Plane Group Symmetry Research Articles

Periodic diffraction from an aperiodic monohedral tiling - the Spectre tiling. Addendum.

This article describes the diffraction pattern (2-periodic Fourier transform) from the vertices of a large patch of the recently discovered `Spectre' tiling - a strictly chiral aperiodic monotile. It was reported recently that the diffraction pattern of the related weakly chiral aperiodic `Hat' monotile was 2-periodic with chiral plane-group symmetry p6 [Kaplan et al. (2024). Acta Cryst. A80, 72-78]. The diffraction periodicity arises because the Hat tiling is a systematic aperiodic deletion of vertices from the 2-periodic hexagonal mta tiling. Despite the similarity of the Hat and Spectre tilings, the Spectre tiling is not aligned with a 2-periodic lattice, and its diffraction pattern is non-periodic with chiral point symmetry 6 about the origin.

Superconductivity in Monolayer Carbon Allotropes with High Thermal Stability.

Intrinsic superconductivity is rarely discovered in sp2-hybridized monolayer carbon allotropes. Here we design a carbon monolayer configured of pentagon, heptagon, and hexagon rings with p2 plane group symmetry. Full-sp2 hybridization is proposed to favor thermal metastability on a low Gibbs free energy. The extremely small thermal expansion coefficient is predicted to the turn negative value to positive with elevating temperature. Carbon polygon structures remain intact at a high thermal temperature of 3,000 K. The high specific surface area is found to approach 2,700 m2/g, with O2-adsorption being advantageous over pristine graphene. We reveal electronic Fermi surfaces mediated by phonon modes of carbon out-of-plane vibrations. By calculating the Eliashberg equation, we evaluate intrinsic superconductivity with a large electron-phonon coupling coefficient. The superconducting transition temperature is estimated to reach 20 K under a high logarithmic average frequency. These first-principles calculations shall stimulate experimentalists' interest in exploring low-dimensional carbon superconductors with gas sensitivity.

A microarchitecture design methodology to achieve extreme isotropic elastic properties of composites based on crystal symmetries

The present contribution describes an optimization-based design technique of elastic isotropic periodic microarchitectures with crystal symmetries aiming at the realization of composites with extreme properties. To achieve this goal, three consecutive procedures are followed: (i) a series of inverse homogenization problems with symmetry constraints, (ii) a correlation analysis between symmetries and effective elastic properties of the attained microarchitectures, and (iii) the pattern resemblance recognition of these topologies and their redesign, by adopting microstructures with two length-scales, through optimized parametric geometries. This paper is devoted to assessing the third procedure because the first two procedures have been evaluated in previous works of the authors, and here they are only summarized. By applying the methodology, two plane group symmetries are assessed to define two families of 2D periodic parameterized microarchitecture. Once the parameters have been optimized, the resulting composites achieve elastic isotropic properties close to the whole range of the theoretically estimated bounds. Particularly, an unprecedented microstructure attaining the theoretical maximum stiffness is reported. Starting from these parameterized topologies, simple, one-length scale, and easily manufacturable geometries are defined. One of the so-designed microarchitectures has been manufactured and tested, displaying an effective Poisson’s ratio of − 0.90 simultaneously with a high shear modulus.

Utilizing plane group symmetry to favor noncentrosymmetry in three-dimensional crystals

Materials that lack inversion symmetry (noncentrosymmetric) demonstrate a diversity of desirable optical and electronic properties in bulk such as second harmonic generation, chiral emission, and piezo-, pyro-, and ferro-electricity. Unfortunately, it is challenging to reliably access noncentrosymmetric packing motifs because the closest packing of molecules is often achieved through inversion symmetry operators, leading to the relatively low occurrence of noncentrosymmetry in organic crystals. In this study, the occurrence of noncentrosymmetry in materials that adopt planar packing motifs is investigated because molecular species achieve closest packing in two dimensions through rotations and (or) glides, symmetry operators that do not individually lead to centrosymmetry. It is found that of the 18 crystal structures investigated here adopting planar packing motifs, 13 structures (72%) are noncentrosymmetric showing in-plane polarization. The 13 noncentrosymmetric crystal structures differ from the centrosymmetric structures by directional halogen bonding interactions or steric collisions that align the polarization directions of neighboring layers, leading to bulk structural polarity. The results from this investigation will be of use for designing noncentrosymmetric materials for application in optical and electronic devices.

