For any queries please contact the Organizing Committee at aicschool@cristallografia.org

Programme

(Wednesday, 7 September, 8.30 am to 5.30 pm)

Matrix representation of symmetry operations; Derivation of the symmetry operation from the matrix vector pair and identification of the corresponding symmetry element; Effect of a change of basis on the representation of the symmetry operations and on the Hermann-Mauguin symbol of the space group; Symmetry reduction (group-subgroup) and its consequences on the symmetry operations and Wyckoff positions.

The first part of the lecture offers a brief revision of representation theory of crystallographic groups. Starting from basic principles, those essential definitions and properties of representations are discussed that are of particular interest in crystallographic applications, namely: (i) definition and basic properties of finite-group representation; (ii) Irreducible and reducible representations, (iii) Subduced and induced representations, (iv) Direct product representations. Illustrative examples and didactic exercises from the representations of the crystallographic point groups will complement the lecture material.

The second part of the lecture will deal with representations of space groups, including: (i) Representations of the translation group; (ii) Symmetry in reciprocal space: Brillouin zones and wave- vector symmetry types; (iii) Little groups and little co-groups; allowed (little-group) representations; (iv) Irreducible representations of space groups and their construction; (v) Subduced and direct-product representations of space groups.

The lecture material will be illustrated with examples from the online databases and programs of the Bilbao Crystallographic server for representations of crystallographic groups such as the database on point-group representations, the Brillouin-zone database and the computer tools for space- group representations.

The lecture is dedicated to a survey of historical ideas, lines of thought, and recent developments in the study of the phenomena that occur during the formation of crystalline solids at a molecular or atomic level, and that are at the roots of stability and polymorphism. Classical nucleation theories, modern multi-step mechanisms and growth units, and theoretical prediction tools are discussed, along with differences between the molecular case and the inorganic and mineralogical environments. Is the construction of a sensible, general, first-principle theory of crystallization possible? Neither thermodynamics not kinetics alone seem to have the last word.

How to construct an energy diagram system and useful information about thermodynamic stability of polymorphs. Monotropic systems and enantiotropic systems. Strategies to obtain metastable phases (quenching, seeding).

The aim of the lecture is to give a short introduction to the crystallographic data of the space groups tabulated in International Tables for Crystallography, Vol. A: Space-group symmetry (ITA). The following main topics are included: (i) symbols of space groups (Schoenflies and Hermann-Mauguin symbolism); (ii) descriptions of space-group symmetry operations (ITA and Seitz symbols of symmetry operations, symmetry-elements and general-position space-group diagrams, 'General-position' presentation of symmetry operations); (iii) asymmetric units and origins of space groups; (iv) general and special Wyckoff positions, site-symmetry groups and multiplicities of Wyckoff positions; (v) coordinate transformations in crystallography, conventional and nonconventional settings of space groups. Illustrative examples and didactic exercises complement the lecture material.

The lecture gives a short outline of the contents and applications of the symmetry relations between space groups listed in Vol. A1 of International Tables for Crystallography (ITA1). After a short introductory remarks on group-subgroup relations of crystallographic groups, the following subjects will be treated: (i) tranlationengleche, klassengleiche and isomorphic maximal subgroups of space groups and their presentation in ITA1; (ii) general subgroups of space groups and the theorem of Hermann for subgroups of space groups; (iii) brief introduction to the domain-structure symmetry analysis: number of domain states, twin and antiphase domains, twinning operations, symmetry groups of the domain states; multiplicity and degeneracy; (iv) supergroups of space groups and their application in structural pseudosymmetry search; (v) Wyckoff-position splittings for group-subgroup related space groups. Illustrative examples and didactic exercises will complement the lecture material.

Types of solid-state phase transitions. Displacive versus order-disorder. Concept of ferroics. introduction to Landau theory. Symmetry principles. Methods of study.

