Types Of Polyhedra Research Articles

Crystal, electronic structure and hydrogenation properties of the Mg5.57Ni16Ge7.43 cluster phase with a new type of polyhedron.

The ternary germanide Mg5.57Ni16Ge7.43 (cubic, space group Fm-3m, cF116) belongs to the structural family based on the Th6Mn23-type. The Ge1 and Ge2 atoms fully occupy the 4a (m-3m symmetry) and 24d (m.mm) sites, respectively. The Ni1 and Ni2 atoms both fully occupy two 32f sites (.3m symmetry). The Mg/Ge statistical mixture occupies the 24e site with 4m.m symmetry. The structure of the title compound contains a three-core-shell cluster. At (0,0,0), there is a Ge1 atom which is surrounded by eight Ni atoms at the vertices of a cube and consequently six Mg atoms at the vertices of an octahedron. These surrounded eight Ni and six Mg atoms form a [Ge1Ni8(Mg/Ge)6] rhombic dodecahedron with a coordination number of 14. The [GeNi8(Mg/Ge)6] rhombic dodecahedron is encapsulated within the [Ni24] rhombicuboctahedron, which is again encapsulated within an [Ni32(Mg/Ge)24] pentacontatetrahedron; thus, the three-core-shell cluster [GeNi8(Mg/Ge)6@Ni24@Ni32(Mg/Ge)24] results. The pentacontatetrahedron is a new representative of Pavlyuk's polyhedra group based on pentagonal, tetragonal and trigonal faces. The dominance of the metallic type of bonding between atoms in the Mg5.57Ni16Ge7.43 structure is confirmed by the results of the electronic structure calculations. The hydrogen sorption capacity of this intermetallic at 570 K reaches 0.70 wt% H2.

Dihedral rigidity in hyperbolic 3-space

We prove a comparison theorem for certain types of polyhedra in a 3-manifold with its scalar curvature bounded below by − 6 -6 . The result confirms in some cases the Gromov dihedral rigidity conjecture in hyperbolic 3 3 -space.

The Synthesis, Crystal Structure, and Magnetic Properties of Mono-Scorpionate Eu(III) Complexes

Three Eu3+ complexes containing a neutral tripodal ligand possessing a predictable coordination mode have been obtained and studied. The trispyrazolylmethane complexes have an aqua ligand in the coordination sphere, forming both the mononuclear species [Eu(HCPz3)H2O(NO3)3] and the dimer [Eu(HCPz3)H2O(CF3SO3)3]2, having a Chinese lantern structure, whereas the use of the methylated tripod leads to the water-free complex, [Eu(HC(PzMe2)3)(NO3)3]. A qualitative analysis of the magnetic susceptibility of polycrystalline samples demonstrated that the magnetic properties can be described using a simple Van Vleck formula with spin–orbit coupling parameter (λ = 383 ÷ 406 cm−1) close to the values for free Eu3+ ions. The stereochemical analysis of the coordination environment of [Ln(HC(PzMe2)3)(NO3)3] has shown that these complexes can be used as diamagnetic model systems to obtain information on the crystal field effects in the paramagnetic monoradical complexes, [LnRad(NO3)3], since both types of compounds have the same type of coordination polyhedron (symmetry point group D3h) and very close Ln–donor atom distances.

Deciphering the Distribution and Crystal-Chemical Environment of Arsenic, Lead, Silica, Phosphorus, Tin, and Zinc in a Porous Ferrihydrite Grain Using Transmission Electron Microscopy and Atom Probe Tomography

