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Publication# Na2/7Gd4/7MoO4: a Modulated Scheelite-Type Structure and Conductivity Properties

Abstract

Scheelite-type compounds with the general formula (A1,A2)(n)(B1,B2)O-4 (2/3

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Related concepts (5)

Related publications (4)

Structure

A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as biological organisms, minerals and chemicals. Abstract structures include data structures in computer science and musical form. Types of structure include a hierarchy (a cascade of one-to-many relationships), a network featuring many-to-many links, or a lattice featuring connections between components that are neighbors in space.

Crystal

A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography.

Crystal structure

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure.

The mathematical facet of modern crystallography is essentially based on analytical geometry, linear algebra as well as group theory. This study endeavours to approach the geometry and symmetry of crystals using the tools furnished by differential geometry and the theory of Lie groups. These two branches of mathematics being little known to crystallographers, the pertinent definitions such as differentiable manifold, tangent space or metric tensor or even isometries on a manifold together with some important results are given first. The example of euclidean space, taken as riemannian manifold, is treated, in order to show that the affine aspect of this space is not at all an axiom but the consequence of the euclidean nature of the manifold. Attention is then directed to a particular subgroup of the group of euclidean isometries, namely that of translations. This has the property of a Lie group and it turns out that the action of its elements, as well as those of its Lie algebra, plays an important role in generating a lattice on a manifold and in its tangent space, too. In particular, it is pointed out that one and only one finite and free module of the Lie algebra of the group of translations can generate both, modulated and non-modulated lattices. This last classification therefore appears continuous rather than black and white and is entirely determined by the parametrisation considered. Since a lattice in a tangent space has the properties of a vector space, it always possesses the structure of a finite, free module, which shows that the assignment of aperiodicity to modulated structures is quite subjective, even unmotivated. Thanks to the concept of representation of a lattice or a crystal in a tangent space, novel definitions of the notions of symmetry operation of a space group and point symmetry operation, as well as symmetry element and intrinsic translation arise; they altogether naturally blend into the framework of differential geometry. In order to conveniently pass from one representation of a crystal in one tangent space to another or to the structure on a manifold, an equivalence relation on the tangent bundle of the manifold is introduced. This relation furthermore allows to extend the concept of symmetry operation to the tangent bundle; this extension furnishes, particularly in the euclidean case, a very practical way of representing symmetry operations of space groups completely devoid of any dependence on an origin, or, in other words, in which each and every point may be considered the origin. The investigation of the group of translations having being completed, the study of the linear parts of the isometries comes naturally. Based on the fact that the set of linear parts possesses the structure of a Lie group, several results are proven in a rigorous manner, such as the fact that a rotation angle of π/3 is incompatible with a three-dimensional cubic lattice. Procedures for determining different crystal systems in function of the type of rotation are laid out by way of the study of orthogonal matrices and their relation to the matrix associated with the type of system. Finally, the description of a crystal by its diffraction patterns is taken on. It is shown that the general aspect of such a pattern is directly linked to the action of that free and finite module of the Lie algebra of translations which generates a lattice on a manifold. In the case of modulated crystals, it is demonstrated that the appearance of supplementary spots is caused by the geometry, i.e. by the parametrisation of the manifold in which the crystal exists and not by the action of the module in the Lie algebra. Thus, there exists a neat separation: the geometrical aspect on the one hand, and the action of the group on the other. As the last topic, other ways of interpreting the diffraction pattern of a modulated structure are laid out in order to argue that mere experimental data do not warrant the uniqueness of a model. The goal of this study is by no means an attempt at overthrowing existing structural models such as the superspace-formalism or at revolutionising the methods for determining structures, but is rather aimed at sustaining that the definition of certain notions becomes thoroughly natural within the appropriate mathematical framework, and, that the term aperiodicity assigned to modulated structures no longer has a true meaning.

