Lattice Metrics Research Articles

A Computer Program for Objective Point Symmetry Classifications of Electron Diffraction Spot Patterns with Apparent Hexagonal or Rectangular-Centered Lattice Metric

A Computer Program for Objective Point Symmetry Classifications of Electron Diffraction Spot Patterns with Apparent Hexagonal or Rectangular-Centered Lattice Metric

Objective Point Symmetry Classifications/Quantifications of an Electron Diffraction Spot Pattern with Pseudo-Hexagonal Lattice Metric

The recently developed information-theoretic approach to crystallographic symmetry classifications and quantifications in two dimensions (2D) from digital transmission electron and scanning probe microscope images is adapted for the analysis of an experimental electron diffraction spot pattern, for the first time. Digital input data are considered in this approach to consist of the pixel-wise sums of approximately Gaussian distributed noise and an unknown underlying signal that is strictly 2D periodic. Structural defects within the crystals or on the crystal surfaces, instrumental image recording noise, slight deviations from zero-crystal-tilt conditions in transmission electron microscopy, inhomogeneous staining in structural biology studies of intrinsic membrane protein complexes in lipid bilayers, and small inaccuracies in the algorithmic processing of the digital data all contribute to a single generalized noise term. The plane symmetry group and projected Laue class(or 2D Bravais lattice type) that is anchored to the least broken symmetries are identified as genuine in the presence of generalized noise. More severely broken symmetries that are not anchored to the least broken symmetries are identified as pseudo-symmetries. Our point symmetry quantification study of an electron diffraction spot pattern is highly topical because a new contrast mechanism for 4D scanning transmission electron microscopy was recently demonstrated by other authors. The usage of objective symmetry quantifications is bound to become the preeminent condition of the establishment of that contrast mode as an industry-wide standard.

Untangling the Structure and Dynamics of Lithium-Rich Anti-Perovskites Envisaged as Solid Electrolytes for Batteries

Lithium-rich anti-perovskites (LiRAPs) have attracted a great deal of attention as they have been praised as another superior group of solid electrolytes that can be used to realize all-solid-state batteries free of flammable liquids. Despite several studies that have reported on the properties of LiRAPs, many questions remain unanswered. In particular, these include fundamental ones concerning the structure, stability, and Li-ion conductivity and diffusivity. Moreover, it is not clear whether some of the previously reported compounds do really exist. To untangle the current picture of LiRAPs, we synthesized “Li3OCl” and Li2OHCl polymorphs and applied a wide spectrum of methods, such as powder X-ray diffraction (PXRD), powder neutron diffraction (PND), nuclear magnetic resonance spectroscopy, and impedance spectroscopy to carefully shed some light on LiRAPs. Here we self-critically conclude that the cubic polymorph of the two compounds cannot be easily distinguished by PXRD alone as the lattice metrics an...

Crystallometric and projective properties of Kikuchi diffraction patterns

Kikuchi diffraction patterns can provide fundamental information about the lattice metric of a crystalline phase. In order to improve the possible precision and accuracy of lattice parameter determination from the features observed in Kikuchi patterns, some useful fundamental relationships of geometric crystallography are reviewed, which hold true independently of the actual crystal symmetry. The Kikuchi band positions and intersections and the Kikuchi band widths are highly interrelated, which is illustrated by the fact that all lattice plane trace positions of the crystal are predetermined by the definition of only four traces. If, additionally, the projection centre of the gnomonic projection is known, the lattice parameter ratios and the angles between the basis vectors are fixed. A further definition of one specific Kikuchi band width is sufficient to set the absolute sizes of all lattice parameters and to predict the widths of all Kikuchi bands. The mathematical properties of the gnomonic projection turn out to be central to an improved interpretation of Kikuchi pattern data, emphasizing the importance of the exact knowledge of the projection centre.

Perovskite to postperovskite transition in NaFeF3.

The GdFeO3-type perovskite NaFeF3 transforms to CaIrO3-type postperovskite at pressures as low as 9 GPa at room temperature. The details of such a transition were investigated by in situ synchrotron powder diffraction in a multianvil press. Fit of the p-V data showed that the perovskite phase is more compressible than related chemistries with a strongly anisotropic response of the lattice metrics to increasing pressure. The reduction in volume is accommodated by a rapid increase of the octahedral tilting angle, which reaches a critical value of 26° at the transition boundary. The postperovskite form, which is fully recoverable at ambient conditions, shows a regular geometry of the edge-sharing octahedra and its structural properties are comparable to those found in CaIrO3-type MgSiO3 at high pressure and temperature. Theoretical studies using density functional theory at the GGA + U level were also performed and describe a scenario where both perovskite and postperovskite phases can be considered Mott-Hubbard insulators with collinear magnetic G- and C-type antiferromagnetic structures, respectively. Magnetic measurements are in line with the theoretical predictions with both forms showing the typical behavior of canted antiferromagnets.