The interaction of contaminants and nutrients with soil constituents is controlled by processes in intergranular and intragranular pore spaces of organic matter or/and common secondary minerals such as ferrihydrite, ∼Fe3+10O14(OH)2. This contribution shows that distribution and clustering of the contaminants As, P, Pb, Si, Sn, and Zn in a porous ferrihydrite grain is greatly affected by the heterogeneous size distribution and chemical composition of the pores as well as the ability of their polyhedra to polymerize with the same type of polyhedron. Transmission electron microscopy (TEM) and atom probe tomography (APT) studies are conducted on focused ion beam (FIB) sections extracted from a porous ferrihydrite grain from the smelter-impacted topsoil in Sudbury, Ontario, Canada. The ferrihydrite grain has pore spaces ranging in diameter from tens to hundreds of nanometers. TEM and scanning-TEM studies indicate that the surfaces of the pore walls are enriched in Si. APT data in conjunction with First Near Neighbor (1NN) analyses indicate different degrees of clustering of Pb, As, Sn, Zn, Si, and P within the sample and selected domains. Careful evaluations of 3D atomic plots and 1NN distances indicates the occurrence of polymerized arsenite-, silica-, Sn-, and Zn-polyhedra within pore spaces of the ferrihydrite. Deciphering adsorption, polymerization, and nucleation processes in porous Fe-(hydr)oxides and other soil constituents requires multianalytical approaches and, in this regard, we discuss the advantages and disadvantages of the combination of TEM and APT for characterizing complex environmental samples at the atomic to nanometer scale.

Characterization of the charge transfer luminescence of the [WO6]6- octahedron in Ca3WO6 and the [WO5]4- square pyramid in Ca3WO5Cl2.

Charge transfer (CT) luminescence of different types of polyhedra, [WO5]4- in Ca3WO5Cl2 and [WO6]6- in Ca3WO6, is characterized by spectroscopic experiments and ab initio calculations. According to the geometry optimization, W6+ ions form five-fold [WO5]4- square pyramids in Ca3WO5Cl2 because of a large interatomic distance between W6+ and Cl- of 3.266 Å. The analysis of the density of electronic states reveals the ionic character of Cl- ions to the W6+ ions in the Ca3WO5Cl2 lattice, resulting in the observed broad luminescence band peak at 488 nm of the single-crystal Ca3WO5Cl2 sample being assigned to the CT transition in the [WO5]4- square pyramid. Compared with the [WO6]6- octahedron in Ca3WO6, the [WO5]4- square pyramid shows an inconsistent CT energy shift: higher CT absorption and lower luminescence energies. The larger bandgap brings about higher absorption energy due to the structural and compositional features of the orthorhombic Ca3WO5Cl2. The redshifted CT luminescence band and small activation energy for the thermal quenching of the Ca3WO5Cl2 sample are explained, assuming that the CT states of the anisotropic [WO5]4- square pyramid take a larger offset in the configurational coordinate diagram than the [WO6]6- octahedron.

Five coordinated Mn in Ba4Mn2Si2Te9: synthesis, crystal structure, physical properties, and electronic structure.

We report the synthesis of single-crystals of a new transition metal-containing quaternary chalcogenide, Ba4Mn2Si2Te9, synthesized by the solid-state method at 1273 K. A single-crystal X-ray diffraction study shows that it crystallizes in the orthorhombic crystal system (space group: Pbam) with cell constants of a = 13.4690(6) Å, b = 8.7223(4) Å, and c = 10.0032(4) Å. The asymmetric unit of the structure consists of eight unique crystallographic sites: one Ba, two Mn, one Si, and four Te sites. In this structure, the two Mn sites, Mn(1) and Mn(2), are disordered, each with fractional occupancy of 50%. The short distance of 2.170(3) Å between Mn(1) and Mn(2) implies that both Mn sites are not occupied simultaneously. The Mn atoms show two types of polyhedra: unique Mn(1)Te5 units along with traditional Mn(2)Te4 tetrahedra. The main motifs of the Ba4Mn2Si2Te9 structure are dimeric Si2Te6 units (with Si-Si single bond), Mn(1)Te5, and Mn(2)Te4 polyhedra. The structure can be described as pseudo-two-dimensional if only Mn(1) atoms are present and one-dimensional when only Mn(2) atoms are filled in the structure. The extended 2∞[Mn(1)Si2Te9]10- layers and 1∞[Mn(2)Si2Te8]8- chains are separated by Ba2+ cations. The direct bandgap for the polycrystalline Ba4Mn2Si2Te9 sample is 0.6(1) eV, as determined from an optical absorption study consistent with the sample's black color. The resistivity study of the polycrystalline Ba4Mn2Si2Te9 also confirms the semiconducting behavior. The thermal conductivity (κ) values are extremely low and decrease with increasing temperature up to 0.46 W m-1 K-1 at 773 K. The DFT studies suggest that the computed bandgap depends on the magnetic ordering of Mn magnetic moments, and the value varies from ∼0.3-1.0 eV. Relative inter-atomic bond strengths of pertinent atom pairs have been analyzed using the crystal orbital Hamilton populations (COHP).

