Abstract

Ever since the discovery of quasicrystals, periodic approximants of these aperiodic structures constitute a very useful experimental and theoretical device. Characterized by packing motifs typical for quasicrystals arranged in large unit cells, these approximants bridge the gap between periodic and aperiodic positional order. Here we propose a class of sequences of 2D quasicrystals that consist of increasingly larger periodic domains and are marked by an ever more pronounced periodicity, thereby representing aperiodic approximants of a periodic crystal. Consisting of small and large triangles and rectangles, these tilings are based on the metallic means of multiples of 3, have a 6-fold rotational symmetry, and can be viewed as a projection of a non-cubic 4D superspace lattice. Together with the non-metallic-mean three-tile hexagonal tilings, they provide a comprehensive theoretical framework for the complex structures seen, e.g., in some binary nanoparticles, oxide films, and intermetallic alloys.

Editor's summary

Quasicrystals differ from traditional incommensurate structures because they have non-crystallographic rotational symmetries. Here the authors introduce a scheme to produce metallic-mean quasicrystals in two dimensions with 6-fold rotational symmetry that can be seen as approximant to periodic tilings.

日本語訳

準結晶の発見以来、近似結晶は実験および理論の両面から有用な道具となっています。準結晶に存在する典型的なクラスターを格子定数の大きな単位胞に並べることによって得られる近似結晶は、周期結晶と準結晶の間の溝の橋渡しを行ってきました。この論文では周期結晶の領域が大きくなってゆく準結晶列を提案し、周期結晶を近似する「近似準結晶」という考え方を提案します。大小の正3角形と長方形で構成されたタイリングは3の倍数の金属比に基づき、六方対称性を持っており、また4次元超格子からの射影と見なせます。金属比ではない六方タイリングと合わせ、二元ナノ粒子や酸化物フィルム、金属間化合物などの様々な複雑構造を特徴付ける一般的な理論的枠組みを提供します。

編集者のサマリー

準結晶は伝統的な不整合構造とは非結晶学的回転対称性を持つため異なる。著者らは２次元六方対称を持った周期結晶の近似準結晶となる金属比準結晶を生成する枠組みを与える。