Non-crystallographic Symmetry Research Articles

Fractal surface states in three-dimensional topological quasicrystals

We study topological states of matter in quasicrystals, which do not rely on crystalline order. In the absence of a band-structure description and spin-orbit coupling, we show that a three-dimensional quasicrystal can nevertheless form a topological insulator. It relies on a combination of noncrystallographic rotational symmetry of quasicrystals and electronic orbital space symmetry, which is the quasicrystalline counterpart of the topological crystalline insulator. The resulting topological state obeys a nontrivial twisted bulk-boundary correspondence and lacks a good metallic surface. The topological surface states, localized on the top and bottom planes respecting the quasicrystalline symmetry, exhibit a different kind of multifractality with probability density concentrated mostly on high-symmetry patches. They form a near-degenerate manifold of immobile states whose number scales proportionally to the macroscopic sample size. This can present an opportunity for a platform for topological surface physics distinct from the crystalline counterpart. Published by the American Physical Society 2024

A Structural Model of Transition of Cubic α-Mn Into a Hexagonal Modification Based on Noncrystallographic Symmetry of Clusters

A Structural Model of Transition of Cubic α-Mn Into a Hexagonal Modification Based on Noncrystallographic Symmetry of Clusters

Unveiling multipole physics and frustration of icosahedral magnetic quasicrystals

Multipolar physics and their hidden orders have been widely discussed in the context of heavy fermions and frustrated magnets. However, despite extensive research, there are few examples of purely multipolar systems in the absence of magnetic dipoles. Here, we show the magnetic behavior of an icosahedral quasicrystal is generally described by multipoles, and in a specific case by pure magnetic octupoles, resulting from the interplay of spin-orbit coupling and crystal field splitting. Importantly, we emphasize that non-crystallographic symmetries of quasicrystals result in multipolar degrees of freedom, in contrast to the conventional crystals. We first classify the characteristics of multipoles and derive the effective spin Hamiltonian. We then explore how frustration and quantum fluctuations induce entangled quantum phases. Our study presents the magnetic icosahedral quasicrystal as a platform for investigating the exotic multipolar physics.

Microstructures and microtextures of melt-spun decagonal Al–Co–Ni and Al–Co

The fabrication and applications of quasicrystals often involve multigrain aggregates. Microstructures and textures of polycrystals can be conveniently studied by orientation mapping. Such studies of quasicrystals have been rare because the widely used electron backscatter diffraction (EBSD) mapping systems do not index patterns from materials with non-crystallographic symmetries. This complication can be circumvented by using a commercial system to detect individual EBSD bands and an in-house software to index the bands and determine the orientations. This approach was applied to decagonal Al–Co–Ni and Al–Co quasicrystals obtained by the melt-spinning technique. The investigated samples showed similar textures and microstructures on the wheel sides of the ribbons. Maps of cross-sections and free sides revealed microstructural differences implying differences in solidification mechanisms. The growth in Al–Co was columnar, whereas ribbons of Al–Co–Ni showed columnar-to-equiaxed transition. The study demonstrates applicability of the conventional orientation mapping based on detection of diffraction reflections to decagonal quasicrystals.

Polytope 240-Based Rod Structures Made of Tetrahedral Atoms: Comparison with Structures Obtained Using a Modular Design Method

Some rod structures consisting of tetrahedral atoms and differing in symmetry are considered. The structures are obtained as a stereographic projection of polytope 240 vertices into three-dimensional Euclidean space when the polytope rolls along a straight line belonging to this space. The axes of the rod structures are the projections of great circles of a three-dimensional sphere containing the polytope 240 vertices. It has been shown that the rod structures obtained earlier by modular design are topologically equivalent to those considered in this paper, but they have a changed symmetry. Three possible conformations following from the results of this work are presented for a CnH2n hydrocarbon chain with close packing of atoms, conservation of the tetrahedral coordination of carbon atoms, and noncrystallographic symmetry.

