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. 71-73

## Section 1.4.5.3. Projections

B. Souvignierd

#### 1.4.5.3. Projections

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As we have seen, a section of a crystal pattern is determined by a vector d and a height s along this vector. Choosing two vectors and perpendicular to d, the points of the section plane at height s are precisely given by the vectors . In contrast to that, a projection of a crystal pattern along d is obtained by mapping an arbitrary point to the point of the plane spanned by and , thereby ignoring the coordinate along the d direction.

#### Definition

In a projection of a crystal pattern along the projection direction d, a point X of the crystal pattern is mapped to the intersection of the line through X along d with a fixed plane perpendicular to d.

One may think of the projection plane as the plane perpendicular to d and containing the origin, but every plane perpendicular to d will give the same result.

Let be the line along d. If a symmetry operation of a space group maps to a line parallel to , then maps every plane perpendicular to d again to a plane perpendicular to d. This means that points that are projected to a single point (i.e. points on a line parallel to ) are mapped by to points that are again projected to a single point and thus the operation gives rise to a symmetry of the projection of the crystal pattern. Conversely, an operation that maps to a line that is inclined to does not result in a symmetry of the projection, since the points on are projected to a single point, whereas the image points under are projected to a line. In summary, the operations of that map to a line parallel to give rise to symmetries of the projection forming a plane group, sometimes called a wallpaper group.

Let be the subgroup of consisting of those mapping the line to a line parallel to , then is called the scanning group along d. The scanning group can be read off a coset decomposition relative to the translation subgroup of . Since translations map lines to parallel lines, one only has to check whether a coset representative maps to a line parallel to . This is precisely the case if the linear part of maps d to d or to −d. Therefore, is the union of those cosets relative to for which the linear part of maps d to d or to −d.

If the operations of a space group are written as augmented matrices with respect to a (usually non-conventional) basis , , d such that and are perpendicular to d, then an operation of the scanning group is of the formwith (just as for planar sections). Then the action of on the projection along d is obtained by ignoring the z coordinate, i.e. by cutting out the upper 2 × 2 block of the linear part and the first two components of the translation part. This gives rise to the plane-group operation

The mapping that assigns to each operation of the scanning group its action on the projection is in fact a homomorphism from to a plane group and the kernel of this homomorphism are the operations of the formi.e. translations along d and reflections with normal vector parallel to d.

#### Definition

The symmetry group of the projection along the projection direction d is the plane group of actions on the projection of those operations of that map the line along d to a line parallel to .

This group is isomorphic to the quotient group of the scanning group along d by the group of translations along d and reflections with normal vector parallel to d.

#### Example

We consider again the space group of type (31) for which the augmented matrices of the coset representatives with respect to the translation subgroup (in the standard setting) are given by

Since the linear parts of all four matrices are diagonal matrices, the scanning group for projections along the coordinate axes is always the full group .

For the projection along the direction [100], one has to cut out the lower 2 × 2 part of the linear parts and the second and third component of the translation part, thus choosing as a basis for the projection plane. This gives as matrices for the projected operationsin which the third and fourth operations are clearly redundant and which is thus a plane group of type p1g1 (plane group No. 4 with short symbol pg).

The projection along the direction [010] gives for the basis , of the projection plane (thus picking out the first and third rows and columns) the matriceswhere the second matrix is the product of the third and fourth. The third operation is a centring translation, the fourth a reflection, thus the resulting plane group is of type c1m1 (plane group No. 5 with short symbol cm).

Finally, the projection along the direction [001] results for the basis of the projection plane in the matriceswhere again the second matrix is the product of two others. The third operation is a glide reflection and the fourth is a reflection, thus the corresponding plane group is of type p2mg (plane group No. 7). Note that in order to obtain the plane group p2mg in its standard setting, the origin has to be shifted to (with respect to the plane basis ).

As for the sectional layer groups, the typical projection directions considered are symmetry directions of the space group , i.e. directions along rotation or screw axes or normal to reflection or glide planes. In order to relate the coordinate system of the plane group to that of the space group, not only the basis vectors perpendicular to the projection direction d have to be given, but also the origin for the plane group. This is done by specifying a line parallel to the projection direction which is projected to the origin of the plane group in its conventional setting. The space-group tables list the plane groups for the projections along symmetry directions of the group in the block Symmetry of special projections'.

It is not hard to determine the corresponding types of plane-group operations for the different types of space-group operations, as is shown by the following list of simple rules:

 (i) a translation becomes a translation (possibly the identity); (ii) an inversion becomes a twofold rotation; (iii) a k-fold rotation or screw rotation with axis parallel to d becomes a k-fold rotation; (iv) a three-, four- or sixfold rotoinversion with axis parallel to d becomes a six-, four- or threefold rotation, respectively; (v) a reflection or glide reflection with normal vector parallel to d becomes a translation (possibly the identity); (vi) a twofold rotation and a screw rotation with axis perpendicular to d become a reflection and glide reflection, respectively; (vii) a reflection or a glide reflection with normal vector perpendicular to d becomes a reflection or glide reflection depending on whether there is a glide component perpendicular to d or not.