Transition of Dielectrophoresis-Assembled 2D Crystals to Interlocking Structures under a Magnetic Field.

Aspherical cubic hematite colloids with cylindrical arms protruding from each face, referred to as "hexapods", were assembled via negative dielectrophoresis and then manipulated using an applied magnetic field. Upon application of an ac electric field, the hexapods aligned in close-packed linear chains parallel to the field direction. The chains then aggregated to the center of the device, with adjacent chains separated by distances approximately equal to twice the arm length. The resulting open packing structure exhibited cmm plane group symmetry due to the obstruction of arms, with a high density of incorporated defects. Subsequent application of a magnetic field to the dielectrophoresis (DEP)-assembled structure was found to anneal the colloidal crystal by reorienting the hexapods to align their intrinsic magnetic dipoles with the magnetic field direction. During reorganization, the colloidal packing density was found to decrease by more than 10% at both the center and edges of the crystal, accompanied by a significant loss of ordering, prior to redensification of the 2D lattice with fewer defects. Reorganization at the edge was 1.5 times faster than at the center, consistent with the need for cooperative colloidal motion to remove defects at the centers of the crystals.

Electric-Field-Induced Reversible Phase Transitions in Two-Dimensional Colloidal Crystals

Two-dimensional colloidal crystals confined within electric field traps on the surface of a dielectrophoretic cell undergo reversible phase transitions that depend on the strength of the applied AC electric field. At low field strengths, the particles adopt a two-dimensional hexagonal close-packed lattice with p6m plane group symmetry and the maximum achievable packing fraction of φ = 0.91. Higher electric field strengths induce dipoles in the particles that provoke a phase transition to structures that depend on the number of particles confined in the trap. Whereas traps containing N = 24 particles transform to a square-packed lattice with p4m symmetry and φ = 0.79 is observed, traps of the same size containing N = 23 particles can also pack in a lattice with p2 symmetry and φ = 0.66. Traps with N = 21, 22, and 25 particles exhibit a mixture of packing structures, revealing the influence of lateral compressive forces, in addition to induced dipole interactions, in stabilizing loosely packed arrangements. These observations permit construction of a phase diagram based on adjustable parameters of electric field strength (0-750 V/cm) and particle number (N = 21-25).

Effects of ferroelectric polarization on surface phase diagram: Evolutionary algorithm study of theBaTiO3(001) surface

We have constructed the surface phase diagram of the BaTiO$_{3}$(001) surface by employing an evolutionary algorithm for surface structure prediction, where the ferroelectric polarization is included as a degree of freedom. Among over 1000 candidate structures explored, a surface reconstruction of (2$\times$1)-TiO is discovered to be thermodynamically stable and have the \emph{p2mm} plane group symmetry as observed experimentally. We find that the influence of ferroelectric polarization on the surface free energy can be either negligibly small or sizably large (over 1 eV per ($2 \times 1$) supercell), depending strongly on the surface structure and resulting in a significant distinction of surface phase diagram with varying ferroelectric polarization. It is therefore feasible to control the surface stability by applying an external electric field. Our results may have important implications in understanding the surface reconstruction of ferroelectric materials and tuning surface properties.