Almost all phase transitions involve at least some degree of lattice distortion. This can be understood in terms of strains coupled to the driving order parameter. The formalities of such coupling make it possible to use changes in lattice parameters to provide information on the evolution of the order parameter(s) themselves. Strain also has a determinative role in suppressing fluctuations and promoting mean field behaviour. In systems with multiple instabilities, common strains which couple to two or more order parameters provide an indirect mechanism for coupling mechanism between the order parameters, such as for antiferromagnetic order plus soft acoustic mode softening in wustite, FeO. Patterns of strain evolution then determine patterns of elastic anomalies but, while the strains vary up to ~3%, elastic constants may vary by 10’s of percent. The overall approach for analyzing strains makes use of Landau theory and is equally valid for minerals and functional materials including, for example, the colossal magnetoresistant perovskite (Pr,Ca)MnO3 and silicate perovskite (Mg,Fe)SiO3 in the Earth’s lower mantle.

What are perovskites and why are the important? X-ray and neutron diffraction to study long-range structures, including Rietveld analysis. Examples of physical properties. Description of basic structure types including cation/anion displacements, tilted octahedra and distortions. Brief introduction to complex perovskites. Tilts versus Jahn-Teller distortions. Phase transitions in perovskites. Irreducible Representations.

Phase transitions are accompanied by diverse and characteristic changes in elastic constants. Because of the fundamental control of structural evolution by strain relaxations these changes are expect to show patterns which conform to Landau theory. Resonant Ultrasound Spectroscopy provides a convenient means of measuring not only the elastic constants but also the anelastic loss which accompanies ferroelastic and coelastic transitions in small experimental samples. Possible mechanisms for anelastic loss include mobile twin walls, high spin/low spin relaxations and interfaces at first order transitions, which may be superimposed on background effect of grain boundaries. Characteristic patterns of elastic softening/stiffening with changing temperature are associated with structural phase transitions, Jahn-Teller transitions, magnetic transitions, metal/insulator transitions, superconductivity, etc., but the fundamental property in common is strain. Similarly, characteristic patterns of acoustic dissipation are indicative of the mobility of microstructures which include some strain component.

Introduction to the definition of polymorphism for organic molecule: the academic and industrial points of view. Conformational polymorphs. Tautomeric polymorphs. Psuedo polymorphs (solvate and hydrate). Characterization of crystal forms: DSC; TGA; Hot stage microscopy; single-crystal X-Ray diffraction; powder X-ray diffraction.

The quantum mechanical aspects of the intermolecular potential are analyzed in comparison with intramolecular bonding. Physical effects are discussed in terms of viable models, including the traditional separation into Coulombic-polarization and London dispersion effects. A broad overview of primary applications is given, in comparison and synergy with experimental evidence on thermophysical properties of the crystalline state. Changes in type and intensity of cohesive energies along phase transitions between condensed phases (amorphous, liquid, and crystalline states) are described and discussed.

Why polymorphs are patentable. How you can protect the crystal forms. Case history on polymorphism and Patents: Ritonavir, Zantac.

Hybrid Photon Counting (HPC) detectors provide distinct advantages for crystallographic data acquisition. Sharp spatial resolution, widest dynamic range, high count rates and zero detector background greatly facilitate any XRD experiment, and in particular increasingly demanding in situ and time-resolved measurements. This presentation will give a brief background on HPC technology and illustrate how a specific detector feature answers a particular requirement of an XRD measurement. Examples from a variety of challenging applications will be presented: accurate structural analysis, polymorphic purity, phase transformations and lowest quantification limits.

Overview of the classification of domain structures. Symmetry reduction at a phase transition and generation of twin (orientation) and antiphase (translation) domains. Reticular characterisation of twin domains (calculation of the twin index and obliquity). Matrix representation of the twin operations.

Diffraction theory and disorder. Methods of studying local order, especially Pair Distribution Function analysis. Case studies from the literature e.g. BaTiO

We will look at the preparation of various organic and metal-organic crystalline solids by high temperature solution methods compared to ambient crystallization and see how this approach can lead to different polymorphs. Variations in solvent and temperature can influence the phase type produced. In the case of organic compounds we describe how some molecules, such as oleanic acid, can form a wide variety of solvated phases, (sometimes called pseudo-polymorphs) and also how this can be controlled. Hybrid solids such as Metal-Organic Framework compounds (MOFs) are an important emerging class of porous materials that can complement and extend the properties of zeolites. Some classical examples of polymorphism in this class will be explored, along with the use of a solvothermal approach to prepare new previously inaccessible framework types. This is the case for [Cd(BzIm)2] which can form a surprisingly large array of polymorphs merely by changing the solvent.

