plane. If the line is along a rational direction, it will be fixed at least by the translations of [{\cal G}] along that direction. However, it may also be fixed by a symmetry operation that reverses the direction of the line. The resulting subgroup of [{\cal G}] that fixes the line is a rod group.

Similarly, a plane having a normal vector along a rational direction is fixed by translations of [{\cal G}] corresponding to a two-dimensional 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 

  • (i) The symmetry group of a planar section of a crystal pattern is the subgroup of the space group [{\cal G}] of the crystal pattern that leaves the section plane invariant as a whole.

    If the section is a rational section, this symmetry group is a layer group, i.e. a subgroup of a space group which contains translations only in a two-dimensional plane.

  • (ii) The symmetry group of a linear section of a crystal pattern is the subgroup of the space group [{\cal G}] of the crystal pattern that leaves the penetration line invariant as a whole.

    If the section is a rational section, this symmetry group is a rod group, i.e. a subgroup of a space group which contains translations only along a one-dimensional line.

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 one-dimensional 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 non-zero 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 [0 \leq s \,\lt\, 1], since for an integer m the sectional layer groups at heights s and s + m are conjugate subgroups of [{\cal G}]. This is a consequence of the orbit–stabilizer theorem in Section 1.1.7[link] , applied to the group [{\cal G}] acting on the planes in [{\bb E}^3]. 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 [{\cal G}]. 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 [{\bf a}'], [{\bf b}'] 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 [x {\bf a}' + y {\bf b}' + s {\bf d}]. In order to determine whether the sectional layer group contains additional symmetry operations which are not translations, the following simple remark is crucial:

Let [\ispecialfonts\sfi g] be an operation of a sectional layer group. Then the rotational part of [\ispecialfonts\sfi g] maps d either to +d or to −d. In the former case, [\ispecialfonts\sfi g] is side-preserving, in the latter case it is side-reversing. Moreover, since the section plane remains fixed under [\ispecialfonts\sfi g], the vectors [{\bf a}'] and [{\bf b}'] are mapped to linear combinations of [{\bf a}'] and [{\bf b}'] by the rotational part of [\ispecialfonts\sfi g]. Therefore, with respect to the (usually non-conventional) basis [{\bf a}'], [{\bf b}'], d of three-dimensional space and some choice of origin, the operation [\ispecialfonts\sfi g] has an augmented matrix of the form[\let\normalbaselines\relax\openup-1pt\pmatrix{r_{11} & r_{12} & 0 {\hskip -4pt}&{\vrule height 8pt depth 5pt}\hfill&{\hskip -4pt}t_1\cr r_{21} & r_{22} & 0{\hskip -4pt}& {\vrule height 8pt depth 5pt}\hfill&{\hskip -4pt}t_2\cr 0 & 0 & r_{33}{\hskip -4pt}& {\vrule height 8pt depth 5pt}\hfill&{\hskip -4pt}t_3\cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 11pt depth 4pt}\hfill&{\hskip -4pt} 1}.]Here, [r_{33} = \pm 1]. Moreover, if [r_{33} = 1], i.e. [\ispecialfonts\sfi g] is side-preserving, then [t_3] is necessarily zero, since otherwise the plane is shifted along d. On the other hand, if [r_{33} = -1], i.e. [\ispecialfonts\sfi g] is side-reversing, then a plane situated at height s along d is only fixed if [t_3 = 2s].

From these considerations it is straightforward to determine the conditions under which a space-group operation belongs to a certain sectional layer group (excluding translations):

The side-preserving operations will belong to the sectional layer groups for all planes perpendicular to d, independent of the height s:

  • (i) rotations with axis parallel to d;

  • (ii) reflections with normal vector perpendicular to d;

  • (iii) glide reflections with normal vector and glide vector perpendicular to d.

Side-reversing operations will only occur in the sectional layer groups for planes at special heights along d:

  • (i) inversion with inversion point in the section plane;

  • (ii) twofold rotations or twofold screw rotations with rotation axis in the section plane;

  • (iii) reflections or glide reflections through the section plane with glide vector perpendicular to d;

  • (iv) rotoinversions with axis parallel to d and inversion point in the section plane.

