Choice Of Unit Cell Research Articles

Topological edge states in bowtie ladders with different cutting edges

We have studied topological edge states in bowtie ladders with various edge truncations. The symmetric bowtie ladder, which comprises two trivial Su–Schrieffer–Heeger (SSH) lattices, exhibits an insulator–metal transition with trivial insulating states. On the other hand, the lattice can be transformed into an extended SSH lattice depending on the edge shapes with non-trivial insulating states in that the winding number is non-zero. The winding numbers are permutationally designated in the phase diagram depending on the choice of unit cell. The topological edge states are affected by the shape of the edge and the corresponding winding number. We also studied general bowtie ladder models with richer phase diagrams using the characteristics of the localization length of the edge states showing state bifurcation.

Size effects in lattice-structured cellular materials: material distribution

This work presents a novel mechanism that causes stiffness size effects in periodic cellular materials in bending, but is absent in other boundary conditions. The work begins by demonstrating that in certain circumstances, a single set of micropolar elastic material properties can predict size effects accurately for either bending or shear but not both. This suggests that, for periodic cellular materials, size effects in shear and bending arise from different mechanisms. This work then presents a novel mechanism causing size effects in bending, called a material distribution effect. A sample with only a few unit cells can have its material concentrated close to the neutral axis, or far away, depending on topology and choice of unit cell. A sample with many cells must have its material spread more evenly. This can cause either a stiffening or softening size effect. This work derives formulas to predict the magnitude and direction of these size effects and shows that these formulas are able to predict size effects for a variety of different periodic cellular materials in different types of bending boundary conditions.

Crystal structure of the (REE)-uranyl carbonate mineral kamotoite-(Y)

AbstractKamotoite-(Y) is a rare supergene product of uraninite hydration–oxidation weathering and its structure is unknown. Based on single-crystal X-ray diffraction data collected with high-redundancy using a microfocus source, kamotoite-(Y) is monoclinic, has space group P21/n,with a = 12.3525(5), b = 12.9432(5), c = 19.4409(7) Å, β = 99.857(3)°, V = 3069.8(2) Å3 and Z = 4. Crystals are pervasively twinned (two-fold rotation around [0.75 0 0.75]), giving a strongly pseudo-orthorhombic diffractionpattern. The pseudoorthorhombic pattern can be described with an orthorhombic super-cell (transformation matrix 0,1,0/1,0,1/3,0,1), approximately four times larger in volume then a true monoclinic unit cell. This unit-cell is the same as the cell given elsewhere for the structure of bijvoetite-(Y),another (REE)-containing uranyl carbonate. The successful structure solution and refinement (R = 0.044 for 6294 unique observed reflections), carried out using our choice of unit cell, as well as the superstructure refinement and comparison of the original structure data forbijvoetite-(Y) reveal that these two crystal structures are identical. The crystal structure of kamotoite-(Y) consists of electroneutral sheets of the bijvoetite-(Y) uranylanion topology and an interlayer with H2O molecules not-coordinated directly to any metal cation. Despite determinationof the kamotoite-(Y) structure and demonstration that bijvoetite-(Y) has the same structure, the identity of these two minerals cannot be proved without additional study of the holotype material.

Structures of the 2:1 Layers of Pyrophyllite and Talc

AbstractTo determine the relationships between the symmetry of the overall pyrophyllite and talc structure and the symmetry of individual layers, the geometry and symmetry of each 2:1 layer of pyrophyllite and talc were analyzed. For each, the previously published, refined unit cell may be rotated clockwise by ~60° for comparison to a layer unit cell. In pyrophyllite, the layer unit cell is ideal and shown to be orthogonal with C2/m symmetry. The agreement between the refined atomic coordinates and those calculated for the layer with C2/m symmetry confirms that the symmetry of the pyrophyllite layer is C2/m. The obliquity of the pyrophyllite refined cell results from the layer stacking and the choice of unit cell, but the interlayer stacking sequence does not disturb the layer symmetry. In contrast, talc has an oblique layer cell, without a mirror plane. For the most part, the distortion of the talc 2:1 layer is probably caused by an elongation of unshared O-O lateral edges around M1 that creates a slight corrugation of the octahedral sheet surface. Perhaps of lesser importance, the distortion of the talc layer cell may result from Coulombic interactions between cations of adjacent layers, and these cation-to-cation distances are sufficiently large (~6–7.5 Å) that the weak van der Waals forces that stabilize the stacking are not overcome. Because pyrophyllite has a vacant octahedral site, similar interactions are not present, and this results in a more idealized layer symmetry.Phyllosilicates consisting of layers with an orthogonal cell and mirror plane (pyrophyllite, kaolinite, sudoite) were shown to have similar stacking faults. In these structures, the 2:1 or 1:1 layers have uniform orientation, and stacking faults occur owing to interstratifications of two alternative interlayer displacements in the same crystal that are related by a mirror plane in the projection on the (001) plane. In talc, stacking faults are associated with layer rotations by ±120°, whereas the lateral displacement between the adjacent tetrahedral sheets across the interlayer region is relatively ordered.