Structural Variations across Wolframite Solid Solutions, (Fe,Mn)WO4

The crystal structure of four samples from natural wolframite solid solutions, (Fe,Mn)WO4, was obtained with synchrotron high-resolution powder X-ray diffraction (HRPXRD) data, Rietveld refinements, space group P2/c, and Z = 2. Wolframite solid solutions extend from ferberite (FeWO4) to hübnerite (MnWO4). The W and (Mn,Fe) cations are in six-fold coordination. This study shows that the unit-cell parameters, a, b, c, and β angle, vary linearly with the unit-cell volume, V, across the wolframite series. The average <Mn,Fe–O> distance increases linearly because of larger Mn2+ (0.83 Å) replacing smaller Fe2+ (0.78 Å) cations, whereas the average <W–O> distance increases slightly because of the higher effective charge of the smaller Fe2+ cation. The distortions of the two types of polyhedra across the series are discussed.

Structural features of the oxidonitridophosphates K3 M III(PO3)3N (M III = Al, Ga).

Cubic crystals of tripotassium aluminium (or gallium) nitridotriphosphate, K3 M III(PO3)3N (M III = Al, Ga), were grown by application of the self-flux method. In their isostructural crystal structures, all metal cations and the N atom occupy special positions with site symmetry 3, while the P and O atoms are situated in general positions. The three-dimensional framework of these oxidonitridophosphates is built up from [M IIIO6] octa-hedra linked together via (PO3)3N groups. The latter are formed from three PO3N tetra-hedra sharing a common N atom. The coordination environments of the three potassium cations are represented by two types of polyhedra, viz. KO9 for one and KO9N for the other two cations. An unusual tetra-dentate type of coordination for the latter potassium cations by the (PO3)3N6- anion is observed. These K3 M III(PO3)3N (M III = Al, Ga) compounds are isostructural with the Na3 M III(PO3)3N (M III = Al, V, Ti) compounds.

3D to 2D Magnetic Ordering of Fe3+ Oxides Induced by Their Layered Perovskite Structure

The antiferromagnetic behavior of Fe3+ oxides of composition RE1.2Ba1.2Ca0.6Fe3O8, RE2.2Ba3.2Ca2.6Fe8O21, and REBa2Ca2Fe5O13 (RE = Gd, Tb) is highly influenced by the type of oxygen polyhedron around the Fe3+ cations and their ordering, which is coupled with the layered RE/Ba/Ca arrangement within the perovskite-related structure. Determination of the magnetic structures reveals different magnetic moments associated with Fe3+ spins in the different oxygen polyhedra (octahedron, tetrahedron, and square pyramid). The structural aspects impact on the strength of the Fe-O-Fe superexchange interactions and, therefore, on the Néel temperature (TN) of the compounds. The oxides present an interesting transition from three-dimensional (3D) to two-dimensional (2D) magnetic behavior above TN. The 2D magnetic interactions are stronger within the FeO6 octahedra layers than in the FeO4 tetrahedra layers.