The relationship between the symmetry operations in three-dimensional space and the corresponding symmetry operations of a projection as listed above can be seen directly in the diagrams of the corresponding groups. In Fig. 1.4.5.4, the top diagram shows the orthogonal projection of the symmetry-element diagram of along the [001] direction and the bottom diagram shows the diagram for the plane group p2mg, which is precisely the symmetry group of the projection of along [001]. Firstly, one sees immediately that in order to match the two diagrams, the origin in the projection plane has to be shifted to (as already noted in the example above). Secondly, keeping in mind that the projection direction d is perpendicular to the drawing plane, one sees the correspondence between the twofold screw rotations in with the twofold rotations in p2mg [rule (iii)], the correspondence between the reflections with normal vector perpendicular to d in and the reflections in p2mg [rule (vii)] and the correspondence between the diagonal glide reflections in (indicated by the dot-dash lines) and the glide reflections in p2mg {rule (vii); note that the diagonal glide vector has a component perpendicular to the projection direction [001]}.

#### Example

Let be a space group of type (117), then the interesting projection directions (i.e. symmetry directions) are [100], [010], [001], [110] and . However, the directions [100] and [010] are symmetry-related by the fourfold rotoinversion and thus result in the same projection. The same holds for the directions [110] and . The three remaining directions are genuinely different and the projections along these directions will be discussed in detail below. The corresponding information given in the space-group tables under the heading Symmetry of special projections' is reproduced in Fig. 1.4.5.5 for .

 Figure 1.4.5.4 | top | pdf |Orthogonal projection along [001] of the symmetry-element diagram for (31) (top) and the diagram for plane group p2mg (7) (bottom).
 Figure 1.4.5.5 | top | pdf |`Symmetry of special projections' block of (117) as given in the space-group tables.

Coset representatives of relative to its translation subgroup can be extracted from the general-positions block in the space-group tables of and are given in Table 1.4.5.2.

• d along [001]: The linear parts of all coset representatives map [001] to ±[001], and therefore the scanning group is the full group . A conventional basis for the translations of the projection is and . The operation acts as a fourfold rotation, acts as a glide reflection with normal vector and as a reflection with normal vector . Thus, the resulting plane group has type p4gm (plane group No. 12). The line parallel to the projection direction [001] which is projected to the origin of p4gm in its conventional setting is the line 0, 0, z.

 Table 1.4.5.2| top | pdf | Coset representatives of (117) relative to its translation subgroup
Coordinate tripletDescription
: Identity
: Twofold rotation with axis along [001]
: Fourfold rotoinversion with axis along [001]
: Fourfold rotoinversion with axis along [001]
: Glide reflection with normal vector [010] and glide component along [100]
: Glide reflection with normal vector [100] and glide component along [010]
: Twofold screw rotation with axis parallel to [110]
: Twofold rotation with axis parallel to

Again, it is instructive to look at the symmetry-element diagrams for the respective space and plane groups, as displayed in Fig. 1.4.5.6. The twofold rotations and fourfold rotoinversions with axis along [001] are turned into twofold rotations and fourfold rotations, respectively [rules (iii) and (iv)]. The glide reflections with both normal vector and glide vector perpendicular to [001] (dashed lines) result in glide reflections [rule (vii)]. The twofold rotations (full arrows) and screw rotations (half arrows) with rotation axis perpendicular to [001] give reflections and glide reflections, respectively [rule (vi)]. Note that the two diagrams can be matched directly, because the line 0, 0, z which is projected to the origin of p4gm runs through the origin of .

 Figure 1.4.5.6 | top | pdf |Orthogonal projection along [001] of the symmetry-element diagram for (117) (left) and the diagram for plane group p4gm (12) (right).
• d along [100]: Only the linear parts of the coset representatives , , and map [100] to ±[100], thus these four cosets form the scanning group (which is of index 2 in ). The operation acts as a translation by , thus a conventional basis for the translations of the projection is and . The operation acts as a reflection with normal vector and acts as the same reflection composed with the translation . The resulting plane group is thus of type p1m1 (plane group No. 3 with short symbol pm). The line which is mapped to the origin of p1m1 in its conventional setting is x, 0, 0.

• d along [110]: Only the linear parts of the coset representatives , , and map [110] to ±[110], thus these four cosets form the scanning group (of index 2 in ). The translation by is projected to a translation by , thus a conventional basis for the translations of the projection is and . The operation acts as a reflection with normal vector , acts as a twofold rotation and acts as a reflection with normal vector . The resulting plane group is thus of type p2mm (plane group No. 6). The line parallel to the projection direction [110] that is mapped to the origin of p2mm (in its conventional setting) is x, x, 0.

Note that for directions different from those considered above, additional non-trivial plane groups may be obtained. For example, for the projection direction , the scanning group consists of the cosets of and . The operation acts as a glide reflection and the resulting plane group is of type c1m1 (plane group No. 5).

### References

Vainshtein, B. K. (1994). Fundamentals of Crystals. Symmetry, and Methods of Structural Crystallography. Berlin, Heidelberg, New York: Springer.