Crystallographic symmetries of ornamental floors in the Baptisteries of San Giovanni in Pisa and Firenze

Ornamental floors of Romanesque-style baptisteries of San Giovanni in Pisa and Firenze contain a great variety of complicated geometric ornaments. The altar of the Pisa baptistery is flanked by a hexagonal interlace related to those in several Islamic localities and by a segment of Cosmatesque mosaic, both analysed in detail here from the symmetry point of view. The floor of the Firenze baptistery contains a wide choice of marble mosaics with periodic patterns, grouped here as those obeying plane group symmetry, two-sided layer symmetry, dichroic symmetries of different kinds, colour symmetry of a so-called sequentially dichroic type, patterns hosting more than one symmetry, and two-dimensional OD (order–disorder) patterns. Artistic canons and practicalities of execution are evaluated using this novel approach.

Crystallization of Micrometer-Sized Particles with Molecular Contours

The crystallization of micrometer-sized particles with shapes mimicking those of tetrabenzoheptacene (TBH) and 1,2:5,6-dibenzanthracene (DBT), both flat polyacenes, in an electric field results in the formation of ordered 2D packings that mimic the plane group symmetries in their respective molecular crystal equivalents. Whereas the particles packed in low-density disordered arrangements under a gravitational gradient, dielectrophoresis (under an ac electric field) produced ordered high-density packings with readily identifiable plane group symmetry. The ordered colloidal assemblies were stable for hours, with the packing density decreasing slowly but with recognizable symmetry for up to 12 h for the TBH-shaped particles and up to 4 h for the DBT-shaped particles. This unexpected stability is attributed to jamming behavior associated with interlocking of the dogbone-shaped (TBH) and Z-block (DBT) particles, contrasting with the more rapid reduction of packing density and loss of hexagonal symmetry for disk-shaped particles upon removal of the electric field. The TBH-shaped and DBT-shaped particles assemble into the p2 plane group, which corresponds to the densest particle packing among the possible close-packed plane groups for these particle symmetries. The p2 symmetry observed for the TBH-shaped and DBT-shaped colloid crystal emulates the p2 symmetry of the (010) layers in their respective molecular crystals, which crystallize in monoclinic lattices. Notably, DBT-shaped particles also form ordered domains with pgg symmetry, replicating the plane group symmetry of the (100) layer in the orthorhombic polymorph of DBT. These observations illustrate that the 2D ordering of colloid particles can mimic the packing of molecules with similar shapes, demonstrating that packing can transcend length scales from the molecular to the colloidal.

Description and crystal structure of albrechtschraufite, MgCa4F2[UO2(CO3)3]2⋅17-18H2O

Albrechtschraufite, MgCa4F2[UO2(CO3)3]2⋅17-18H2O, triclinic, space group Pī, a = 13.569(2), b = 13.419(2), c = 11.622(2) A, α = 115.82(1), β = 107.61(1), γ = 92.84(1)° (structural unit cell, not reduced), V = 1774.6(5) A3, Z = 2, D c = 2.69 g/cm3 (for 17.5 H2O), is a mineral that was found in small amounts with schrockingerite, NaCa3F[UO2(CO3)3](SO4)⋅10H2O, on a museum specimen of uranium ore from Joachimsthal (Jachymov), Czech Republic. The mineral forms small grain-like subhedral crystals (≤ 0.2 mm) that resemble in appearance liebigite, Ca2[UO2(CO3)3]⋅ ~ 11H2O. Colour pale yellow-green, luster vitreous, transparent, pale bluish green fluorescence under ultraviolet light. Optical data: Biaxial negative, nX = 1.511(2), nY = 1.550(2), nZ = 1.566(2), 2 V = 65(1)° (λ = 589 nm), r < v weak. After qualitative tests had shown the presence of Ca, U, Mg, CO2 and H2O, the chemical formula was determined by a crystal structure analysis based on X-ray four-circle diffractometer data. The structure was later on refined with data from a CCD diffractometer to R1 = 0.0206 and wR2 = 0.0429 for 9,236 independent observed reflections. The crystal structure contains two independent [UO2(CO3)3]4− anions of which one is bonded to two Mg and six Ca while the second is bonded to only one Mg and three Ca. Magnesium forms a MgF2(Ocarbonate)3(H2O) octahedron that is linked via the F atoms with three Ca atoms so as to provide each F atom with a flat pyramidal coordination by one Mg and two Ca. Calcium is 7- and 8-coordinate forming CaFO6, CaF2O2(H2O)4, CaFO3(H2O)4 and CaO2(H2O)6 coordination polyhedra. The crystal structure is built up from MgCa3F2[UO2(CO3)3]⋅8H2O layers parallel to (001) which are linked by Ca[UO2(CO3)3]⋅5H2O moieties into a framework of the composition MgCa4F2[UO2(CO3)3]⋅13H2O. Five additional water molecules are located in voids of the framework and show large displacement parameters. One of the water positions is partly vacant, leading to a total water content of 17-18H2O per formula unit. The MgCa3F2[UO2(CO3)3]⋅8H2O layers are pseudosymmetric according to plane group symmetry cmm. The remaining constituents do not sustain this pseudosymmetry and make the entire structure truly triclinic. A characteristic paddle-wheel motif Ca[UO2(CO3)3]4Ca relates the structure of albrechtschraufite partly to that of andersonite and two synthetic alkali calcium uranyl tricarbonates.