Structural Investigations of Novel Indoloacridines

5,6-Dihydro-9-chloro-7-phenylindolo[3,2-c]acridinium sulfates (3a-c) were formed from 2,3,4,9-tetrahydro-1H-carbazol-1-ones (1a-c) on reaction with 2-amino-5-chlorobenzophenone. 3b and 3c are isostructural and all three sulfates crystallize in the triclinic system with the space group P $$ \bar{1} $$ each with two crystallographically independent organic cations and one sulfate anion in the asymmetric unit. 3a exhibits the parameters a = 9.8896(17), b = 14.165(2) and c = 14.583(2) A, α = 83.184(3)°, β = 78.985(3)° and γ = 75.903(3)°; 3b a = 10.5713(9), b = 14.5667(12) and c = 14.9947(13) A, α = 90.916(1)°, β = 97.153(1)° and γ = 97.208(1)°; and 3c a = 10.5505(10), b = 14.6090(13) and c = 15.2960(14) A, and α = 91.4310(10)°, β = 98.8630(10)° and γ = 98.9100(10)°. The structure of 3c exhibits voids of 286 A3 of ill defined and disordered solvate molecules. The structure of 3a is both non-merohedrally twinned and also exhibits an interesting type of pseudosymmetry. All but the sulfate ion and the ethylene moieties also fit with a unit cell half the size with only one independent cation, and disordered ethylene moieties and sulfate anions. The choice of unit cell was made based on the diffraction pattern and the absence of disorder in the larger cell, pointing towards alternating ordered ethylene and sulfate units rather than randomly disordered ones. [The structures of three indoloacridines 3a, 3b, and 3c with similar packing motifs are described. The structure of 3a is both non-merohedrally twinned and also exhibits an interesting type of pseudosymmetry by being a double volume ordered supercell of an alternative cell with disordered ethylene and sulfate anion moieties.]

Angular momentum of optical vortex arrays

Guided by the aim to construct light fields with spin-like orbital angular momentum (OAM), that is light fields with a uniform and intrinsic OAM density, we investigate the OAM of arrays of optical vortices with rectangular symmetry. We find that the OAM per unit cell depends on the choice of unit cell and can even change sign when the unit cell is translated. This is the case even if the OAM in each unit cell is intrinsic, that is independent of the choice of measurement axis. We show that spin-like OAM can be found only if the OAM per unit cell vanishes. Our results are applicable to the z component of the angular momentum of any x- and y-periodic momentum distribution in the xy plane, and can also be applied other periodic light beams, arrays of rotating massive objects and periodic motion of liquids.

Fast clustering of equivalent structures in crystal structure prediction

Most methods of crystal structure prediction generate many trial structures. Because these may differ in choice of unit cell, it is not always immediately obvious whether or not two such structures are equivalent. A method to answer this question is described for the case where the asymmetric unit contains one molecule in a general position, defined by the rotation and translation of that molecule with respect to some reference geometry. In the comparison of two structures, the rotation needed to transform one orientation into the other is determined first. Then it is checked whether this rotation corresponds to a transformation that is compatible with the imposed space group symmetry. A final test compares the cell lengths, the cell angles, and the molecular centers of gravity after the transformation of one structure into the other. The method is implemented for triclinic, monoclinic, and orthorhombic systems and is found to be very fast in tests on hypothetical crystal structures of acetic acid. © 1997 John Wiley & Sons, Inc. J Comput Chem 18:1036–1042, 1997

Redetermination of the crystal structure of Na 3Sc 2(PO 4) 3

The crystal structure of Na 3Sc 2(PO 4) 3 at room temperature has been reinvestigated. The space group is R 3 c with a = 8.930 (2) Å and c = 22.245 (4) Å. By transforming the data of a single crystal relative to one choice of unit cell into another choice of unit cell we obtain the three structures of Na 3Sc 2(PO 4) 3 reported in the literature.