Volesti: Volume Approximation and Sampling for Convex Polytopes in R

Sampling from high dimensional distributions and volume approximation of convex bodies are fundamental operations that appear in optimization, finance, engineering, artificial intelligence and machine learning. In this paper we present volesti, an R package that provides efficient, scalable algorithms for volume estimation, uniform and Gaussian sampling from convex polytopes. volesti scales to hundreds of dimensions, handles efficiently three different types of polyhedra and provides non existing sampling routines to R. We demonstrate the power of volesti by solving several challenging problems using the R language.

Frank-Kasper Z16 local structures in Cu-Zr metallic glasses

Although previous molecular dynamics studies proposed the existence of Zr-centered Frank-Kasper Z16 structures in Cu-Zr metallic glasses, it cannot be concluded yet, owing to the degeneracy problem of the Voronoi index and the artifact in sets of potentials for Cu-Zr systems. We solve both problems by combining a recently developed Finnis-Sinclair potentials to generate realistic atomic structures and the ${p}_{3}$ code to properly differentiate local structures. In previous studies, the Voronoi index $\ensuremath{\langle}0,0,12,4\ensuremath{\rangle}$ was typically associated with Frank-Kasper Z16 local structures. However, we demonstrate that this index includes two types of polyhedra, only one of which is associated with the Frank-Kasper Z16 structures. We reveal that the Z16 structures are more frequent than the other structures and that the two have different behaviors. The tendency of Zr-centered Z16 local structures to be neighbors of Cu-centered icosahedral local structures is also confirmed. Our findings illustrate the importance of properly differentiating local structures in order to elucidate the behavior of metallic glasses at the atomic scale.

Structural study of Ni67Zr33 amorphous alloy: Interatomic space analysis approach

The structure of Ni67Zr33 amorphous alloy was studied in terms of interatomic space. The three-dimensional atomic configuration of this amorphous alloy was analysed by using a method of decomposing the whole volume into basic polyhedra with only triangular faces formed by nearest neighbour distances. Various types of polyhedra were identified through the analysis. The tetrahedra account for approximately 60% of the whole volume, and most of them are densely packed but some are loosely packed. The large polyhedra have empty spaces commensurate with their own sizes but they are often deformed and non-spherical in shape. That is, spherical empty spaces are not as large as atomic volumes and there is no space in which any atoms enter. These low density polyhedra are more likely to gather together and they form the loosely packed clusters about the size of several nanometers.

ОБ ОДНОМ КЛАССЕ СИЛЬНО СИММЕТРИЧНЫХ МНОГОГРАННИКОВ

We prove the completeness of the list of closed convex polyhedra in E 3 , that are strongly symmetric with respect to the rotation of the faces . Polyhedron is called symmetric if it has at least one non-trivial rotation axis. All axes intersect at a single point called the center of the polyhedron. All considered polyhedra are polyhedra with the center. A convex polyhedron is called a strongly symmetrical with respect to the rotation of the faces, if each of its faces F has an rotation axis L , intersects the relative interior of F , and L is the rotation axis of the polyhedron. It is obvious that the order of rotation axis of L does not necessarily coincide with the order of this axis, if the face of F regarded as a figure separated from the polyhedron. It has previously been shown, that the requirement of global symmetry of the polyhedron faces the rotation axis can be replaced by the weaker condition of symmetry of the star of each face of the polyhedron: to polyhedron was symmetrical with respect to the rotation of the faces, it is necessary and sufficient that some nontrivial rotation axis of each face, regarded as a figure separated from the polyhedron, is the rotation axis of the star of face. Under the star of face F is understood face itself and all faces have at least one common vertex with F . Given this condition, the definition of the polyhedron strongly symmetric with respect to the rotation of the faces is equivalent to the following: the polyhedron is called a strongly symmetrical with respect to the rotation of the faces , if some non-trivial rotation axis of each face, regarded as a figure separated from the polyhedron, is the rotation axis of the star of face. In the proof of the main theorem on the completeness of the list of this class of polyhedra using the result of the complete listing of the so- called polyhedra of 1st and 2nd class [1]. In this paper we show that in addition to the polyhedra of the 1st and 2nd class, listed in [1], only 8 types of polyhedra belongs to the class of polyhedra stronghly symmetric with respect to the rotation of faces. Seven of this eighteen types are not combinatorially equivalent regular or semi-regular (Archimedean). One type of eight is combinatorially equivalent Archimedean polyhedra, but does not belong to polyhedra of 1st or 2nd class. Turning to the polyhedra, dual strongly symmetrical about the rotation of faces, that is, to the polyhedra, stronghly symmetric about the rotation of polyhedral angles, we get their complete listing. It follows that there are 7 types of polyhedra, highly symmetric with respect to the rotation of polyhedral angles which are not combinatorially equivalent to Gessel bodies. Class of polyhedra stronghly symmetric with respect to the rotation of faces, as well as polyhedra 1st and 2nd class mentioned above can be viewed as a generalization of the class of regular (Platonic) polyhedra. Other generalizations of regular polyhedra can be found in [3],[4], [12]-[15].

