Phason elasticity and atomic dynamics of quasicrystals

As the order of a quasicrystal is quasiperiodic, it can be described as an irrational cut through a periodic structure in a higher-dimensional space. This mathematical trick has important consequences for the low energy excitations that can occur. Translating the cut space to a different position is a symmetry operation, which changes the quasicrystal structure, but not its energy. A small breaking of this symmetry, by chosing a cut of small and slowly varying slope (with respect to the ideal orientation) therefore leads to low-energy Goldstone modes, called phasons. In many respects, phasons are analogous to phonons, which are small and slowly varying distortions of a structure in physical space. The distortions related to phasons rest in the complementary, internal space needed for the embedding of the quasicrystal in the higher-dimensional crystal. In much the same way as there is an elastic energy for phonon type distortions of a solid, there is an effective elasticity theory for the phason degrees of freedom, which moreover is coupled to the phonon elasticity. A non-zero phason strain, i.e., a non-zero slope of the cut space, will cost energy. At higher temperatures, however, phason excitations, which correspond to a fluctuating phason strain, will become possible. This is analogous to phonons. There is one important difference, however. Whereas phonons are usually propagating modes, phasons are believed to be diffusive. Phason strain in a quasicrystal is connected with a rearrangement of certain local atomic configurations. In an elementary form these rearrangements are called phason flips. Understanding the dynamics of phason flips and other atomic rearrangements of the structure of a quasicrystal is essential for the understanding of the formation and stability of quasicrystals, and also for many of their physical properties. Perfect quasicrystals are usually obtained by high temperature annealing after solidification. During this process, many defects initially present are eliminated. It is therefore necessary that phason flips, atomic diffusion, and other dynamical processes are possible and effective at these temperatures. Atomic diffusion and phason mobility are also important for the mobility of dislocations. Unlike in a crystal, in a quasicrystal a moving dislocation leaves a phason wall in its wake. This phason wall must be smoothed out and finally eliminated by phason flips and diffusion processes, for otherwise the material would harden very quickly. Mobile phason flips are therefore necessary for the ductility of quasicrystals observed at high temperatures. There are several other interesting consequences …