Two-Dimensional Directed Assembly of Dicolloids

The assembly of ordered dicolloid monolayers is directed by an electric field. The dicolloid particles are polystyrene latex with a maximum equatorial diameter 3.45 μm and length 4.63 μm. The monolayer structure is characterized using small-angle light scattering and bright-field microscopy. With increasing field strength from 26.7 to 200 V(RMS)/cm, a transition from a disordered monolayer, to first orientationally ordered, and then translationally ordered two-dimensional (2D) arrays occurs. A c2mm plane group symmetry dominates the ordered structure but is present alongside structures with p2 symmetry, leading to a spread in the angular distribution of the light scattering peaks. The order-disorder transition dependence on field strength and frequency is similar to that observed for colloidal spheres; at higher frequencies, stronger fields are required to assemble particles. Optimal ordered structures reflect a balance between inducing sufficiently strong interparticle interactions while limiting the rate of formation to ensure the growth of large crystalline domains.

Modified Shepard interpolation of gas-surface potential energy surfaces with strict plane group symmetry and translational periodicity

A new formulation of modified Shepard interpolation of potential energy surface data for gas-surface reactions has been developed. The approach has been formulated for monoatomic or polyatomic adsorbates interacting with crystalline solid surfaces of any plane group symmetry. The interpolation obeys the two dimensional translational periodicity and plane group symmetry of the solid surface by construction. The interpolation remains continuous and smooth everywhere. The interpolation developed here is suitable for constructing potential energy surfaces by sampling classical trajectories using the Grow procedure. A model function has been used to demonstrate the method, showing the convergence of the classical gas-surface reaction probability.

Two-Dimensional Crystallization of Carboxylated Benzene Oligomers

Various carboxylic acid substitution patterns on the 1,3,5-triphenylbenzene nucleus were explored, and their influence on the symmetry of the resulting two-dimensional (2D) crystal structures was assessed. The symmetry of 1,3,5-benzenetribenzoic acid (H(3)BTB) was reduced by modifying the substitution pattern of the arene and/or adding an additional carboxylic acid. Four analogues belonging to various point groups were studied. Comparison of the monolayers of the analogues to that of H(3)BTB shows that plane group symmetry and molecular symmetry are not correlated: H(3)BTB and its analogues exhibit the same plane group p2 at the heptanoic acid/graphite interface. The 2D crystal structure of the H(3)BTB analogues is more strongly controlled by the geometry of hydrogen-bonding interactions rather than molecular symmetry. Other significant observations in this study include porosity, uncommon hydrogen-bonding motifs, and an unusually high number of inquivalent molecules (Z' = 3) present in the 2D crystal of the lowest symmetry analogue. This research demonstrates that reduction of molecular symmetry based on geometric modification of noncovalent interactions allows for control over porosity of the 2D crystals (close-packed structures to nanoporous networks) without changing the core shape of the molecule.