One-dimensional ion-conductive grain-aligned polycrystalline lithium titanogallate fabricated by combined techniques of reactive diffusion and ion exchange

Grain-aligned polycrystalline lithium titanogallate, Li0.91Ti0.48Ga4.40O8 (LTGO), was successfully prepared by the combined techniques of reactive diffusion and ion exchange. We firstly synthesized the b-axis-oriented polycrystal of sodium titanogallate (NTGO) by the reactive diffusion between NaGaO2 and Ga2TiO5 at 1323K for 24h, followed by the ion-exchange reaction of Li+ for Na+ at 773K for 125h. The crystal structure of resulting LTGO (space group C2/m) was isomorphous with that of the original NTGO. Both crystal structures were built up of five types of polyhedra; [MO7] (M=Li or Na), [(Ti,Ga)O6], [Ga1O6], [Ga2AO4], and [Ga2BO6]. There was a marked difference between the two structures for the coordination environment of M; the centroid-to-M distance of [LiO7] (0.026nm) was significantly larger than that of [NaO7] (0.008nm). The lithium-ion conductivity along the b-axis steadily increased from 3.41×10−4 to 6.21×10−4S cm−1 with increasing temperature from 773 to 1073K. The activation energy of conduction was 0.22eV. The combination of the two distinct techniques would enable us to synthesize the grain-aligned polycrystalline materials with various constituent compounds and chemical compositions.

Quaternary rare-earth sulfides RE3M0.5GeS7 (RE=La–Nd, Sm; M=Co, Ni) and Y3Pd0.5SiS7

The two metal-deficient series of quaternary Ge-containing sulfides RE3M0.5GeS7 (RE = La–Nd, Sm; M = Co, Ni), as well as the related Si-containing sulfide Y3Pd0.5SiS7, were prepared by reactions of the elements at 1050°C. Single-crystal X-ray diffraction analysis performed on all compounds confirmed noncentrosymmetric hexagonal structures (space group P63, Z =2) with cell parameters in the ranges of a =10.0–10.3Å and c =5.7–5.8Å for RE3Co0.5GeS7 and RE3Ni0.5GeS7, or a =9.7891(3) Å and c =5.6840(4) Å for Y3Pd0.5SiS7. They are classified as La3Mn0.5SiS7-type structures, with M atoms centred within octahedra (in contrast to La3CuSiS7-type structures in which M atoms occupy trigonal planar sites) and Ge atoms centred within tetrahedra, both types of polyhedra being arranged in one-dimensional stacks aligned along the c-direction. Charge balance requirements dictate half-occupancy of the M sites. However, bond valence sum arguments indicated that the M atoms are somewhat underbonded within these octahedral sites, so that there is evidence that in some compounds, they can also enter the trigonal planar site at low occupancy (~5%). Magnetic measurements on RE3Co0.5GeS7 (RE = Ce, Pr, Sm) revealed paramagnetic behaviour for the Ce and Pr members and apparent antiferromagnetic ordering (TN =14K) for the Sm member; fitting to the Curie-Weiss law gave effective magnetic moments consistent with the presence of RE3+ and Co2+ species. Band structure calculations on ordered models of La3M0.5GeS7 (M = Co, Ni) showed that the Fermi level cuts through M 3d states in the DOS curve and supported the presence of strong M–S and Ge–S bonding interactions.

