Christopher J. Kimmer, Ph.D.

IU Southeast Informatics

Quasiperiodic Structures

The observation of an icosahedrally symmetric diffraction pattern from an AlMn alloy in 1984 by Shechtman, et al[ 1 ] demonstrated that long-range orientational order existed in materials with symmetries that were known to be incompatible with periodicity. Almost simultaneously with the announcement of the alloy's unusual diffraction pattern, Mackay[ 2 ] and Levine and Steinhardt[ 3 ] hypothesized that phases of matter that exhibited quasiperiodic translational order with perfect orientational order could exist. Levine and Steinhardt dubbed this new class of hypothetical materials quasicrystals and proposed that the AlMn alloy observed by Shechtman, et al was indeed an icosahedral quasicrystal.

After their initial discovery, quasicrystals became the subject of intense study, and the high-resolution X-ray investigation of the icosahedral quasicrystal Al ₆ Mn revealed that its diffraction peaks could be indexed by six integers[ 4 ]. This indexing reflected the already-known fact that quasiperiodic functions could be represented as cuts of higher-dimensional periodic functions[ 5 ], and similar ideas were employed to produce an icosahedral diffracton pattern as a cut through or projection from a subset of a six-dimensional periodic lattice to a three-dimensional quasiperiodic point set, or quasilattice[ 6 , 7 , 8 , 9 ].

Both Mackay and Levine and Steinhardt's original inspirations for the definition and concept of a quasicrystal was the Penrose tiling[ 10 , 11 ], a two-dimensional quasiperiodic tiling with decagonal symmetry. A variety of tilings made up of different basic tiles, or prototiles, are called Penrose tilings, but more specifically, the prototiles can be chosen (up to symmetry) to be one fat and one skinny rhombus. The tiling can be assembled by following local matching rules that restrict the permissible configurations of tiles about any given vertex. The matching rules may take the form of a decoration of each tile's edge so that only edges with the same decoration may be aligned; any infinite assemblage of these tiles obeying these matching rules will be a quasiperiodic tiling. The constraints matching rules place on local environments in a Penrose tiling gave rise the idea that a small number of energetically-favorable arrangments of atoms in a quasicrystal can give rise to an energetically-stabilized quasiperiodic structure, and this idea was indeed one of the predominant early paradigms for quasiperiodic ordering.

Such a rigid picture of quasicrystalline ordering leaves no room for the disorder that must accompany the materials at nonzero temperature, and the role of entropy in the stabilization of a quasicrystal's structure is the guiding light behind the study of ensembles of random tilings[ 12 ]. Here, the central view is that there are several energetically degenerate or nearly-degenerate local configurations of tiles which may be assembled with more freedom than matching rules would allow. In this way, an ensemble of tilings is defined where individual members are related by tile rearrangements, and the physical properties of the system are defined by ensemble averages. The state with maximum entropy yields the crystallographically forbidden symmetry.

An energetically-stabilized quasicrystal is then a ``perfect'' quasicrystal analagous to a crystalline structure with no disorder, and such phases are well-modeled by quasiperiodic tilings like the Penrose tiling. A random tiling can be thought of as a higher-temperature version of a tiling with disorder in the tiling's local configurations. Even the best crystalline structure's diffraction patterns exhibit disorder, however, and the diffraction pattern of a quasicrystal is no exception. Peak width, location, or intensity may vary due to thermal effects, point defects, and dislocations, but there are also kinds of disorder not seen in periodic crystals. To understand these kinds of disorder, the six integers required to index the Bragg peaks can be thought of as reciprocal wavevectors of a periodic lattice in six dimensions, and the actual three-dimensional wavevectors become projections of the higher-dimensional lattice into the usual 3D reciprocal space of a periodic crystal. The 6D space can then be seen as containing 3D reciprocal space and another orthogonal, three-dimensional subspace known as ``perp'' or ``phason'' space.

The higher-dimensional formalism that is useful in describing the diffraction pattern of a quasicrystal is also useful in characterizing quasiperiodic tilings. For instance, the Penrose tiling may be constructed as a two-dimensional cut through a five dimensional cubic lattice. From this viewpoint, the nonperiodicity of the tiling is a result of the cut-plane's totally irrational orientation with respect to the 5D lattice, while the decagonal symmetry is a result of the subspace's invariance under a particular subgroup of the higher-dimensional lattice's space group that contains five-fold rotations. The complete irrationality of the cut means that it passes through at most one lattice point, and the vertices of the tiling are produced by intersections of the cut plane with 3D perp-space decorations known as acceptance domains (ADs) at each lattice point. The cut plane may be translated in perp space, and the vertex set of the tiling will change since ADs that intersects the cut before translation may not intersect the cut after translation. If the cut no longer intersects a certain AD, it will intersect a different AD, displacing the corresponding vertex in physical space. Furthermore, boundaries of multiple ADs are co-planar in perp space so that many vertex jumps occur simultaneously when the cut plane is moved. These discontinuous motions, also called ``phason jumps'', sometimes manifest themselves as chains of tile flips called ``worms'' consisting of one basic tile rearrangement that is propagated along infinite strips of the tiling [ 13 ].

Finally, the quasiperiodicity of the Penrose tiling may be demonstrated by considering inflation rules where each tile is scaled up in size by the golden mean τ = , and re-tiled by original-sized copies of the prototiles. This process can be continued ad infinitum to fill space, forming a tiling where there are τ fat rhombi for every skinny rhombus.

The Penrose tiling was extended by Kramer and Neri to three dimensions even before the announcement of the first quasicrystal[ 7 ]. Like the Penrose tiling, the three-dimensional Penrose tiling (3DPT) contains two types of tiles (up to symmetry), both of which are rhombohedra with one prolate and the other oblate. Local matching rules exist that govern the assembly of patches of tiles[ 14 , 15 ], and inflation rules allow copies scaled by τ ³ to be re-tiled in a configuration compatible with the matching rules. Unfortunately, these constructs for the 3DPT are more cumbersome to handle, mainly due to the complications of the extra dimension of these tilings compared to the 2D Penrose tiling; there are more than two tiles when decorated with matching rules, and the representation and visualization of inflation rules is considerably easier in two dimensions. More effectively, the 3DPT may be constructed as a cut through a six dimensional cubic lattice. The cut plane contains the 3D physical space, and the acceptance domain is a triacontahedron lying in the three perp space dimensions. As the cut plane is moved and leaves infinitely many ADs, it enters others that are glued to the faces of each AD that it just left. As in the Penrose tiling, there is one basic tile rearrangement that occurs during a phason jump and propogates along parallel faces forming in two directions forming a sheet-like analog of worm flips.

The higher-dimensional picture is still valid for random tilings, for the tiling can be viewed as a non-planar cut through the higher-dimensional lattice. The non-planar cut can create local configurations of tiles not found in the perfect tiling formed by a planar cut, and these deviations from the perfect tiling are reflections of ``phason strain'' resulting from the non-planar cut deviating from the cut used for the perfect tiling. Consequently, the tiling picture is an effective way to visualize quasiperiodic ordering, whether a perfect tiling or random tiling picture is used. Nonetheless, the atomic structure of an icosahedral quasicrystal is still an area of active research interest. Various schemes have been employed where the vertices of the 3DPT, random tilings, or other vertex sets are decorated with clusters of atoms, but none has completely reproduced the diffraction pattern. Tilings' predominant power has so far lain in their ability to serve as model systems for which the dynamics and ordering of quasicrystals can be studied.

Chris Kimmer 2011-06-01

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