Metallic-mean quasicrystals as aperiodic approximants of periodic crystals

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 2-D 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 4-D 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.

How to assign a (3 + 1)-dimensional superspace group to an incommensurately modulated biological macromolecular crystal

Periodic crystal diffraction is described using a three-dimensional (3D) unit cell and 3D space-group symmetry. Incommensurately modulated crystals are a subset of aperiodic crystals that need four to six dimensions to describe the observed diffraction pattern, and they have characteristic satellite reflections that are offset from the main reflections. These satellites have a non-integral relationship to the primary lattice and requireqvectors for processing. Incommensurately modulated biological macromolecular crystals have been frequently observed but so far have not been solved. The authors of this article have been spearheading an initiative to determine this type of crystal structure. The first step toward structure solution is to collect the diffraction data making sure that the satellite reflections are well separated from the main reflections. Once collected they can be integrated and then scaled with appropriate software. Then the assignment of the superspace group is needed. The most common form of modulation is in only one extra direction and can be described with a (3 + 1)D superspace group. The (3 + 1)D superspace groups for chemical crystallographers are fully described in Volume C ofInternational Tables for Crystallography. This text includes all types of crystallographic symmetry elements found in small-molecule crystals and can be difficult for structural biologists to understand and apply to their crystals. This article provides an explanation for structural biologists that includes only the subset of biological symmetry elements and demonstrates the application to a real-life example of an incommensurately modulated protein crystal.

Study on the Spread of the Energy Gap in Nanostructure System

We present a discussion of the size-, potential-dependence of the confinement energy in the nanostructure, as well the blue shift due to quantum confinement effect. In this case, we solve the Schrödinger equation by employing two simple models with one-dimensional periodic crystal potential. Results show that the confinement energy increases abruptly as the size of nanostructures decreases. Importantly, the confinement energy no longer strictly follows the size-dependent inverse square formula given by Brus. Furthermore, the band gap and blue shift depend on the crystal potential in the nanostructure, and the confinement energy decreases with the increase of the potential. We also find that the surface bond constriction plays an important role of the confinement energy.