Note that, because of the periodicity along d, a side-reversing operation that occurs at height s gives rise to a side-reversing operation of the same type occurring at height [s+\textstyle{{1 \over 2}}]: if [\ispecialfonts\sfi g] is a side-reversing symmetry operation fixing a layer at height s, then [\ispecialfonts\sfi g] maps a point in the layer at height [s + \textstyle{{1 \over 2}}] with coordinates [x,y,s+\textstyle{{1 \over 2}}] (with respect to the layer-adapted basis [{\bf a}', {\bf b}', {\bf d}]) to a point with coordinates [{x}',{y}',s-\textstyle{{1 \over 2}}] and hence the composition [\ispecialfonts{\sfi t}_{{\bf d}} {\sfi g}] of [\ispecialfonts\sfi g] with the translation by d maps [x,y,s+\textstyle{{1 \over 2}}] to [{x}',{y}',s+\textstyle{{1 \over 2}}], i.e. it fixes the layer at height [s + \textstyle{{1 \over 2}}]. This shows that the composition with the translation by d provides a one-to-one correspondence between the side-reversing symmetry operations in the layer group at height s with those at height [s + \textstyle{{1 \over 2}}].

If a section allows any side-reversing symmetry at all, then the side-preserving symmetries of the section form a subgroup of index 2 in the sectional layer group. Since the side-preserving symmetries exist independently of the height parameter s, the full sectional layer group is always generated by the side-preserving subgroup and either none or a single side-reversing 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 [{\cal G}] of type [Pmn2_1] (31). In its standard setting, the cosets of [{\cal G}] relative to the translation subgroup are represented by the operations given in Table 1.4.5.1[link].

Table 1.4.5.1| top | pdf |
Coset representatives of [Pmn2_1] (31) relative to its translation subgroup

Seitz symbolCoordinate tripletDescription
[\{1|0\}] [x,y,z] Identity
[\{2_{001}|\textstyle{\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}}\}] [\bar{x}+\textstyle{{1 \over 2}},\bar{y},z+\textstyle{{1 \over 2}}] Twofold screw rotation with axis along [001]
[\{m_{010}|\textstyle{\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}}\}] [x+\textstyle{{1 \over 2}},\bar{y},z+\textstyle{{1 \over 2}}] n-glide reflection with normal vector along [010]
[\{m_{100}|0\}] [\bar{x},y,z] Reflection with normal vector along [100]

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

  • d along [100]: A point [x,y,z] in a plane perpendicular to the coordinate axis along [100] is mapped to a point [x',y',z'] in the same plane if [x' = x], i.e. if [x' - x = 0].

    [Figure 1.4.5.2]

    Figure 1.4.5.2 | top | pdf |

    Symmetry-element diagrams for the space group [Pmn2_1] (31) for orthogonal projections along [100] (left), [010] (middle) and [001] (right).

    A general operation from the coset of [\{2_{001}|\textstyle{\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}}\}] maps a point with coordinates [x, y, z] to a point with coordinates [x' = \bar{x}+\textstyle{{1 \over 2}}+u_1, y'=\bar{y}+u_2, z'=z+\textstyle{{1 \over 2}}+u_3] for integers [u_1], [u_2], [u_3]. One has [x' - x = -2x + \textstyle{{1 \over 2}} + u_1] which becomes zero for [x = \textstyle{{1 \over 4}}] (and [u_1 = 0]) and [x = \textstyle{{3}\over{4}}] (and [u_1 = 1]), thus operations from the coset of [\{2_{001}|\textstyle{\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}}\}] fix planes at heights [s = \textstyle{{1 \over 4}}] and [\textstyle{{3}\over{4}}]. In the left-hand diagram in Fig. 1.4.5.2[link], the symmetry elements to which these operations belong are indicated by the half-arrows, the label [\textstyle{{1 \over 4}}] indicating that they are at level [x = \textstyle{{1 \over 4}}] and [x = \textstyle{{3}\over{4}}].

    An operation from the coset of [\{m_{010}|\textstyle{\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}}\}] maps x, y, z to [x' = x+\textstyle{{1 \over 2}}+u_1, y'=\bar{y}+u_2, z'=z+\textstyle{{1 \over 2}}+u_3] and one has [x' - x] = [\textstyle{{1 \over 2}} + u_1]. Since this is never zero, no operation from this coset fixes a plane perpendicular to [100].

