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Although W. L. Bragg's law can be easily derived for beginners in the field of crystallography, its interpretation however seems to cause some difficulties which lies essentially in the relation between the concept of lattice planes and the unit cell constants characterizing the lattice periodicity of the crystal structure. Our approach is certainly not new and is based on a more physical approach where every single point in the crystal participates in the diffraction process. From the early stages of developing a model of diffraction, we make abundant use the dual reference frames namely the direct and reciprocal reference frames. With this approach, W. L. Bragg's law can be reformulated directly in terms of the reciprocal unit cell constants avoiding thus the necessity to introduce a priori the notion of lattice planes. Following the derivation of the diffraction law, different steps and methods leading to the complete determination of a crystal structure are derived. We present also some simulation tools to explain in particular the crystal diffraction phenomenon based on the Ewald sphere and the solution of crystalline structures based on the dual space iteration techniques which are currently used.

An efficient method for modelling a polytypic family is presented with the example of cimetidine in the form C polymorph. The method exploits the (3 + 1)-dimensional superspace model, which is a powerful tool for the description, prediction and understanding of polytype modifications in small-molecule crystallography, as illustrated with this pharmaceutical example.

The incommensurately modulated scheelite-like KSm(MoO4)2 structure has been refined in the monoclinic superspace group I2/b(ab0)00 by the Rietveld method on the basis of synchrotron radiation powder diffraction data. The systematic broadening of satellite reflections has been accounted for by applying anisotropic microstrain line-broadening. The microstructure has been studied by transmission electron microscopy (TEM). The partial disorder of the K and Sm cations in the A position is best approximated by a combination of harmonic and complex crenel functions with (0.952Sm + 0.048K) and (0.952K + 0.048Sm) atomic domains. This combination yields a compositional wave distribution from {KMoO4} to {SmMoO4} observed in the ab structure projection along q. The specific features of KSm(MoO4)2 and degree of the A-cation ordering are discussed in comparison with the previously reported structure of KNd(MoO4)2.

SUPERFLIP is a computer program that can solve crystal structures from diffraction data using the recently developed charge-flipping algorithm. It can solve periodic structures, incommensurately modulated structures and quasicrystals from X-ray and neutron diffraction data. Structure solution from powder diffraction data is supported by combining the charge-flipping algorithm with a histogram-matching procedure. SUPERFLIP is written in Fortran90 and is distributed as a source code and as precompiled binaries. It has been successfully compiled and tested on a variety of operating systems.

Scheelite related compounds with general formula Mn(XO4)m are of interest owing to their optical properties, stability and relatively simple preparation. Eu3+-containing scheelites are considered as red emitting phosphors and the main factors affecting their luminescence are thought to be chemical composition and particle size while the influence of their structure is generally ignored. Here we report eight compounds from the NaxEu(2_x)/3MoO4 series prepared by conventional solid-state reaction and present a detailed analysis of their crystal structures. Six of them have modulated structures, a common feature of SRCs, in which dopant Eu3+ ions are orderly distributed. Moreover, different amounts of Eu3+ dimers are detected in the modulated structures, characterized by weak satellite reflections appearing in the lower angle part of the XRD patterns. These reflections are indexed and incorporated into Rietveld’s refinement using superspace (3 + 1)-dimension symmetry. The remarkable feature of the compounds is that the characteristic luminescence parameters, overall and intrinsic quantum yields, Eu lifetimes, and sensitization efficiencies, correlate with the number of Eu3+ aggregates, but not directly with the composition x of the materials. This provides an efficient tool for understanding and controlling the luminescence properties of scheelite related compounds.