Mechanical characterisation of novel aperiodic lattice structures

This paper compares the mechanical properties of a class of lattice metamaterials with aesthetically-pleasing patterns that are governed by the mathematics of aperiodic order. They are built up of ordered planar rod networks and exhibit higher non-crystallographic rotational symmetries. However, they lack the translational symmetry associated with periodic lattice metamaterials. We present schematics illustrating their development based on pattern-unique mathematical substitution rules and exploit a numerical framework from previous work to demonstrate that they exhibit fascinating near-isotropic properties. The lattice structures are compared to the well-known hexagonal lattice with respect to their elastic anisotropy measure and the proximity of their bulk and shear moduli to Hashin–Shtrikman-Walpole limits. The study lends insight into the cost-constrained benefits of introducing additional connectivities between aperiodically-ordered point sets. The results show that aperiodic lattices have the potential to yield superior mechanical properties to periodic ones subject to the mechanical rigidity of the underlying shapes that constitute the pattern. The inherent ‘near-isotropy’ associated with these aperiodic structures, even with uniform strut thicknesses and at low fractional densities, and the ordered and varied orientation of their lattice struts present them as promising mesoscale architecture for solving complex multi-axially loaded structural optimization problems, providing inspiration for this study.

Direct Phasing of Coiled-Coil Protein Crystals

Coiled-coil proteins consisting of multiple copies of helices take part in transmembrane transportation and oligomerization, and are used for drug delivery. Cross-alpha amyloid-like coiled-coil structures, in which tens of short helices align perpendicular to the fibril axis, often resist molecular replacement due to the uncertainty to position each helix. Eight coiled-coil structures already solved and posted in the protein data bank are reconstructed ab initio to demonstrate the direct phasing results. Non-crystallographic symmetry and intermediate-resolution diffraction data are considered for direct phasing. The retrieved phases have a mean phase error around 30∼40°. The calculated density map is ready for model building, and the reconstructed model agrees with the deposited structure. The results indicate that direct phasing is an efficient approach to construct the protein envelope from scratch, build each helix without model bias which is also used to confirm the prediction of AlphaFold and RosettaFold, and solve the whole structure of coiled-coil proteins.

Crystal structure of the second extracellular domain of human tetraspanin CD9: twinning and diffuse scattering.

Remarkable features are reported in the diffraction pattern produced by a crystal of the second extracellular domain of tetraspanin CD9 (deemed CD9EC2), the structure of which has been described previously [Oosterheert et al. (2020 ▸), Life Sci. Alliance, 3, e202000883]. CD9EC2 crystallized in space group P1 and was twinned. Two types of diffuse streaks are observed. The stronger diffuse streaks are related to the twinning and occur in the direction perpendicular to the twinning interface. It is concluded that the twin domains scatter coherently as both Bragg reflections and diffuse streaks are seen. The weaker streaks along c* are unrelated to the twinning but are caused by intermittent layers of non-crystallographic symmetry related molecules. It is envisaged that the raw diffraction images could be very useful for methods developers trying to remove the diffuse scattering to extract accurate Bragg intensities or using it to model the effect of packing disorder on the molecular structure.

Variations on a theme: crystal forms of the amino-acid transporter MhsT.

The bacterial amino-acid transporter MhsT from the SLC6A family has been crystallized in complex with different substrates in order to understand the determinants of the substrate specificity of the transporter. Surprisingly, crystals of the different MhsT-substrate complexes showed interrelated but different crystal-packing arrangements. Space-group assignment and structure determination of these different crystal forms present challenging combinations of pseudosymmetry, twinning and translational noncrystallographic symmetry.

Expression, purification, crystallization and structure of Glyceraldehyde-3-Phosphate Dehydrogenase from Candida albicans, main causative agent of candidiasis

Glyceraldehyde-3-phosphate dehydrogenase from Candida albicans (CaGAPDH) is considered a therapeutic target for candidiasis. In this context, we report here the heterologous production, the three dimensional (3D) structure solution and some structural analyses and comparisons of CaGAPDH. The enzyme was expressed, purified and crystallized to collect X-ray diffraction data at the limit of 2.68 Å resolution, presenting some degree of translational non-crystallographic symmetry. Compared with other GAPDH sequences, CaGAPDH presents two paired mutations, Glu296 in place of a common Ala, and Ala258 in place of common Lys/Asn, which stand face to face, therefore, hydrophilic/hydrophobic residues in interchanged positions in CaGAPDH.

Crystal structure of the PX domain of Vps17p from Saccharomyces cerevisiae.

The structure determination of the PX (phox homology) domain of the Saccharomyces cerevisiae Vps17p protein presented a challenging case for molecular replacement because it has noncrystallographic symmetry close to a crystallographic axis. The combination of diffraction-quality crystals grown under microgravity on the International Space Station and a highly accurate template structure predicted by AlphaFold2 provided the key to successful crystal structure determination. Although the structure of the Vps17p PX domain is seen in many PX domains, no basic residues are found around the canonical phosphatidylinositol phosphate (PtdIns-P) binding site, suggesting an inability to bind PtdIns-P molecules.