The physical and chemical robustness of solids with organic components is discussed. Compared to purely inorganic phases such compounds are thought of as thermally unstable, but some solids are surprisingly robust to 500°C. Analytical tools such as DSC, TGA and VT-pXRD can help establish their compositional and structural integrity versus temperature, as well as any structural changes via phase transition behaviour. The use of VT-pXRD can be used to study phase changes for hybrid solids and MOFs and give an indication of their structural robustness, for example in the case of desolvation of their channels. Some diverse organic and metal-organic examples with order-disorder and symmetry-based phase transitions will be featured, including the anti-malarial compound artemisone and the novel quartz-like MOFs [M(2-EtIm)2].

Presently available models, methods and computer packages for the computational simulation of the crystalline state are reviewed. First-principles quantum mechanical methods are exemplified by a succinct description of the CRYSTAL environment (University of Torino). Static methods of simulation (atom-atom, distributed dipole, PIXEL) are compared with iterative (i.e. Monte Carlo-type) or fully dynamical treatments of phase space sampling. The presentation aims at giving a realistic picture of the achievements and of the many extant problems in the theoretical prediction and control of crystal structure and of its polymorphism.

General definitions. Global, local and partial operations. The concept of family structure. Nonspacegroup systematic absences.

The Rietveld method is the main tool employed for the analysis of powder diffraction data. Its success and speed rely on the major assumption that an infinite lattice, (space) group theory and atomic coordinates are sufficient to reproduce the position and intensity of the observed lines. Only ideal powders free of any defect are compatible with this view. To include the non-ideal nature of the instrument and common defects, the fundamental parameter approach and the Whole Powder Pattern Modelling have been developed, resulting in physical shapes and breadths of the diffraction peaks. The finite size of the domains, their shape and distributions, as well as the presence of a

Layered systems, heavily faulted structures, composite crystals, domain structures, OD structures in general and all the cases where multiple lattices are coherently coexisting in the diffracting domains cannot be properly handled using the Rietveld approach, as the knowledge of the unit cell and its symmetry are insufficient to describe the measured specimens. The Rietveld method is however currently employed also in those cases by considering the material as an incoherent mixture of phases or by introducing unphysical extra phases needed to account for the observed features. This is definitely wrong, as the material is no longer deterministic and the short and long range albeit related, can differ: each independent crystalline domain in the powder is potentially different from the others and can give a completely different powder pattern; the resulting powder pattern is the incoherent sum of all of them. The increased entropy and complexity of the system calls for a large number of parameters to describe the local structure of the material and its arrangement in the measured specimen.

It will be shown that in most practical cases, the introduction of concepts from the information theory, the OD theory, the matrix method and the Whole Powder Modelling still allow the generation of the powder pattern in a fast and accurate way. The specimen is described without recurring to different short and long range structures as usually proposed in the analysis of the PDF obtained by Fourier transforming the data. It will be shown that the calculation agrees with the results of the atomistic Debye scattering equation that, in those cases, results prohibitive in terms of computing requirements.

Brillouin spectroscopy is an optical technique based on the inelastic scattering of light (photons) by thermally generated acoustic vibrations (phonons) which can be used to measure the acoustic velocities and elastic properties of crystalline solids, glasses and liquids. It is the most frequently used technique for measurements of transparent single-crystals of materials of geophysical interest because the elastic properties of a material determine how much this will change reversibly its volume under an applied stress and therefore they can be used to calculate the seismic wave velocities (seismic properties of the Earth) and the change in density that occurs when minerals are under extreme conditions (pressure and/or temperature). The lecture will be divided in three parts: in the first part the physical principle of the Brillouin effect in solid will be presented together with the basic experimental setup for Brillouin measurements. In the second part the relation between the elastic tensor of an anisotropic material and the Brillouin frequency shifts will be derived and the general problems associated with such procedure will be underlined. Finally, the lecture will be concluded with a couple of working examples.