The comparable order of magnitude between interatomic distances in a crystal and the wavelength of X-rays make X-ray crystallography the ideal analytical tool to gain insight into the structure of crystalline material, including biomolecules. Nevertheless, biomolecular crystallography has until now relied on the successful growth of single crystals of suitable size and quality. These remain the exception rather than the rule, since biomolecules often produce polycrystalline precipitate instead. Yet, an interest in making use of the once-discarded polycrystalline material, through the technique of powder diffraction, has only recently emerged. This can be accounted for by the information deficit which powder diffraction data suffers from in comparison with that of single crystal. The paucity of information in powder data stems from the compression of the three-dimensional reciprocal space onto the one-dimension of a powder pattern. In spite of this, powder diffraction holds the potential for application in biomolecular crystallography as is shown in the two different studies presented herein. Both studies were carried out with methods which do not rely on employing previously determined crystal-structures as molecular models. This therefore allowed the objective assessment of the quality of information that powder diffraction data can contribute to the structural investigation of biomolecules. In the first project, the traditional single-crystal structure-solution process is applied to data extracted from protein powder diffraction patterns measured on a synchrotron source. The use of models is avoided by employing the de novo phasing method of isomorphous replacement. With two protein test-cases, namely hen egg white lysozyme and porcine pancreatic elastase, it is demonstrated that protein powder diffraction data can afford structural information up to medium resolution. Indeed, a single isomorphous replacement analysis generated molecular envelopes accurately describing the crystal packing of both protein systems, while a multiple isomorphous replacement experiment, carried out only on lysozyme, revealed an electron density map in which elements of the secondary structure could be located. In fact, the resolution of the latter was discovered to be sufficient to determine the chirality of the protein molecule it represented. In addition to being encouraging, these results do not reflect the full potential of biomolecular powder diffraction, due, in large part, to the ultimately unsuitable nature of one type of phasing method, single crystal, being applied to another type of data, powder. An alternative approach to extract information from protein powder diffraction data is to employ powder-specific structure-solution techniques, such as global optimization methods. Although these methods make use of a starting "model", it is a molecular description of the system under study based on known chemical quantities rather than a related molecular configuration based on a previously determined crystal structure. Since their conception, global optimization methods have continuously been developed to enable the tackling of increasingly complex crystal structures. However, the immense complexity of biomacromolecules has kept proteins well out of reach of such methods. In an attempt to further reduce the gap separating the two levels of complexity, the second study reported herein puts forth the implementation of Ramachandran plot restraints into the algorithm of a global optimization method, i.e. that of simulated annealing. More specifically, the Ramachandran plot was approximated using a two-dimensional Fourier series which was subsequently expressed as a penalty function and incorporated into the search algorithm.

Nicolas Alexandre Serge Leclaire

Cyanine dyes are organic semiconductor compounds with light absorption and emission properties useful for emerging technologies such as solar cells and light-emitting devices. The characteristics of these materials in the solid state depend on their organization of the constituting building blocks. This thesis focuses on controlling the morphology of cyanine dye thin films at different length scales and clarifying the resulting properties.
When microstructures present features whose size matches visible light wavelengths, new properties may arise from light-matter interactions. Here the properties resulting from the light-matter interactions of cyanine droplet films cast from solution are studied. Based on experimental evidence, it is shown that dye droplet ensembles scatter light with different efficiencies and wavelength ranges depending on their dimensions. FDTD simulations are used to show that this effect results fromscattering enhancement at the absorption edge of the dye where the refractive index varies considerably. Simulations also provide a better understanding of individual dropletsâ interaction with light. While earlier work had hypothesized that the observed scattering phenomenon were due to crystalline clusters within the droplets, this work highlights the contribution of the dye filmmorphology.
Cyanines also form single crystals whose fabrication induces molecular-scale order in the material. Previous work demonstrated that thin single crystals could be grown by solvent vapor annealing of dye droplets. Here it is shown that in uncontrolled conditions, cyanine single crystals destabilize to formdendritic crystals. In-situ microscopy observations highlight the solute reservoir role of the droplet distribution surrounding a growing crystal; when the
distance between droplets and the crystal front is large, the solute supply is diffusion-limited. Moreover variations in local pressure equilibrium between the droplets and crystal front lead to advection fluxes which perturb the crystal growth. These observations help design configurations to either prevent crystal destabilization or take advantage of the dendritic growth in a controlled manner.
In addition, the patterning of crystals on a substrate is relevant for their application in devices. A practical challenge is to induce single crystal growth at specific locations. Here, surfaces patterned with SAMs of hydrophilic and hydrophobic thiols are used to create dye droplet arrays from which crystals can be grown. This method is shown to yield local crystallization of the dye and to prevent crystal destabilization through better control of the droplet distribution. By varying the dye solution concentration, partial control over crystal density is achieved, however it proved difficult to control the number of nuclei per droplet. A more controlled evaporation and solvent vapor annealing system might be necessary to master the nucleation process.
Finally the structure and optical properties of cyanine single crystals are addressed. The crystal structure was determined by X-ray diffraction. Structural aspects are shown to lead to excitonic couplings, which are evidenced by orientation-dependent spectroscopic measurements of single crystals. Although further investigation of the absorption band structure is necessary, the results are promising for photovoltaic devices as they might improve exciton transport compared to amorphous layers.