In this work we describe the structure and the magnetic properties of a new copper(II)-NIT complex, where NIT stands for nitronyl nitroxide radical. This was achieved using (NBu4)4[Cu2(bopba)] as a precursor (where NBu4+ is the tetrabutylammonium cation and bopba is bis-o-phenylenebis(oxamate)), whose crystal structure was resolved and its magnetics properties studied. From the reaction of this precursor with a cationic nitronyl nitroxide radical, a new compound was prepared and magnetically characterized, namely (pEtRad)4[Cu2(bopba)] (where pEtRad+ stands for the cationic para-2-(4-pyridil)-4,4,5,5-tetramethylimidazoline-1 -oxyl-3-oxide radical). The magnetic properties suggest that in this six-spin complex there is a competition between short range ferromagnetic and antiferromagnetic interactions. The ferromagnetic interaction can be explained by the orthogonality of the involved magnetic orbitals. Neste trabalho é descrito o estudo das propriedades magnéticas de um novo complexo cobre(II)-NIT, onde NIT representa um radical do tipo nitronil nitróxido. O precursor (NBu4)4[Cu2(bopba)] (onde NBu4+ é o cátion tetrabutilamônio e bopba é o bis-o-fenilenobis(oxamato)) teve a sua estrutura cristalina determinada por difração de raios X e também foi caracterizado magneticamente. Através da complexação deste precursor com o radical catiônico nitronil nitróxido, um novo complexo foi obtido, o (pEtRad)4[Cu2(bopba)] (onde pEtRad+ é o radical catiônico para-2-(4-piridil)-4,4,5,5-tetrametilimidazolina-1-oxil-3 -óxido). Os estudos magnéticos sugerem que neste complexo existe uma competição entre interações ferromagnéticas e antiferromagnéticas. A interação ferromagnética pode ser explicada com base na ortogonalidade dos orbitais magnéticos envolvidos.

A method based on the superspace approach is presented with the aim of generating a family of modular structures from a single incommensurately modulated structure. This approach based on the variation of the modulation vector q is applied to the generation of the K5Yb(MoO4)4, potassium ytterbium tetramolybdate, family of modular structures. The $\beta$ coefficient of the modulation vector q = $\beta b*$ is a temperature-dependent variable which determines the modification. Our method gives a unified frame to describe and explain the three temperature-dependent phases of K5Yb(MoO4)4. Phase can be represented as a polytypic modification with 1/2b*\le q\le 2/3b*; phases $\gamma$ (q = 1/2b*) and \alpha (q = 1b*) are the lowest and the highest temperature members of the K5Yb(MoO4)4 family, respectively.

The superspace concept is a new crystallographic approach for the generalisation of symmetry, structural models and structure dependent properties. Although the original motivation for the superspace was intended to describe symmetry of aperiodic crystals, the concept gained general acceptance in a wider range of applications. In particular, the method is able to combine structures into a single model, thus offering a clearer view of their hidden relationships. While the individual members of the family can have different space group symmetry and be both periodic or aperiodic, the generalization in superspace leads to a common crystallographic structure model and a common superspace group reuniting all of them. The benefits of such model are best demonstrated, if applied to an extensive and diverse family of compounds. Hexagonal ferrites is such an extensive family and a prominent example. Ferrites are widely known for their applications in motors, consumer electronics, microwave technology and, recently, in medicine. In general ferrites are complex oxides composed of various metals and oxygen. "Hexagonal" ferrites, named in opposition to cubic or spinel ferrites, form a particular group closely related to the mineral magnetoplumbite with the approximate chemical composition PbFe12O19. Their high uniaxial magnetocrystalline anisotropy renders them particularly useful for electronics. Currently the family includes more than 60 members of magnetic materials with unit cell dimension extending up to 1577 Å. The study of compounds approaching such a "biological magnitude" is necessarily complicated by different symmetries of individual structures and by the large number of possible structure models, generated by different stacking of several basic units. Fortunately, the generalization in higher dimensional space offers a mechanism for solving these difficulties. The description of the family is not only more elegant but also reveals characteristic relations which are not easily observable while dealing with individual structures. Understanding the "code" of such features permits a direct readout of crystal structures and unique structure solutions for this realm of inorganic species. In this work, the structures of the hexagonal ferrites are treated in superspace as a structural modulation of a common underlying average structure. The average layer-to-layer distance is directly related to the average structure periodicity along the layer stacking direction. In a first approximation series of equidistant layers are stacked along a specific direction. A modulation is introduced by varying the type of each layer. The fundamental atomic modulation is therefore of occupational nature and can be described by means of step-like functions, which define discontinuous atomic domains in superspace. The higher-dimensional description leads to the diffraction pattern indexed with four indices hklm. However, this indexing is not unique, providing room for various models. An appropriate choice of the model is based on common structural features of the described compounds and their embedding in superspace. In order to simplify the search for a common superspace group for a set of given space groups, especially for future superspace embeddings, a database providing analytical relations between (3+1)-dimensional superspace groups and three-dimensional space groups has been created and is now available on the World Wide Web. The first complete derivation of a subgroup-supergroup tree for (3+1)-dimensional symmetry made in the course of this thesis provided an informational ground for such database. The results obtained in the present thesis demonstrate that the superspace approach is an appropriate and powerful tool for analysing compounds sharing common structural features. Both periodic and aperiodic ferrite structures, which were investigated in the course of this work, can be better understood by the description in (3+1)-dimensional superspace.