Quasicrystal

A quasicrystal is a peculiar form of solid in which the atoms of the solid are arranged in a seemingly regular, yet non-repeating structure.

Patterns in Quasicrystals

In a normal crystalline solid the positions of atoms are arranged in a periodic[?] lattice of points, which repeats itself in 3 dimensions the same way that a honeycomb structure repeats itself: each cell has an identical pattern of cells surrounding it. In a quasicrystal, the pattern of atoms is only quasiperiodic[?]. The local arrangements of atoms are fixed, and in a regular pattern, but are not periodic: each cell has a different configuration of cells surrounding it.

There is a strong analogy between the quasicrystal and the Penrose tiling of Roger Penrose. In fact, some quasicrystals can be sliced such that the atoms on the surface follow the exact pattern of the Penrose tiling.

Quasicrystals are remarkable in that some of them display five-fold symmetry. Prior to the discovery of quasicrystals, it was thought that five-fold crystal symmetry was impossible, because there are no space-filling periodic tilings which have five-fold symmetry.

The Geometric Interpretation

For a periodic pattern, if you fill all of space with the pattern, you can slide the pattern a certain distance in a certain direction, and every atom will lie exactly where an atom lay in the original pattern.

For a quasiperiodic pattern, if you fill space with it, there is no distance you can slide the pattern to make every atom lie exactly where an atom lay in the original pattern. However, you can take a bounded region, no matter how large, and slide it to match up exactly with some other part of the original pattern.

There is actually a simple relationship between periodic and quasiperiodic patterns. Any quasiperiodic pattern of points can be formed from a periodic pattern in some higher dimension.

For example, to create the pattern for a three-dimensional quasicrystal, you can start with a regular grid of points in six-dimensionsal space. Let the 3D space be a hyperplane that passes through 6D space at an angle. Take every point in the 6D space that is within a certain distance of the 3D hyperplane. Project those points into the hyperplane. If the angle and distance were chosen correctly, those projected points will form a quasiperiodic pattern.

Every quasiperiodic pattern can be generated this way. Every pattern generated this way will be either periodic or quasiperiodic.

This geometric approach is a useful way to analyze physical quasicrystals. In a crystal, flaws are locations where the pattern is interrupted. In a quasicrystal, flaws are locations where the 3D "hyperplane" is bent, or wrinkled, or broken as it passes through the higher-dimensional space.

Resources

• D. P. DiVincenzo and P. J. Steinhardt, eds. 1991. Quasicrystals: The State of the Art. Directions in Condensed Matter Physics, Vol 11. ISBN 9810205228.