08 | Periodic materials and translational symmetry

Word count: 0 words
Reading Time: 0 minutes

A very large number of materials can, to varying degrees, be described as periodic. Crystals are the epitome of the periodic structure with atoms arranged precisely in three dimensions over long ranges to within a fraction of an Angstrom. Polymers very often display some degree of crystallinity as well, though typically over shorter length scales and/or few dimensions. Nonetheless, much can be understood about polymers by appreciating their local translational symmetries.

There are materials that lack translational symmetry. Conventional glasses, glassy polymers, gels, and quasicrystals all typically lack translational symmetry. However, as materials the long range interactions in these compounds remain important for understanding their properties. What we know of high symmetry systems can still be qualified and applied to these less ordered or more complex systems. It is the symmetry of periodic materials that allows us to analyze a material with using a repeat unit with a manageable one to a few dozen atoms from which the properties that emerge in the bulk material with 10²⁰ atoms can be calculated and understood.

Translation

Pure translation is the simplest translational symmetry operation. In the ideal infinite periodic system there is always a constant distance b which we can move an find our selves at a location indistinguishable from our starting position. An example of a 1D crystal is shown in Fig. 8.

Fig. 8 (Click image) Translational symmetry operation described as translate right by 1 unit cell length, \(a\).

Light is periodic and this constant distance for a monochromatic ray is called the wavelength. In polymers, this distance would correspond the the length of the monomer. In crystals this distance corresponds to the lengths of the edges of the unit cell. Crystals can be 1D, 2D, and most often 3D. In the 3D case, there would be 3 wavelengths (unit cell lengths) required to describe the periodicity of the crystal.

Matrix transformations that describe translation

Previously used matrix multiplication to linearly transform a set of coordinates (atom positions) from some reference state to another.

\[ \tilde{x} = \mathbf{W}x+\mathbf{w} \]

For operations of point symmetries the inclusion of \(\mathbf{w}\) was trivial since the translational component for symmetry operations about a point is always zero. As before, if \(\mathbf{W}\) and \(\mathbf{w}\) leave an object indistinguishable from its reference state then it can be described as a symmetry operation.

Here, \(\tilde{x}(W,w)\) moves all input coordinates by 1 unit along \(x\), 2 units along \(y\), and 3 units along \(z\).

\[\begin{split} \begin{bmatrix} x+1\\y+2\\z+3 \end{bmatrix} = \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix} + \begin{bmatrix} 1\\2\\3 \end{bmatrix} \end{split}\]

Glide

The glide operation combines a reflection with translation. And example of an \(a\)-glide \(\perp\) \(b\) is shown in Fig. 9. Glide planes can be described in both 2D and 3D systems. The symmetry of a glide operation is called a glide plane. However to define a glide operation both the axis of translation and the axis perpencular of the plane (or line in 2D) or reflection must be defined. In in Fig. 9 the \(a\)-glide \(\perp\) \(b\) describes a translation along the \(a\) direction coupled with a reflection accross a line perpendicular to the vertical axis, \(b\).

Fig. 9 (Click image) Glide symmetry operation described as translate right by 1 unit cell length, \(a\) and reflect across line \(\perp b\).

Matrix transformations that describe glides

Below is a an example for a 3D unit cell of a \(c\)-glide perpendicular to \(a\). That is a reflection of the \(x\) coordinate followed by a translation (glide) along the \(z\) axis.

\[\begin{split} \begin{bmatrix} -x\\y\\z+0.5 \end{bmatrix} = \begin{bmatrix} -1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix} + \begin{bmatrix} 0\\0\\0.5 \end{bmatrix} \end{split}\]

In this example the glide plane is clearly centered at the origin but with the inclusion of a translation vector (\(w\)) in our transform function \(\tilde{x}(\mathbf{W},\mathbf{w})\) this need not be the case. For example here is an example of the same glide transformation with the mirror plane located at x = 0.5

\[\begin{split} \begin{bmatrix} 1-x\\y\\z+0.5 \end{bmatrix} = \begin{bmatrix} -1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix} + \begin{bmatrix} 1\\0\\0.5 \end{bmatrix} \end{split}\]

Screw

A screw operation is combination of rotation and translation.

Fig. 10 (Click image) Screw symmetry operation described as translate right along \(a\) and turn 90º about \(a\). This operation is named a \(4_1\)-screw.

Matrix transformations that describe screws

Here is an example of a 180° screw along the \(z\)-axis.

\[\begin{split} \begin{bmatrix} -x\\-y\\z+0.5 \end{bmatrix} = \begin{bmatrix} -1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix} + \begin{bmatrix} 1\\0\\0.5 \end{bmatrix} \end{split}\]

Exercises

Define \(W\) and \(w\) for the following transformations

1. 90º screw along the \(x\)-axis coupled with at 0.25 unit translation

2. a pure translation by 0.5 units along both the \(x\)-axis and \(y\)-axis.

Describe in words the following transformation:

\[\begin{split} W= \begin{bmatrix} -1&0&0\\ 0&-1&0\\ 0&0&-1\\ \end{bmatrix};\; w=\begin{bmatrix} 0.5 \\ 0.5 \\ 0 \end{bmatrix} \end{split}\]