Penrose Matching Rules from Realistic Potentials in a Model System

After the first couple years of quasicrystal papers, and especially after the discovery of long-range-ordered equilibrium quasicrystals, theorists began to address their stabilisation. But this was mostly in terms of abstract tilings – with inter-tile matching rules (as in Penrose’s tiling), or without (as in the random-tiling models); it was implicitly understood that the tiles’ interactions were induced by those of the atoms. Only towards 1990 did people begin to ask which interatomic potentials actually favored specific (hopefully realistic) quasicrystal atomic structures [1–4] Two competing scenarios emerged for the origins of quasicrystal long-range-order. The matching-rule model posits that atomic interactions implement something like Penrose’s arrows: then there exists an ideal quasicrystal structure (in analogy to ideal crystal structures). This paradigm was esthetically attractive, since it would imply (i) mathematically beautiful symmetries under “inflation” by powers of τ ≡ (1+ √ 5)/2 with interesting consequences for physical properties; (ii) special conditions on the atomic structures when represented as a cut through a higher-dimensional space. [5] The random-tiling scenario, on the other hand, posits that long-range order is emergent, [6, 7] with the quasicrystal phase well represented as a high-entropy ensemble of different tile configurations. This had an esthetic advantage in the sense of Occam’s razor, in demanding fewer coincidences from the interactions. Furthermore, the known structures appeared to be made from highly symmetrical clusters, which tends to imply random-tiling type interactions (matching rule “markings” always entail a partial spoiling of the tiles’ symmetries). In the case of icosahedral quasicrystals, the random-tiling scenario seems to be the most plausible. First of all, no simple matching rules are known [5, 8]. More importantly, diffraction experiments have shown diffuse 1/|q|2 tails around Bragg peaks which are well fitted by the quasicrystal elastic theory with its “perp” space displacements complementary to the usual kind [9]; such gradient-squared elasticity is not expected in matching-rule based models. For decagonal quasicrystals, however, the basis for a random-tiling description has been weaker. It is much easier to implement matching rules from local interactions – particularly for the Penrose pattern, as it is rigorously known that other (“local isomorphism”) classes of decagonal pattern demand longerrange interactions in order to force the correct structure [10] or cannot stabilise an ideal quasicrystal [11]. Furthermore, a three-dimensional decagonal random-tiling theory is required – only extensive entropy can stabilise a bulk system [7] – but the theory for stacked decagonal random tilings was never completed [12]. As to experiments, the 1/|q|2 tails have not been observed in decagonals, whereas it is claimed (from high-