11 | Plane and space groups

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Just as there are groups of symmetry elements arranged about a point in space (point groups), elements of symmetry can also be arrange along a line (line group), across a plane (plane group), and throughout space (space group). However, in these cases where all symmetry operations (not including the identity operation) do not coincide with a common point it, any object describe by this group can be generated in an infinite number of new locations.

Exercise

1. Draw two parallel mirror plane and an object anywhere else on the plane.

2. Transform the object through the first mirror plane and draw it in its new location.

3. Transform the object through the second mirror plane and draw it in its new location.

4. Go to step 2 and repeat until you are satisfied an infinite periodic array of objects has been generated.

Line Groups

Line groups can be mathematically described and certainly can be observed in chemistry but practically are not commonly encountered in chemistry research nor commonly used as a thinking tool for understanding chemical structure. However note that should you find your self regularly working with 1D crystals, their symmetry can certainly be described by a line group and doing so is no more difficult than describign 2D or 3D crystals in Hermann-Mauguin notation.

Plane Groups

While there are 32 crystallographic points groups there are only 17 plane groups the occupy the 4 crystal systems available in two-dimensions. Sometimes these are referred to as wallpaper groups. These groups are compiled in the International Tables of Crystallography Volume A Section 6.1 (2006), Wikipedia, and elsewhere.

In 2D in inversion operation is indistinguishable from a 2-fold proper rotation and is there for not typically included as a symmetry operation when defining these groups. Likewise a screw axis is indistinguishable from a glide plane and also not included a symmetry operation in these groups to avoid redundancy.

The symmetry operation of plane groups are visulaized by drawing all of the symmetry element that residew within the bounds of a single unit cell. For example, Fig. 21 shows the symmetry diagram for one plane group called \(pmm\) in Hermann-Mauguin notation. This unit cell is rectangular (\(\beta=90°,\, a\neq b\)) and the group can be generated from two perpendicular mirror planes. The lattice points sit at the intersection of these two mirror planes. By apply both mirror operations is equivalent to just applying one 2-fold proper rotation and is denoted as a diamond in Fig. 21. by iteratively applying these symmetry operations in the lattice all of the elements shown for \(pmm\) will, eventually, be generated.

Fig. 21 Symmetry diagram for a unit cell in the plane group \(pmm\). Lattice points are shown in red. Mirror planes are shown as pink, purple, and orange lines, the assymetric unit is highlighted in yellow and the letter F is draw to illustrate how an object transforms within the unit cell of this group.

Plane group symbols

Plane groups symbols like point group symbols in Hermann-Mauguin notation contain the elements required to generated the entire symmetry of the group. Unlike point groups however they also much not denote the centering of the lattice. The crystal system of the plane group can be immediate inferred from the the plane group symbol and there are naming conventions for choosing a consistent set of generators and their ordering within the plane group symbol.

Consider \(pmm\) The first position informs

Assigning plane groups

1. Find lattice points

2. Define a unit cell

3. Draw all the key elements of symmetry

4. Identify which of the 17 groups is described

Space groups

1. There are 230 space groups

2. They are tabulated in the international tables

3. We can illustrate their symmetry operations in 2D like the plane groups using multiple projections of the unit cell and designating elements offset perpendicular to the page.

Space group symbols

1. First letter denotes the centering (P, A, B, C, I, F, R)

2. Remaining symbols list the symmetry elements that can generate all other symmetry elements in the unit cell.

3. The conventions for listing these generators are unique for each crystal system

4. You should be able to easily identify the crystal system and centering of a crystal from the space groups symbol.

5. The point group of a crystal can be identified by simply replace all translational symmetry elements in the space group symbols with their purely rotational components; i.e. glides –> \(m\), \(2_1\) screw –> \(2\) rotation, translation -> identity (\(1\)).

6. Polar crystals like polar molecules contain a single axis through which all other symmetry elements in the crystal’s point group intersect.