next up previous
Next: Recent Experiments Up: No Title Previous: Introduction

A New Penrose Tiling Picture

A new Penrose tiling picture of quasicrystals emerges in which the structure is determined entirely by a single repeating cluster which overlaps (shares atoms with) neighbor clusters according to simple energetics. We first discuss the the structure in terms of somewhat artificial overlap rules, which play the same role as Penrose edge-matching rules in forcing a unique structure isomorphic to Penrose tiling. We then discuss how the overlap rules may arise from physically plausible energetics.

In this paper, we will focus on the case of decagonal quasicrystals whose quasiperiodic layers have the same symmetry as two-dimensional Penrose tilings. The two-dimensional analogue for the overlapping cluster model consists of decagonal tiles which overlap to form a covering of the two-dimensional plane. Petra Gummelt[15] first conjectured that decagonal tiles with appropriately chosen overlap rules can force a perfect quasiperiodic tiling and she outlined a proof. Jeong and Steinhardt[16] provided a simple, alternative proof which makes clear the isomorphism to two-tile Penrose tilings.

In the new paradigm, the atomic structure of the decagonal phase is determined entirely by the atom decoration of the overlapping decagon tiles. The decagon tiles represent decagonal cluster columns in the three-dimensional quasicrystals structure. For the case of a perfect quasiperiodic structure, the overlapping cluster can be dubbed a ``quasi-unit cell," since it is analogous to the conventional unit cell in a perfect periodic crystal. However, an important difference is that the atomic decoration of the quasi-unit cell is constrained: the atom configuration inside the quasi-unit cell must have the property that neighboring clusters can share atoms without significant distortion of their atomic arrangements.


 
Figure 1: A quasiperiodic tiling can be forced using marked decagons shown in (a). Matching rules demand that two decagons may overlap only if shaded regions overlap. This permits two possibilities in which the overlapped area is either small (A -type) or large (B -type), as shown in (b)> If each decagon is inscribed with a large obtuse rhombus, as shown in (c), a tiling of overlapping decagons (d, left) is converted into a Penrose tiling (d, right).
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=5.0 in \epsfbox{stein1.ps}\end{center}\end{figure}

The decagonal quasi-unit cell is shown in Fig. 1(a) with a decoration consisting of kites and star-like shapes designed as a mnemonic for the overlap rules. To force a perfect quasiperiodic covering isomorphic to a Penrose tiling, the decagons are permitted to overlap only in two ways, A - or B -type, as shown in Fig. 1(b). With these overlap rules, kite regions always overlap kite regions and star-like regions always overlap star regions. The isomorphism between decagons and Penrose tilings can be realized by inscribing each decagon with a large Penrose obtuse rhombus tile marked with single- and double-arrows, as illustrated in Fig. 1(c). The decagon tiling in Fig. 1(d) is thereby converted into a Penrose tiling. (Spaces are left where acute Penrose rhombi can be inserted.)

The reduction of a Penrose tiling or decagonal phase to a single-repeating cluster is a remarkable simplification. In terms of atomic modeling, it means that the entire structure is defined by the atomic decoration of the quasi-unit cell, similar to the familiar case of periodic crystals. However, as an explanation for why quasicrystals form, a worrisome aspect is the overlap rules, which appear to require complex energetics. Hence, a second important discovery[16] for the new paradigm has been that it is possible to avoid matching or overlap rules altogether. Instead, a perfect Penrose tiling can arise simply by maximizing the density of some chosen atom cluster, C .

To illustrate the result, we idealize the discussion by considering arrangements of obtuse and acute rhombi, each representing some atom cluster. We assume no matching or overlaps rules. Without any further specifications, there are an infinite number of distinct tilings possible, including Penrose tiling, periodic tilings, and random arrangements. If there is no energetics to distinguish among the possibilities, the ground state is degenerate.


  
Figure 2: Given obtuse and acute rhombi and no matching rules, the Penrose tiling is configuration with the highest density of C -clusters. (a) Shows a C -cluster; if a decagon is circumscribed about the central 7 rhombi (dotted line), the decagons from an overlapping decagon tiling. (b) Shows the two kinds of overlaps between C -clusters which bring the centers of the C -clusters closest together. (The A -types have the same separation between centers.) If decagons are circumscribed about the central 7 rhombi of each C -cluster, the A - and B -type overlaps between C -clusters transform into the precisely the A - and B -types overlaps between decagons in the quasi-unit cell description.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=4.0 in \epsfbox{stein2.ps}\end{center}\end{figure}

In realistic models, it is natural to suppose that the tiles represent atom clusters and that some tile cluster C is low energy compared to the others. The degeneracy may, thereby, be broken. Jeong and Steinhardt[16] have shown that, for an appropriately chosen C cluster, the Penrose tiling emerges as the unique ground state. That is, if one imagines that the chosen cluster of tiles represents some energetically preferred atomic cluster, then minimizing the free energy would naturally maximize the cluster density. Jeong and Steinhardt have shown that the Penrose tiling is the unique configuration with the maximum density of C clusters.

An example is the cluster C shown in Fig. 2, although other choices are possible.[17] Two neighboring C 's can share tiles. The greatest overlaps correspond precisely to the A and B overlaps of the central decagonal region of C . It is clear from this that the C -cluster is inspired by the decagon covering described above, although the precise relationship is quite subtle. For example, if there are no explicit overlap rules to forbid certain arrangements for the C cluster, the hexagon tabs must introduced to prevent undesirable overlaps. The non-trivial result is that maximizing the density of C clusters automatically leads to a structure in which the C -clusters are in one-to-one correspondence with the quasi-unit cells in a single decagon tiling. In particular, of all possible arrangements of obtuse and acute tiles, the Penrose tiling is the unique arrangement of C in which every C has an A or B overlap with its neighbors. Jeong and Steinhardt have also shown that the ground state remains unique for a wide range of assignments of energies to clusters.


next up previous
Next: Recent Experiments Up: No Title Previous: Introduction

4/14/1998