Day 6

Whew ... let's recap the previous few days:

We observed the consistency of interfacial angles in all crystals of a given mineral. Based on this observation we concluded that minerals are built according to a pattern. We identified vectors called translations that represent the magnitudes of offset that are characteristic of the repetition in the pattern. We defined the lattice of a crystal as the set of points that are equivalent by translation. Three non-coplanar translations were used to define a unit cell, a parallelepiped that contains all the symmetry elements of the pattern. We considered three-dimensional lattices and forms in the context of the triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and isometric systems.

Now then ....

In the context of a pattern of infinite extent, the operation of repeating the unit cell by offset (translation) is a symmetry operation: the pattern is unchanged by the operation of translation.

Therefore...

Because patterns (as opposed to objects) have transitional symmetry, they may also possess symmetry elements that are combinations of translation and reflection (glide planes) or translation and rotation (screw axes). Two-dimensional patterns can contain glide planes and many popular wallpaper patterns repeat motifs by glide reflection. Three dimensions are needed for the screw rotation that leads, for example, to the helix popular among organic molecules.

Glide planes repeat motifs by (a) translation parallel to the glide plane, followed by (b) reflection across the glide plane. Typically, the magnitude of the translation part of the glide reflection is one half the lattice translation parallel to the glide reflection. In three dimensions, the glide plane is identified by the unit cell translation that is parallel to the glide translation (e.g. a-, b-, or c-glide). Diagonal glides are also possible. Glide reflection is an enantiomorphous operation.

Screw axes repeat motifs by (a) translation parallel to the screw axis, followed by (b) rotation about the screw axis. Typically, the magnitude of the translation part of the screw axis is equal to the lattice translation parallel to the screw axis divided by the "fold" of the rotation part of the screw direction. Thus, a 6-fold screw axis would translate a motif 1/6 of the lattice translation parallel to the screw axis and rotate the motif 60° about the screw axis. Screw rotation is a congruent operation. Screw axes may be either right-handed or left-handed.

Because all symmetry elements of a pattern repeat one another, only certain collections of symmetry elements are possible for patterns. These collections are the 17 plane groups in two dimensions and the 230 space groups in three dimensions. All two-dimensional patterns belong to one of the 17 plane groups described in the handout. All minerals have crystal structures that belong to one of the 230 space groups described in the International Tables for X-ray Crystallography. This means that it is not necessary to identify all the symmetry elements of a two-dimensional pattern or crystal structure from scratch (be thankful for the little things). Only enough symmetry elements need to be identified to place the pattern or structure in one of the possible groups.

As an example of the interaction of symmetry elements that limits the number of possible space groups, consider that only 1-, 2-, 3-, 4-, and 6- fold rotation axes are possible in minerals. This is proven mathematically on pages 115-116 of the Klein and Hurlburt text. 5-fold rotation axes of symmetry do exist for some objects (echinoderms, some flowers, beer mugs, etc.) but not in the long-range order of minerals. Nor are there any axes of rotation higher than 6-fold.