Dissymmetrization of artificial and natural diamonds

The occurrence rates of combinatorial types of simple polyhedra {111} are analyzed for natural and artificial diamonds. The empirical occurrence rates of 14 possible polyhedra in an isotropic environment are obtained based on numeral simulation of growth forms of octahedral crystals by the Monte-Carlo method. The phenomenon of dissymmetrization by Curie’s principle related to the crystallization conditions is established for artificial and natural diamonds.

Supramolecular chemistry: Unexplored territory for self-assembly.

Cage-like structures can self-assemble from suitable metal ions and organic linkers, but the size of the assemblies was limited. The surprise discovery of a new series of cages opens up fresh horizons for self-assembly. See Letter p.563 Rational control of self-assembly remains a formidable challenge. It is thought to become increasingly difficult, even impossible, as the number of individual constituents increases. For example, no one has ever been able to design discrete molecules that self-assemble from more than 100 components. When exploring design principles appropriate for dealing with the self-assembly of large numbers of components, Daishi Fujita et al. discovered a spherical structure with a topology not previously reported at the molecular level. Using mathematics describing this class of structures, the authors then targeted and characterized an even larger structure containing 48 palladium ions coordinated by 96 bent organic ligands—by far the largest number of components observed in a self-assembled molecular structure to date. The structure has the topology of a tetravalent Goldberg polyhedron, a type of convex polyhedron made from squares and triangles—a twist on the original Goldberg polyhedra that were first described by Michael Goldberg in 1937.

Identifying icosahedron-like clusters in metallic glasses

Since the discovery of the first metallic glass (MG) with the composition of Au75Si25 in 1960, vast efforts have been devoted to understanding the mechanisms of glass formation in metals, because this class of glassy alloy usually possesses unique properties that may have the potential application as engineering material. As is well known, structure determines properties of material. Therefore, understanding the glass formation of MG from the structural perspective is helpful for guiding researchers in developing more MGs. So far, icosahedral clusters are regarded as the preferred clusters contributing to the formation of amorphous structure due to its five-fold symmetrical feature and high atomic packing. However, it has been found that an ideal icosahedron usually does not have a high concentration in many MG compositions. Thus, we wonder whether icosahedral clusters are popular in microstructures of amorphous alloys. In this work, a feasible scheme for identifying the icosahedron-like clusters in MGs is developed to address this issue. It is found that icosahedron-like clusters are popular structural units in amorphous structure indeed, contributing to the glass formation in alloy. A projection method of reflecting the styles of shell-atom connections in Voronoi-tessellation indexed clusters is developed in detail, so that all clusters can be further geometrically indexed as different projected types of polyhedra. It is revealed that there are three kinds of clusters (0, 2, 8, 1, 0, 2, 8, 2 I-type, and 0, 1, 10, 2) which have the most similar geometrical features to that of the so-called ideal icosahedron, 0, 0, 12, 0. Therefore, besides the ideal icosahedron, these three types of clusters can be regarded as the icosahedron-like clusters. The ideal icoshahedron (0, 0, 12, 0) has a coordination number (i.e., the number of shell atoms) of 12, while these three icosahedron-like clusters have coordination numbers ranging from 11 to 13, so that structural balance between the geometrical atomic stacking and the chemical interactions among various elements in MGs (especially multicomponent MGs) is more easy to achieve. Furthermore, structural models of three selected ZrCu compositions are studied, which are obtained by systematic experimental measurements combined with reverse Monte Carlo simulation. It is found that both the icosahedron-like cluster and the ideal icosahedron have the similar values of some structural parameters, in terms of high atomic packing efficiency, high cluster regularity, fruitful five-fold symmetrical feature, etc. In addition, it is revealed that these ideal icosahedra and icosahedron-like clusters can contain almost all the atoms in these structural models, enhancing the space filling efficiency. In conclusion, these identified icosahedron-like clusters should be the popular building blocks, contributing to the glass formation in alloy. This work provides an insight into the glass formation in alloy from the cluster-level structural angle and will shed light on developing more MGs.