Modeling and Structure Determination of Homo-Oligomeric Proteins: An Overview of Challenges and Current Approaches.

Protein homo-oligomerization is a very common phenomenon, and approximately half of proteins form homo-oligomeric assemblies composed of identical subunits. The vast majority of such assemblies possess internal symmetry which can be either exploited to help or poses challenges during structure determination. Moreover, aspects of symmetry are critical in the modeling of protein homo-oligomers either by docking or by homology-based approaches. Here, we first provide a brief overview of the nature of protein homo-oligomerization. Next, we describe how the symmetry of homo-oligomers is addressed by crystallographic and non-crystallographic symmetry operations, and how biologically relevant intermolecular interactions can be deciphered from the ordered array of molecules within protein crystals. Additionally, we describe the most important aspects of protein homo-oligomerization in structure determination by NMR. Finally, we give an overview of approaches aimed at modeling homo-oligomers using computational methods that specifically address their internal symmetry and allow the incorporation of other experimental data as spatial restraints to achieve higher model reliability.

Structural Units for Characterizing the Noncrystallographic Symmetry of Hydrocarbon Chains as Components of Phospholipid Molecules

Structural Units for Characterizing the Noncrystallographic Symmetry of Hydrocarbon Chains as Components of Phospholipid Molecules

Lessons in translational non-crystallographic symmetry: solving the crystal structure of a putative protease from Gemella haemolyans

Lessons in translational non-crystallographic symmetry: solving the crystal structure of a putative protease from <i>Gemella haemolyans</i>

Are the St John's wort Hyp-1 superstructures different?

Two commensurately modulated structures (PDB entries 4n3e and 6sjj) were solved using translational noncrystallographic symmetry (tNCS). The data required the use of large supercells, sevenfold and ninefold, respectively, to properly index the reflections. Commensurately modulated structures can be challenging to solve. Molecular-replacement software such as Phaser can detect tNCS and either handle it automatically or, for more challenging situations, allow the user to enter a tNCS vector, which the software then uses to place the components. Although this approach has been successful in solving these types of challenging structures, it does not make it easy to understand the underlying modulation in the structure or how these two structures are related. An alternate view of this problem is that the atoms and associated parameters are following periodic atomic modulation functions (AMFs) in higher dimensional space, and what is being observed in these supercells are the points where these higher dimensional AMFs intersect physical 3D space. In this case, the two 3D structures, with a sevenfold and a ninefold superstructure, seem to be quite different. However, describing those structures within the higher dimensional superspace approach makes a strong case that they are closely related, as they show very similar AMFs and can be described with one unique (3+1)D structure, i.e. they are two different 3D intersections of the same (3+1)D structure.

Detection of translational noncrystallographic symmetry in Patterson functions.

Detection of translational noncrystallographic symmetry (TNCS) can be critical for success in crystallographic phasing, particularly when molecular-replacement models are poor or anomalous phasing information is weak. If the correct TNCS is detected then expected intensity factors for each reflection can be refined, so that the maximum-likelihood functions underlying molecular replacement and single-wavelength anomalous dispersion use appropriate structure-factor normalization and variance terms. Here, an analysis of a curated database of protein structures from the Protein Data Bank to investigate how TNCS manifests in the Patterson function is described. These studies informed an algorithm for the detection of TNCS, which includes a method for detecting the number of vectors involved in any commensurate modulation (the TNCS order). The algorithm generates a ranked list of possible TNCS associations in the asymmetric unit for exploration during structure solution.

Size-dependent strain in fivefold twins of gold

Multiple twinned structures are common in low-dimensional materials. They are intrinsically strained due to the geometrical constraint imposed by the non-crystallographic fivefold symmetry. In this study, the strain distributions in sub-10 nm fivefold twins of gold have been analyzed by combining aberration-corrected transmission electron microscopy and first-principles calculations. Bending of atomic planes has been measured by both experiments and calculations, and its contribution to the filling of the angular gap was shown to be size-dependent.

