Quasicrystals: Solids with Unusual Symmetry

Post-publication activity

Quasicrystals: Solids with Unusual Symmetry refers to the unexpected, noncrystalline molecular structures of these equilibrium solids.

Discovery of quasicrystals

Experimental evidence for the existence of materials with unusual molecular structure, now called quasicrystals, was published by Shechtman et al in 1984 (Shechtman et al 1984). The unusual structure was detected by (electron and X-ray) scattering patterns of sharp Bragg peaks, with rotational symmetries which are mathematically impossible for crystals. This was quite surprising as it had been generally believed that solid matter in thermal equilibrium had to be crystalline (Radin 1987).

The original material, an aluminum-manganese alloy produced by a rapid quench, was only a metastable state with imperfect order (as evidenced by the width of the diffraction peaks), rather than true thermal equilibrium. However, a consensus was eventually reached that similar alloys with much more perfect noncrystalline structure really did exist in thermal equilibrium. We leave discussion of this issue to other articles and concentrate here on one of the standard models for this class of materials, which may be thought of as based on cluster shapes. With suitable atomic interactions, it might be imagined that clusters of atoms of definite shapes might predominate, and could thus be thought of as "building blocks" for the material. More specifically, soon after the experimental discovery Levine and Steinhardt (Levine, Steinhardt 1984), inspired by Penrose tilings of the plane (discussed below), argued that molecules with appropriate shapes could produce the apparent scattering patterns. This article is concerned with the structure of these tiling models within the framework of classical statistical mechanics; once this is clarified all other material properties (for instance scattering intensity (Dworkin 1993)) can be explored in a traditional manner.

Modelling quasicrystals within statistical mechanics

Since we are interested in modelling an equilibrium (solid) phase we will at some point have to consider infinite volume limits, for practical reasons (Fisher, Radin 2006). We begin with the grand canonical ensemble of a system of classical particles in a cubic box $$B\ ,$$ interacting through a potential energy function $$E_{pot}\ .$$ The particles will be treated as immutable molecules of one or several (chemical) species, with shapes which are modelled by a hard core in the interaction. If there are particles with (center of mass) positions $$x_j\ ,$$ with (linear and angular) momenta $$p_j\ ,$$ and they are of (molecular) types $$w_j\ ,$$ all abbreviated as states $$s_j=(x_j,p_j,w_j)\ ,$$ then the potential energy of interaction of the system, with collective state $$\bar s\ ,$$ will be denoted $$E_{pot}(\bar s)\ .$$ $$w_j$$ will usually consist of the shape and orientation of the molecule in space with respect to some fixed reference axes, so that a state $$s=(x,p,w)$$ models a molecule with center of mass at $$x\ ,$$ momentum $$p$$ and shape and orientation $$w\ .$$ We assume there are a small number $$\nu$$ of different chemical species, with only one shape for each. The interaction represents the shape through a hard core$E_{pot}(\bar s)=\infty$ if within $$\bar s$$ there are any overlapping molecules.

The phase space $$\Omega^B$$ of such a model consists of all ways to put molecules in the box $$B\ .$$ It can be conveniently decomposed into subspaces $$\Omega^B_{\bar N}$$ corresponding to a fixed finite number of each of the different species, labelled by a vector $$\bar N=N_1,\cdots,N_\nu$$ listing the particle numbers for each species$\Omega^B=\cup_{\bar N}\Omega^B_{\bar N}\ .$

For notational convenience we allow the molecules to overlap in $$\Omega^B\ ,$$ and use the hard core in the interaction to disallow overlap. The grand canonical ensemble with parameters $$\beta$$ (inverse temperature) and $$\bar \mu=\{\mu_k\}$$ (chemical potentials) assigns relative probability density

$\tag{1} \exp{-\beta[E(\bar s)-\sum_k\mu_kN_k(\bar s)]}$

to a point $$\bar s=\{s_j\}$$ in $$\Omega^B$$ such that $$\bar N(\bar s)>\bar 0\ ,$$ and 1 to the vacuum ($$\bar N(\bar s)=\bar 0$$), where $$E(\bar s)$$ is the total (kinetic and potential) energy. So the normalization constant (or partition function) is

$\tag{2} Z^B_{\beta,\bar \mu}=1+ \sum_{\bar N\ne \bar 0} \int_{\Omega^B} \exp{-\beta[E(\bar s)-\sum_k\mu_k N_k(\bar s)]}\, d\bar s.$

We assume, as is common in statistical mechanics models, that $$E(\bar s)=E_{kin}(\bar s)+E_{pot}(\bar s)\ ,$$ with the kinetic energy $$E_{kin}(\bar s)$$ only depending on the momenta $$\bar p$$ (and easily computed) and the potential energy $$E_{pot}(\bar s)$$ only depending on the position/geometric variables $$\bar x, \bar w\ ,$$ so that the probability density factors into two terms: one easily understood term depending only on $$\bar p$$ and one complicated term depending only on $$\bar x, \bar w\ .$$ This factorization holds as well for the partition function, allowing us to concentrate below on the reduced distribution on the $$\bar x$$ and $$\bar w$$ variables alone. From here on, the state of a particle will no longer contain momentum information$s=(x,w)\ .$