Single-crystal diffraction and Raman spectroscopy of hedenbergite up to 33 GPa

Pyroxenes are important minerals in Earth’s upper mantle and subducting plate. Here, we report results of high-pressure single-crystal X-ray diffraction and Raman spectroscopy experiments conducted on natural Ca, Fe pyroxene hedenbergite up to ~33 GPa in diamond anvil cell. Unit cell parameters a, b, c, β and V, as well as bond lengths of hedenbergite are reported within the studied pressure range. Cell parameters exhibit continuous decrease on compression. Axial compressibilities of a, b and c are calculated to be 1.7(2), 4.9(5) and 2.13(9) × 10−3 GPa−1, respectively. Bulk modulus and its pressure derivative are determined to be 131(4) GPa and 3.8(3) by fitting third-order Birch–Murnaghan equation of state. Compression mechanism is dominated by polyhedral and bond compression trends typical of clinopyroxenes. In general, shorter bonds show lower compressibility, and SiO4, the smallest polyhedron, shows the lowest compressibility. Angle and elongation distortions are reported for the three types of polyhedra at high pressure. Thirteen vibrational modes are observed with Raman spectroscopy up to ~33 GPa. All observed mode frequencies increase as pressure increases.

Aperiodic Crystals at the Mesoscale

Materials with mesoscale structural characteristics have attracted great attention across the fields of chemistry, physics, and materials science. A typical example is mesoporous silica, which are synthesized in water/surfactant/silica systems, and has well-defined mesopores resulting in high surface area. Mesoporous silicas have two defining structural characteristics: (i) disorder at the atomic scale, i.e. only short-range order; and (ii) distinct order at the mesoscale, i.e. long-range order. Atomic-scale structural characterization by common diffraction techniques, such as X-ray single crystal diffraction, is challenging for these partially ordered materials. This is because of the difficulty in obtaining large (> 10 µm) single crystals, and because large-distance periodic features cause diffraction intensities to fall off rapidly with scattering angle, so that only limited small-angle data are available. On the other hand, transmission electron microscopy (TEM) is a powerful tool for structural characterization at the mesoscale level due to the stronger interaction of electrons with matter compared to X-rays, enabling the use of very small crystals. In particular, high-resolution TEM (HRTEM) images give the phase and the amplitude of the crystal structure factors experimentally, leading to a 3D structural model by electron crystallography. Cage-type anionic-surfactant-templated mesoporous silicas display rich structural diversity. Among them, cage-type mesoporous silica with tetrahedrally close-packed (TCP) structures can be described by four types of polyhedra, 5^12, 5^12 6^2, 5^12 6^3, and 5^12 6^4.[1] A variety of structural polymorph have been observed and characterized by TEM. Their structures show a close resemblance to the Frank-Kasper phases, which are well known in intermetallic compounds. We found mesoporous silica with dodecagonal quasicrystalline ordering as one of the TCP structures (Figure).[2] In this presentation, I will discuss structural characterization of aperiodic crystals at the mesoscale, such as mesoporous silicas and binary colloidal crystals, by electron microscopy.