An uncomplicated method for growing nano-quasicrystalline structures in the AlCuFeB quaternary alloy system: A short-time milling

Quasicrystals have been comprehensively studied since their discovery in 1984 by Daniel Shechtman to dissolve their complex structures concerning their physical properties [1–5]. The quasi-periodic well-ordered atomic structures made QCs as a novel type of solids differing from the crystalline or amorphous materials [5–7]. The QC alloys have non-crystallographic rotational symmetries, such as five-fold, eight-fold, and even ten-fold rotational axes that are prohibited in ordinary periodic crystals [5–7]. The QC alloys have been recently considered for several applications owing to a series of the unusual combination of exclusive physicomechanical properties for metallic-based alloys [5–7]. Accordingly, the properties of these alloys can be pointed out to their high strength, elevated hardness, desirable corrosion and wear resistance, minute adhesion, low values of thermal and electrical conductivity, and superior optical properties [5–7].•This paper focuses on the synthesis, structural and microstructural evolutions, thermal stability, microhardness, and electrical and optical properties of the Al59Cu25.5Fe12.5B3 nanoquasicrystalline alloy for solar selective absorber usages. Accordingly, there are various advanced techniques for synthesizing the nanostructured quasicrystals, such as laser ablation, sol-gel, electros pining, mechanical alloying, and hydrothermal methods.•The structural and microstructural evolutions of the mechanically alloyed AlCuFeB powders were investigated by X-ray diffractometry and field-emission scanning electron microscopy. The thermal stability of the AlCuFeB powders was recorded by differential thermal analysis.•The nanostructured Al59Cu25.5Fe12.5B3 stable quasicrystalline phase was synthesized by short-time milling procedure in 1 h. It was found that the presence of the quasicrystalline phase in the AlCuFeB alloy prominently improves the microhardness, electrical resistivity, and sunlight absorptance.

Spiral tetrahedral packing in the β-Mn crystal as symmetry realization of the 8D E8 lattice.

Experimental values of atomic positions in the β-Mn crystal permit one to distinguish among them a fragment of the helix containing 15 interpenetrating distorted icosahedra, 90 vertices and 225 tetrahedra. This fragment corresponds to the closed helix of 15 icosahedra in the 4D {3, 3, 5} polytope. The primitive cubic lattice of these icosahedral helices envelopes not only all atoms of β-Mn, but also all tetrahedra belonging to the tiling of the β-Mn structure. The 2D projection of all atomic positions in the β-Mn unit cells shows that they are situated (by neglecting small differences) on three circumferences containing 2D projections of 90 vertices of the {3, 3, 5} polytope on the same plane. Non-crystallographic symmetry of the β-Mn crystal is defined by mapping the closed icosahedral helix of the {3, 3, 5} polytope into 3D Euclidean space E3. This interpretation must be correlated also with the known previous determination of non-crystallographic symmetry of the β-Mn crystal by mapping into the 3D E3 space system of icosahedra from the 6D cubic B6 lattice. The recently proposed determination of non-crystallographic symmetry of the β-Mn crystal actually uses the symmetries of the 8D E8 lattice, in which both the 4D {3, 3, 5} polytope and cubic 6D B6 lattice can be inserted.

Cluster-Precursors and Self-Assembly Li36Ca4Sn24-oS64 and LiMgEu2Sn3-oS28 Crystalline Structures

Using computer methods (the ToposPro software package), the combinatorial-topological analysis and modeling of self-assembly of the crystal structure of Li36Ca4Sn24-oS64 (space group (sg) Cmcm, a = 4.640 A, b = 27.112 A, c = 11.491 A, V = 1445.5 A3) and LiMgEu2Sn3-oS28 (sg Cmcm, a = 4.782 A, b = 20.717 A, c = 7.743 A, V = 767.1 A3). For the Li36Ca4Sn24 intermetallic compound a new type of 11-atom cluster K11 is installed, formed from double pentagonal rings: K11 = 0@11 (Li5)Ca(Sn5). The maximum cluster symmetry K11 and primary strand of translationally related K11 clusters corresponds to noncrystallographic symmetry 5m. Primary chains of linked K11 clusters preserving only symmetry m are located in the direction [100] and the distance between the centers of the clusters determines the length of vector a = 4.640 A. The layer is formed when the primary chains located antiparallel are bound. The tetrahedral clusters are located between the primary chains K4 = 0@4 (Li3Sn), forming chains in the direction [100]. The symmetry and topological code of the self-assembly processes of the 3D structure of the LiMgEu intermetallic compound LiMgEu2Sn3-oS28 is reconstructed from clusters K4 = 0@4 (LiMgEuSn) and K3 = 0@3 (Sn2Eu). The layer is formed when the parallel chains of K4 + K3 are bound.