    Finally, an operation from the coset of [\{m_{100}|0\}] maps x, y, z to [x' = \bar{x}+u_1], [y'=y+u_2], [z'=z+u_3] and one has [x' - x] = [-2x + u_1], which becomes zero for x = 0 (and [u_1 = 0]) and [x = \textstyle{{1 \over 2}}] (and [u_1 = 1]). Thus, operations from the coset of [\{m_{100}|0\}] fix planes at heights s = 0 and [\textstyle{{1 \over 2}}]. The symmetry elements of these reflections with mirror plane parallel to the projection plane are indicated by the right-angle symbol in the upper left corner of the left-hand diagram in Fig. 1.4.5.2[link].

    The sectional layer groups are thus layer groups of type pm11 (layer group No. 4 with symbol p11m in a non-standard setting) for s = 0 and [s = \textstyle{{1 \over 2}}], of type [p112_1] (layer group No. 9 with symbol [p2_111] in a non-standard setting) for [s = \textstyle{{1 \over 4}}] and [s =\textstyle{{3}\over{4}}] and of type p1 (layer group No. 1) for all other s between 0 and 1. The side-preserving operations are in all cases just the translations.

    It is worthwhile noting that in many cases most of the information about the sectional layer groups can be read off the space-group diagrams. In the present example, the left-hand diagram in Fig. 1.4.5.2[link] displays the twofold screw rotation at height [s = \textstyle{{1 \over 4}}] (and thus also at [s = \textstyle{{3}\over{4}}]) and the reflection at height s = 0 (and thus also at [s = \textstyle{{1 \over 2}}]). On the other hand, the n glide, indicated by the dashed-dotted lines in the diagram, does not give rise to an element of the sectional layer group, because its glide vector has a component along the [100] direction and can thus not fix any layer along this direction.

  • d along [010]: A point [x,y,z] in a plane perpendicular to the coordinate axis along [010] is mapped to a point [x',y',z'] in the same plane if [y' = y], i.e. if [y' - y = 0].

    From the calculations above one sees that for operations in the coset of [\{m_{100}|0\}] one has [y' - y = u_2], hence operations in this coset fix the plane for any value of s and are side-preserving operations. In the middle diagram in Fig. 1.4.5.2[link] the sym­metry elements for these reflections are indicated by the horizontal solid lines.

    For the operations in the coset of [\{2_{001}|\textstyle{\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}}\}] one has [y' - y = -2y + u_2], and so these operations fix planes only for s = 0 and [s = \textstyle{{1 \over 2}}]. The same is true for the operations in the coset of [\{m_{010}|\textstyle{\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}}\}], because here one also has [y' - y = -2y + u_2]. The symmetry elements to which the screw rotations belong are indicated by the half arrows in the middle diagram of Fig. 1.4.5.2[link], and the symmetry elements for the glide reflections are symbolized by the right angle with diagonal arrow in the upper left corner, indicating that the geometric element is a diagonal glide plane.

    The sectional layer groups are thus of type [pmn2_1] (layer group No. 32 with symbol [pm2_1n] in a non-standard setting) for [s = 0, \textstyle{{1 \over 2}}] and of type pm11 (layer group No. 11) for all other s. The group of side-preserving operations is in all cases of type pm11.

    In Fig. 1.4.5.3[link] the diagram of the symmetry elements for the layer group [pm2_1n] (layer group No. 32) is displayed. It coincides with the middle diagram in Fig. 1.4.5.2[link] (up to the placement of the symbol for the diagonal glide plane), showing that in this case the sectional layer groups can also be read off directly from the space-group diagrams.

    [Figure 1.4.5.3]

    Figure 1.4.5.3 | top | pdf |

    Symmetry-element diagram for the layer group [pm2_1n] (32).

  • d along [001]: A point [x,y,z] in a plane perpendicular to the coordinate axis along [001] is mapped to a point [x',y',z'] in the same plane if [z' = z], i.e. if [z' - z = 0].

    As in the case of d along [010], operations in the coset of [\{m_{100}|0\}] fix such a plane for any value of s, since [z' - z = u_3]. Again, these are side-preserving operations. The symmetry elements to which these reflections belong are indicated by the horizontal solid lines in the right-hand diagram in Fig. 1.4.5.2[link].

    For the operations in the cosets of [\{2_{001}|\textstyle{\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}}\}] and [\{m_{010}|\textstyle{\textstyle{{1 \over 2}},0,\textstyle{{1 \over 2}}}\}] one has [z' - z = \textstyle{{1 \over 2}} + u_3], which is never zero (for an integer [u_3]), and so operations in these cosets never fix a plane perpendicular to [001].

    Thus, for any value of s the sectional layer group is of type pm11 (layer group No. 11) and contains only side-preserving operations.








































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