Acta Crystallographica Section A: Foundations and Advances最新文献

Due to the short de Broglie wavelength of electrons compared with X-rays, the curvature of their Ewald sphere is low, and individual electron diffraction patterns are nearly flat in reciprocal space. As a result, a reliable unit-cell determination from a set of randomly oriented electron diffraction patterns, an essential step in serial electron diffraction, becomes a non-trivial task. Here we describe an algorithm for unit-cell determination from a set of independent electron diffraction patterns, as implemented in the program PIEP (Program for Interpreting Electron diffraction Patterns), written in the early 1990s. We evaluate the performance of the algorithm by unit-cell determination of two known structures - copper perchlorophthalocyanine (CuPcCl₁₆) and lysozyme, challenging the algorithm by high-index zone patterns and long crystallographic axes. Finally, we apply the procedure to a new, structurally uncharacterized five amino acid peptide.

Bloch waves are often used in dynamical diffraction calculations, such as simulating electron diffraction intensities for crystal structure refinement. However, this approach relies on matrix diagonalization and is therefore computationally expensive for large unit cell crystals. Here Bloch wave theory is re-formulated using the physical optics concepts underpinning the multislice method. In particular, the multislice phase grating and propagator functions are expressed in matrix form using elements of the Bloch wave structure matrix. The specimen is divided into thin slices, and the evolution of the electron wavefunction through the specimen calculated using the Bloch phase grating and propagator matrices. By decoupling specimen scattering from free space propagation of the electron beam, many computationally demanding simulations, such as 4D STEM imaging modes, 3D ED precession and rotation electron diffraction, phonon and plasmon inelastic scattering, are considerably simplified. The computational cost scales as {cal O}({N^2} ) per slice, compared with {cal O}({N^3} ) for a standard Bloch wave calculation, where N is the number of diffracted beams. For perfect crystals the performance can at times be better than multislice, since only the important Bragg reflections in the otherwise sparse diffraction plane are calculated. The physical optics formulation of Bloch waves is therefore an important step towards more routine dynamical diffraction simulation of large data sets.

The double-layer honeycomb with hexagonal cells, three rhombic faces between the two layers and p3m1 layer space-group symmetry, used universally by honeybees, is often considered to be the most efficient (from the point of view of wax economy) and the only honeycomb manufactured by bees. However, another variant of a symmetric and periodic double-layer hexagonal honeycomb with two hexagons and two rhombi between the two layers and slightly better wax economy was discovered theoretically in 1964 by Fejes Tóth and found in nature some years later. The present work shows that there is yet another possibility, with the interface formed by one hexagon and two quadrangles, in addition to the trivial case with flat hexagonal cell bottoms and very poor wax economy. Moreover, we demonstrate that the geometry of the Fejes Tóth honeycomb can be optimized for even better wax economy. All the theoretical honeycomb types are derived using the principle of Dirichlet-domain construction and shown to have more and less symmetric variants. Wax economy is calculated for each case, confirming that indeed the modified Fejes Tóth honeycomb is the most efficient, while the trivial flat-bottom case is the least.

A systematic description of 1-periodic polycatenanes is given. The description uses piecewise-linear embeddings (straight edges) and is limited to structures with symmetry-related vertices (isogonal). Components linked are polygons, including knotted polygons and polyhedra. The structures described are generally those with the order of rotational symmetry up to 10. An account is given of transitivity and intransitivity in patterns of links.

We report symmetric (vertex- and arc-transitive) embeddings of catenated rings, polyhedra and rods. Linked triangles form infinite families of structures, and we limit this report to only structures with each ring linked to three or six others. For linked squares, hexagons, tetrahedra, octahedra, cubes and rods, only a small number of symmetric structures were found, and all are reported.

This paper discusses the geometric properties and symmetries of general moiré patterns generated by homophase bilayers twisted by rotation 2δ. These patterns are generically quasiperiodic of rank 4 and result from the interferences between two basic periodicities incommensurate to each other, defined by the sites in the layers that are kept invariant through the symmetry operations of the structure. These invariant sites are distributed on the nodes of a set of lattices called Φ-lattices - where Φ runs on the rotation operations of the symmetry group of the monolayers - which are the centers of rotation 2δ + Φ transforming a lattice node of the first layer into a node of the second. It is demonstrated that when a coincidence lattice exists, it is the intersection of all the Φ-lattices of the structure.

Twisted homophase bilayers, stacks of two rotated monolayers such as graphene, exhibit remarkable physical properties absent in their constituent monolayers. The structure of bilayer systems is dominated by a moiré effect and critically depends on the twist angle. Quiquandon & Gratias [Acta Cryst. (2025), A81, 94-106] develop a crystallographic framework for rigorous description of the structure of bilayers, including systems without a coincidence lattice. They offer a set of tools that can describe the structure of any arbitrary bilayer system and enable the connection with its physical properties.

Group-theoretical and linear-algebraic methods and tools have recently been developed that aim to exhaustively identify the small-angle rotational rigid-unit modes (RUMs) of a given framework material. But in their current form, they fail to detect RUMs that require a compensating lattice strain which grows linearly with the amplitude of the rigid-unit rotations. Here, we present a systematic approach to including linear strain compensation within the linear-algebraic RUM-search method, so that any geometrically possible small-angle RUM can be detected.

This work analyzes the rules governing the growth of the numbers of vertices, edges and faces in all possible periodic tessellations of the 2D Euclidean space, and encodes those rules in several types of polynomial growth functions. These encodings map the geometric, combinatorial and topological properties of the tessellations into sets of integer coefficients. Several general statements about these encodings are given with rigorous mathematical proof. The variation of the growth functions is represented graphically and analyzed in orphic diagrams, so named because of their similarity to orphic art. Several examples of 3D space groups are included, to emphasize the complexity of the growth functions in higher dimensions. A freely available Python library is presented to facilitate the discovery of the growth functions and the generation of orphic diagrams.

Ladder graphs admit a maximum-symmetry embedding in which edges coincide. In folding ladders, there are no zero-length edges. We give examples of high-symmetry 3-periodic ladders, particularly emphasizing the structures of 3-periodic vertex- and edge-transitive folding ladders. For these, the coincident-edge configuration is one of maximum volume for fixed edge length and has the same coordinates as (is isomeghethic to) a higher-symmetry 3-periodic graph.