Two-dimensional crystallization conditions of human leukotriene C4 synthase requiring adjustment of a particularly large combination of specific parameters

Human leukotriene C(4) synthase (LTC(4)S) forms highly ordered two-dimensional (2D) crystals under specific reconstitution conditions. It was found that control of a larger number of parameters than is usually observed for 2D crystallization of membrane proteins was necessary to induce crystal formation of LTC(4)S. Here, we describe the parameters that were optimized to yield large and well-ordered 2D crystals of LTC(4)S. Careful fractioning of eluates during the protein purification was essential for obtaining crystals. While the lipid-to-protein ratio was critical in obtaining order, four parameters were decisive in inducing growth of crystals that were up to several microns in size. To obtain a favorable diameter, salt, temperature, glycerol, and initial detergent concentration had to be controlled with great care. Interestingly, several crystal forms could be grown, namely the plane group symmetries of p2, p3, p312, and two different unit cell sizes of plane group symmetry p321.

Biochemical and Structural Analysis of Bacterial O-antigen Chain Length Regulator Proteins Reveals a Conserved Quaternary Structure

Lipopolysaccharide (LPS) is a major component of the Gram-negative outer membrane and is an important virulence determinant. The O-antigen polysaccharide of the LPS molecule provides protection from host defenses, and the length of O-antigen chains plays a pivotal role. In the Wzy-dependent O-antigen biosynthesis pathway, the integral inner membrane protein Wzz determines the O-antigen chain length. How these proteins function is currently unknown, but the hypothesis includes activities such as a "molecular ruler" or a "molecular stopwatch," and other possibilities may exist. Wzz homologs are membrane proteins with two transmembrane helices that flank a large periplasmic domain. Recent x-ray crystallographic studies of the periplasmic portions of Wzz proteins found multiple oligomeric forms, with quaternary structures favoring the "molecular ruler" interpretation. Here, we have studied full-length Wzz proteins with the transmembrane portions embedded in lipid membranes. Using electron microscopy and image analysis we find a unique hexameric state rather than differing oligomeric forms. The data suggest that in vivo Wzz proteins determine O-antigen chain length via subtle structure-function relationships at the level of primary, secondary, or tertiary structure within the context of a hexameric complex.

Non-integral hybrid ions in tourmaline: buffering and geo-thermometry

The tourmaline group of minerals is indeed an enigma. Experimental data from optical spectroscopy, electron microscopy, and Mossbauer spectroscopy reveal a host of physical properties that lack a common structural clarification. For example, tourmaline samples change colour when irradiated with X-ray and gamma-ray radiations some reverting back when heated in air; exhibit simultaneous oxidation and reduction on annealing in an atmosphere of H-2; display different plane group symmetries under TEM; possess the most complicated Mossbauer spectra of all Fe-bearing silicates. In this study, four Brazilian samples were chosen for detailed study by Mossbauer spectroscopy to find out a common structural factor to the physical anomalies reported in the literature. It was found out that the tourmaline group of minerals contain multi-valence elements that are involved in electron exchange between the edge-sharing asymmetric Y and Z crystallographic sites. It is conceivable that the host of physical properties recorded in the literature could be due to the inherent structural misfit between the Y and Z sites and the mechanisms adopted to reduce the strain associated along the shared edges. The complexity of non-integral oxidation states possible - due to electron sharing among the different multi-valence elements present in the structure - further enhances the diverse physical properties observed. Moreover, on heating in air, no net oxidation or reduction takes place in the tourmaline group of minerals over a temperature range as long as there are electron donors and acceptors left in the structure, serving simultaneously as potential single-phase buffers and geo-thermometers.