Binomial options pricing has no closed-form solution

We set a lower bound on the complexity of options pricing formulae in the lattice metric by proving that no general explicit or closed form (hypergeometric) expression for pricing vanilla European call and put options exists when employing the binomial lattice approach. Our proof follows from Gosper's algorithm.

Space of symmetry matrices with elements 0, ±1 and complete geometric description; its properties and application

A fixed set, that is the set of all lattice metrics corresponding to the arithmetic holohedry of a primitive lattice, is a natural tool for keeping track of the symmetry changes that may occur in a deformable lattice [Ericksen (1979). Arch. Rat. Mech. Anal. 72, 1-13; Michel (1995). Symmetry and Structural Properties of Condensed Matter, edited by T. Lulek, W. Florek & S. Walcerz. Singapore: Academic Press; Pitteri & Zanzotto (1996). Acta Cryst. A52, 830-838; and references quoted therein]. For practical applications it is desirable to limit the infinite number of arithmetic holohedries, and simplify their classification and construction of the fixed sets. A space of 480 matrices with cyclic consecutive powers, determinant 1, elements from {0, ±1} and geometric description were analyzed and offered as the framework for dealing with the symmetry of reduced lattices. This matrix space covers all arithmetic holohedries of primitive lattice descriptions related to the three shortest lattice translations in direct or reciprocal spaces, and corresponds to the unique list of 39 fixed points with integer coordinates in six-dimensional space of lattice metrics. Matrices are presented by the introduced dual symbol, which sheds some light on the lattice and its symmetry-related properties, without further digging into matrices. By the orthogonal lattice distortion the lattice group-subgroup relations are easily predicted. It was proven and exemplified that new symbols enable classification of lattice groups on an absolute basis, without metric considerations. In contrast to long established but sophisticated methods for assessing the metric symmetry of a lattice, simple filtering of the symmetry operations from the predefined set is proposed. It is concluded that the space of symmetry matrices with elements from {0, ±1} is the natural environment of lattice symmetries related to the reduced cells and that complete geometric characterization of matrices in the arithmetic holohedry provides a useful tool for solving practical lattice-related problems, especially in the context of lattice deformation.

Crystal engineering of lattice metrics of perhalometallate salts and MOFs

The synthesis of the salt 3 and metallo-organic framework (MOF) [{(4,4(')-bipy)CoBr(2)}(n)] 4 by a range of solid state (mechanochemical and thermochemical) and solution methods is reported; they are isostructural with their respective chloride analogues 1 and 2. 3 and 4 can be interconverted by means of HBr elimination and absorption. Single phases of controlled composition and general formula [4,4(')-H(2)bipy][CoBr(4-x)Cl(x)] 5(x) may be prepared from 2 and 4 by solid--gas reactions involving HBr or HCl respectively. Crystalline single phase samples of 5(x) and [{(4,4(')-bipy)CoBr(2-x)Cl(x)}(n)] 6(x) were prepared by solid-state mechanochemical routes, allowing fine control over the composition and unit cell volume of the product. Collectively these methods enable continuous variation of the unit cell dimensions of the salts [4,4(')-H(2)bipy][CoBr(4-x)Cl(x)] (5(x)) and the MOFs [{(4,4(')-bipy)CoBr(2-x)Cl(x)}(n)] (6(x)) by varying the bromide to chloride ratio and establish a means of controlling MOF composition and the lattice metrics, and so the physical and chemical properties that derive from it.

Incommensurately modulated LT″-Ni 1+δSn ( δ=0.60, 0.63): Rietveld refinement, line-broadening analysis and structural relation with LT- and LT′-Ni 1+δSn

X-ray powder diffraction data of NiAs/Ni 2In-type Ni 1.60Sn and Ni 1.63Sn alloys annealed at or below about 573 K reveal the development of an incommensurately ordered phase called LT″. In this phase Ni(2) atoms occupy partially the trigonal–bipyramidal interstices formed by five Sn within an NiAs-type arrangement Ni(1)Sn. The modulated occupational ordering of Ni(2) in the LT″ phase can be described in the superspace group Cmcm( α00)0 s 0, and the parameters describing this occupational modulation were refined together with atomic displacement modulations using the Rietveld method. The structure parameters revealed close structural analogies of the LT″ phase with the previously reported commensurate LT-Ni 1+ δ Sn and incommensurate LT′-Ni 1+ δ Sn phases (A. Leineweber, J. Solid State Chem. 177 (2004) 1197–1212), which both occur for lower Ni contents than the LT″ phase. The 1st-order satellite reflections visible in the powder-diffraction patterns exhibit, with respect to the fundamental reflections, a considerable diffraction-line broadening, caused by a small size of the particularly ordered domains. This small-domain-size broadening was successfully described by a recently developed reflection-index ( hklm) dependent (anisotropic) line-broadening model (A. Leineweber, V. Petricek, J. Appl. Crystallogr. 40 (2007) 1027–1034) designed to consider the effect of fluctuations of the lattice metrics on the peak widths in powder diffraction patterns of incommensurately modulated crystal structures. The small domain sizes encountered for the LT″ phase indicate that domain coarsening is much more difficult than for the LT and LT′ phases. This special feature of the LT″ phase goes along with a compared to the LT and LT′ phases absent orthorhombic distortion and the low ordering temperature, which are discussed as a consequence of the ordering patterns due to the Ni(2) atoms.