Intuitively, if there is an attractive part to the interaction, and $$\bar \mu$$ is held fixed as $$\beta\to \infty\ ,$$ then the (reduced) probability distribution will concentrate on particle configurations of optimally low potential energy. (This is actually more delicate than it appears (Bellissard et al 2009), as it is necessary to take the infinite volume limit before taking the limit $$\beta\to \infty\ .$$) In most models which have been solved, such energy ground state configurations are periodic in space, like perfect crystals. Similarly, if $$\beta$$ is held fixed and any $$\mu_k\to -\infty$$ then, because of the hard core, the (reduced) probability distribution will concentrate on particle configurations of optimally high volume fraction, which again, for most solved examples, are periodic in space. For both asymptotic regimes one might hope that the qualitative situation carries over to $$\beta$$ and $$\bar \mu$$ with values nearby, not just equal to, the asymptotic values. For instance if there is only one chemical species and if the physical space is replaced by a cubic lattice (in any dimension), then with mild assumptions one can prove this to hold: near the optimal parameter values one has a periodic phase, separated by a sharp transition from the usual disordered phase of low $$\beta$$ and/or high $$\mu$$ (Ginibre 1969). We are interested here in a variation of this approach, to model quasicrystals based on advances in the mathematics of dense packings of spheres and other shapes.

Aperiodic tilings

A basic mathematical fact, first published by Berger in 1966 (Berger 1966), is the existence of finite (prototile) collections of polyhedral shapes, in 2 or higher dimensions, each with the following property. Given an unlimited number of each shape in the collection one can fit them together without overlap, like jigsaw puzzle pieces, to fill up all of space without gaps (ie to tile space); furthermore, for each collection one can in fact create infinitely many noncongruent tilings, but none of them has any translational symmetry - thus the name aperiodic tilings. (It is important to distinguish this from the simpler situation where, from a given collection of shapes, one could make tilings with, and tilings without, translational symmetry.) The best known example is the prototile set consisting of the kite and dart (Figure 1) invented by Penrose about 35 years ago (Gardner 1977); see Figure 2 for a kite and dart tiling.

Figure 1: Kite and dart tiles

It is important, for applications, to think of a tiling of space by such shapes as a configuration of optimally high volume fraction (namely volume fraction 1) occupied by the nonoverlapping shapes. The physical interpretation of this requirement is that the protrusions and recesses of the tile edges correspond to local energetic interactions between the atomic clusters represented by the tiles. It then easily follows, using the two kite and dart shapes as defining two species in the above grand canonical formalism, that the asymptotic probability distribution corresponding to $$\mu_k= -\infty$$ would be concentrated on the asymmetric configurations of molecules as in . (As noted above one should take the infinite volume limit before taking the limit in the $$\mu_k\ ,$$ and this causes difficulties which are well understood but which we do not discuss here.)

Figure 2: Kite and dart tiling

It will also be useful to note the older aperiodic prototile examples, similar to that of Berger, in which all the shapes are small perturbations of squares or cubes, which can only tile space when they abut one another full-face to full-face. More specifically, consider the grand canonical ensemble in which physical space is replaced by a lattice, say $$\Z^2\ ,$$ and consider any of the 2-dimensional examples of aperiodic prototile sets of modified squares, for instance one with 16 prototiles due to Ammann (Grunbaum, Shephard 1986; see Figure 3). One could use such a prototile set to define a nearest-neighbor interaction between particles of 16 different species, one for each prototile, living on all the lattice sites. For instance one could limit a two-body potential to value $$+1$$ for neighbors that do not fit and value $$-1$$ for neighbors that fit. In this model the support of the probability distribution is all configurations for all parameter values of $$\beta\ ,$$ and $$\bar \mu\ ,$$ but reduces to the aperiodic configurations if $$\beta\to \infty$$ and $$\bar \mu = 0\ .$$ This type of statistical mechanics model was introduced, with proofs in different degrees of generality, in (Ruelle 1978, Radin 1985).

Figure 3: Ammann tiles

Away from the ground state

We have discussed how to use an aperiodic set of prototiles to produce an interaction for which the grand canonical distribution is supported only on asymmetric configurations, at least for appropriate asymptotic parameter values of temperature or chemical potential. There remains an important question: does this method actually produce models showing a quasicrystal-like phase away from the asymptotic values, as is the case for the simple lattice gas models noted above (Ginibre 1969). There has been very little progress on this matter. One work of note is a Monte Carlo simulation by Leuzzi and Parisi (Leuzzi, Parisi 2000) of a complicated lattice model based on Ammann's prototiles, which gives evidence for the existence of a phase transition with a strong peak in the specific heat at strictly positive temperature. The periodic or aperiodic nature of the low temperature equilibrium state was not investigated in that work, but in further simulations on the same model (Koch, Radin 2010, Aristoff, Radin 2011) the structure was shown to be aperiodic and a possible order parameter identified. It is an important open problem for modelling quasicrystals to prove such features.

References

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