Self-assembly of bipyridine derivatives at solid/liquid interface: Effects of the number of peripheral alkyl chains and metal coordination on the two-dimensional structures

Self-assembled two-dimensional structures of bipyridine derivatives ( bpys) were observed by scanning tunneling microscopy at HOPG/1-phenyloctane interface. Two types of bpy molecules were used in this study. One is bpy 1, which has two alkyl chains on each end, and the other is bpy 2, which holds single alkyl chain on each end. The bpy 1 formed a monolayer, which is composed of a bent molecular structure with interdigitated alkyl chains. In the case of bpy 2, the structure exhibited pm plane group symmetry. A pair of molecules formed a columnar structure, and the alkyl chains aligned in a tail-to-tail orientation. Metal coordination of both bpy samples increased the intermolecular distance at the π-conjugated units, resulting in the formation of lamellar structure. Although both bpys showed straight form, the alkyl chain unit of complexed bpy 1 was not interdigitated, whereas that of complexed bpy 2 was interdigitated. Thus, the metal–metal distances could be tuned by changing the number of alkyl chain unit.

Two-dimensional crystallization of human vitamin K-dependent γ-glutamyl carboxylase

Planar-tubular two-dimensional (2D) crystals of human vitamin K-dependent γ-glutamyl carboxylase grow in the presence of dimyristoyl phosphatidylcholine (DMPC). Surprisingly, these crystals form below the phase transition temperature of DMPC and at the unusually low molar lipid-to-protein (LPR) ratio of 1, while 2D crystals are conventionally grown above the phase transition temperature of the reconstituting lipid and significantly higher LPRs. The crystals are up to 0.75 μm in the shorter dimension of the planar tubes and at least 1 μm in length. Due to the planar-tubular nature of the crystals, two lattices are present. These are rotated by nearly 90° in respect to each other. The ordered arrays exhibit p12 1 plane group symmetry with unit cell dimensions of a = 83.7 Å, b = 76.6 Å, γ = 91°. Projection maps calculated from images of negatively stained and electron cryo-microscopy samples reveal the human vitamin K-dependent γ-glutamyl carboxylase to be a monomer.

PH-induced structural change in a sodium/proton antiporter from Methanococcus jannaschii

Na+/H+ antiporters are pH-dependent membrane transport proteins that maintain the homeostasis of H+ and Na+ in living cells. MjNhaP1 from Methanococcus jannaschii, a hyperthermophilic archaeon that grows optimally at 85 degrees C, was cloned and expressed in Escherichia coli. Two-dimensional crystals were obtained from purified protein at pH 4. Electron cryomicroscopy yielded an 8 A projection map. Like the related E. coli antiporter NhaA, MjNhaP1 is a dimer, but otherwise the structures of the two antiporters differ significantly. The map of MjNhaP1 shows elongated densities in the centre of the dimer and a cluster of density peaks on either side of the dimer core, indicative of a bundle of 4-6 membrane-spanning helices. The effect of pH on the structure of MjNhaP1 was studied in situ. A major change in density distribution within the helix bundle, and an approximately 2 A shift in the position of the helix bundle relative to the dimer core occurred at pH 6 and above. The two conformations at low and high pH most likely represent the closed and open states of the antiporter.

Experimental verification of the existence and structure of ζ Pu6Fe in a Pu–Ga alloy

Abstract A phase belonging to the space group I4/mcm was identified in a Pu–Ga alloy containing trace amounts of Fe and Ni using electron diffraction and energy-dispersive X-ray spectroscopy (EDXS) in a transmission electron microscope. The plane group symmetry of six experimental diffraction patterns showed that the structure of this phase was at least body-centered orthorhombic. Simulated diffraction patterns, generated from the body-centered tetragonal structure of ζ Pu6Fe with the space group I4/mcm, match the experimental diffraction patterns closely. These results present the first crystallographic evidence for the existence of ζ Pu6Fe in a dilute Pu–Ga alloy. The Pu/Fe ratio of the phase yielded by EDXS was 12.5 at.% and the Pu/(Fe + Ni) ratio was 15.9 at.%. These results suggest that Ni substitutes for Fe in the ζ, Pu6Fe lattice.