Ambiguities in Powder Indexing: the Impact of a Quaternary Lattice Metric Singularity on the Characterization of Mawsonite and Chatkalite.

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d-spacings. The minerals Mawsonite and Chatkalite are of especial interest as both are characterized by tetragonal unit cells that correspond to the second member of a quaternary lattice metric singularity. This singularity includes lattices that are Cubic I, Tetragonal P, Orthorhombic F, and Orthorhombic P. The Mawsonite and Chatkalite lattices are unique in that they are highly specialized. In each case: (1) the determinative c/a ratio is very near 1/√2, (2) the symmetrical scalars of the reduced form [a · a : b · b : c · c = 1:2:2] have greater specialization than required for the given reduced form type, (3) the tetragonal lattice has derivative lattices of higher symmetry, and (4) the powder pattern is highly compressed. Mawsonite and Chatkalite serve as exemplar-type compounds. Their tetragonal structure has important implications in structure determination using powder diffraction data. First, any cubic I lattice — established solely on the basis of indexing procedures — may actually be tetragonal or orthorhombic. Second, in establishing the lattice of an unknown, results from powder data indexing require routine confirmation by other techniques (e.g., single crystal, optical, etc.).

Ambiguities in Powder Indexing: Conjunction of a Ternary and Binary Lattice Metric Singularity in the Cubic System.

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d-spacings. The existence of such singularities, therefore, has a practical and theoretical impact on the indexing of powder patterns. For example, in experimental practice an indexing program may find only the lower symmetry member of a singularity. Obviously, it is important to recognize such cases and know how to proceed. Recently, we described: a binary singularity involving a monoclinic and a rhombohedral lattice in a subcell-supercell relationship anda second type of singularity—a ternary singularity–in which two of the three lattices are in a derivative composite relationship. In this work, we describe a ternary lattice metric singularity involving a cubic P, a tetragonal P, and an orthorhombic C lattice. Furthermore, there is a binary singularity, involving a hexagonal P and orthorhombic P lattice, which is characterized by a set of unique d-spacings very close to that of the ternary singularity. The existence of such singularities is more common than once thought and requires a paradigm shift in experimental practice. In addition singularities provide opportunities in material design as they point to highly specialized lattices that may be associated with unusual physical properties.

A Variational Model for Reconstructive Phase Transformations in Crystals, and their Relation to Dislocations and Plasticity

1We study the reconstructive martensitic transformations in crystalline solids (i.e., martensitic transformations in which the parent and product lattices have arithmetic symmetry groups admitting no finite supergroup), the best known example of which is the bcc–fcc transformation in iron. We first describe the maximal Ericksen-Pitteri neighborhoods in the space of lattice metrics, thereby obtaining a quantitative characterization of the “weak” transformations, which occur within these domains. Then, focusing for simplicity on a two-dimensional setting, we construct a class of strain-energy functions admitting large strains in their domain, and which are invariant under the full symmetry group of the lattice. In particular, we exhibit an explicit energy suitable for the square-to-hexagonal reconstructive transformation in planar lattices. We present a numerical scheme based on atomic-scale finite elements and, by means of our constitutive function, we use it to analyze the effects of transformation cycling on a planar crystal. This example illustrates the main phenomena related to the reconstructive martensitic phase changes in crystals: in particular, the formation of dislocations, vacancies and interstitials in the lattice.

The median function on distributive semilattices

A median of a k-tuple π=( x 1,…, x k ) of elements of a finite metric space ( X, d) is an element x for which ∑ i=1 k d( x, x i ) is minimum. The function m with domain the set of all k-tuples with k>0 and defined by m(π)={x : x is a median of π} is called the median function on X. Continuing with the program of characterizing m on various metric spaces, this paper presents a characterization of the median function on distributive semilattices endowed with the standard lattice metric.

Engineering crystal symmetry and polar order in molecular host frameworks.

A crystal design strategy is described that produces a series of solid-state molecular host frameworks with prescribed lattice metrics and polar crystallographic symmetries. This represents a significant advance in crystal engineering, which is typically limited to manipulation of only gross structural features. The host frameworks, constructed by connecting flexible hydrogen-bonded sheets with banana-shaped pillars, sustain one-dimensional channels that are occupied by guest molecules during crystallization. The polar host frameworks enforce the alignment of these guests into polar arrays, with properly chosen guests affording inclusion compounds that exhibit second harmonic generation because of this alignment. This protocol exemplifies a principal goal of modern organic solid-state chemistry: the precise control of crystal symmetry and structure for the attainment of a specific bulk property.

Metric engineering of soft molecular host frameworks.

The self-assembly and solid-state structures of host-guest inclusion compounds with lamellar architectures based on a common building block, a resilient hydrogen-bonded sheet consisting of guanidinium ions and sulfonate moieties of organodisulfonate "pillars", are described. The pillars connect adjacent sheets to generate galleries with molecular-scale cavities occupied by guest molecules. The size, shape, and physicochemical character of the inclusion cavities can be systematically adjusted by interchanging framework components while maintaining the lamellar architecture, enabling prediction and control of crystal lattice metrics with a precision that is unusual for "crystal engineering". The reliability of the lamellar architecture is a direct consequence of conformational flexibility exhibited by these hosts that, unlike rigid systems, enables them to achieve optimal packing with guest molecules. The adaptability of these hosts is further reflected by an architectural isomerism that is driven by guest templating during assembly of the inclusion compounds. Host frameworks constructed with various pillars display metric interdependences among specific structural features that reveal a common mechanism by which these soft frameworks adapt to different guests. This unique feature facilitates structure prediction and provides guidance for the design of inclusion compounds based on these hosts.

The median function on distributive semilattices

Abstract Continuing with the program to characterize the median function (f(x 1 ,…,x k ) = {x : ∑ i = 1 k d(x, x i ) is minimum }) on various finite metric spaces, we present a characterization when the metric space is a finite distributive semilattice endowed with the standard lattice metric.

Phenomenological model of anisotropic peak broadening in powder diffraction

Anisotropic line-shape broadening (peak width which is not a smooth function ofd-spacing) is frequently observed in powder diffraction patterns, and can be a source of considerable difficulty for whole-pattern fitting or Rietveld analysis. A model of the multi-dimensional distribution of lattice metrics within a powder sample is developed, leading naturally to a few parameters which can be varied to achieve optimal line-shape fits. Conditions on these parameters are derived for all crystal systems, and the method is illustrated with two examples: sodiump-hydroxybenzoate and rubidium fulleride.

Changing lattice metrics of synthetic cordierites: The metastable hexagonal to orthorhombic transformation sequence during isothermal annealing

Starting from glasses heated at 1290°C, the irreversible transition of initially formed hexagonal high-into stable orthorhombic low-cordierite was studied for pure Mg 2 Al 4 Si 5 O 18 as well as for potassium-doped cordierites according to the scheme KAlSi −1 with 0.10 and 0.25 K per formula unit, using the Guinier method and synchrotron X-ray powder diffraction, respectively. Instead of the classical distortion index Δ, a 2θ-range at higher angles was monitored. Compared to a theoretically calculated continuous change of lattice metrics during the transformation, the data obtained after annealing for various lengths of time clearly show a discontinuous development with well Al,Si-ordered orthorhombic cordierites forming in small amounts after only four hours and increasing in mass with continued heating at the expense of hexagonal cordierite. This metastable coexistence of two cordierite polymorphs is caused by slow reaction kinetics in the intimately intergrown polycrystalline aggregates formed by devitrification and contrasts with the abrupt transition observed in single crystals under identical heating conditions (Wunder et al., 1991). For K-doped cordierite aggregates, the kinetically induced phase coexistence is conserved for longer heating periods the higher the amounts of potassium incorporated, and cordierites with intermediate lattice metrics are observed periodically. On the basis of these results, previous interpretations of NMR- and IR-spectra, as well as of calorimetric data that were thought to have been obtained on single-phase cordierites must be reconsidered, because they were also made on polycrystalline, polyphase aggregates. Thus, it is not clear as to whether or not the Al,Si-ordering process in cordierite is truly continuous. For all polyphase cordierite products formed after intermediate heating times, the classical distortion index Δ is an artefact that yields at most the approximate mass ratio between the hexagonal and orthorhombic polymorphs. Nevertheless, it may still be a useful qualitative parameter indicating the stage of transition. For truly single-phase orthorhombic synthetic or natural cordierites. Δ remains an important rapid indicator of lattice metrics which in turn are largely dependent on the chemical composition