XRPD is often combined with other techniques to overcome its well-known limitations. The combined experiment can be “separated” or simultaneous. Pushed by the increased XRPD detectors speed and by the easiness of use of complementary probes, several experimental facilities combining simultaneously XRPD with other techniques are permanently installed and available in many synchrotron facilities. These factors allow new experiments to study phase transitions and transformations, and require new data analysis tools to manage the huge amounts of (possibly combined) data, collected in a few days of experimental work. Some examples of combined experiments will be presented, together with recently developed tools, based on statistical methods such as Principal Component Analysis (PCA) for the analysis of in situ XRPD data.

The Bilbao Crystallographic Server is a web site with crystallographic databases and programs. It can be used free of charge from any computer with a web browser via Internet. The aim of the tutorial is to give a practical guide to some of the computer tools available on the Bilbao Crystallographic Server for treating problems of theoretical crystallography, solid-state physics and crystal chemistry. The following topics will be discussed in more detail: (i) coordinate transformations in crystallography with special emphasis on the tools for crystal-structure descriptions and for crystal-structure comparisons; (ii) symmetry relations between crystal structures and their descriptions by Baernighausen tree (family tree of structures); (iii) tools for the evaluation of structural pseudosymmetry with applications in the search for ferroelectric and ferroelastic materials; (iv) online tools for group-theoretical analysis of structural phase transitions including the study of structural relationships and the symmetry-mode analysis of the phase transition.

Bilbao Crystallographic Server web site

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Main computer programs for the hands-on tutorials of the Bilbao Crystallographic Server

COMPSTRU (J Appl. Cryst. (2016), v. 49, 653-664)

A quantitative comparison of similar crystal structures is important for a cross-checking of different experimental and/or theoretical structural models of the same phase coming from different sources, or for the identification of different phases with the same symmetry, and it is fundamental for the still open problem of the classification of structures into structure types. In most cases, even if the setting of its space group is fixed, there is more than one equivalent description for a given structure. The existence of various equivalent structure descriptions makes the comparison of different structural models a non-trivial task in general. To deal with it, the program COMPSTRU has been developed. The program measures the similarity between two structures having the same space-group symmetry (or space groups that form an enantiomorphic pair) with the same or different compositions, and under the condition that the sequence of the occupied Wyckoff positions is the same in both structures (isopointal structures). During the hands-on tutorial, the efficiency and utility of the program will be demonstrated by a number of illustrative examples.

PSEUDO (Z. Krist. (2011), v.226(2), 186-196)

The program PSEUDO provides tools for the systematic search of structural pseudosymmetry, based on group-supergroup relations between space groups. For a crystal structure, specified by its space group, cell parameters and the coordinates of the atoms in the asymmetric unit, the program searches for pseudosymmetry among all minimal supergroups of the structure space group. The interpretation of a structural pseudosymmetry as a small distortion of a higher symmetric (prototype) structure allows: (i) the prediction of phase transitions at higher temperature, if the distortion is small enough; (ii) the search for new ferroelastic and ferroelectric materials; (iii) the detection of false symmetry assignments (overlooked symmetry) in crystal structure determination. During the tutorial, after defining the procedures for the detection and quantification of pseudosymmetry, the different options of the use of PSEUDO will be demonstrated by illustrative examples, including worked cases of polar structures which are either known or reported as possible ferroelectrics.

AMPLIMODES (J. Appl. Cryst. (2009), v. 42, 820-833)

AMPLIMODES is a computer program that can perform a symmetry-mode analysis of any distorted structure of displacive type. The analysis consists in decomposing the symmetry- breaking distortion present in the distorted structure into contributions from different symmetry-adapted modes. Given the high- and the low-symmetry structures, AMPLIMODES determines the atomic displacements that relate them, defines a basis of symmetry-adapted modes, and calculates the amplitudes and polarization vectors of the distortion modes of different symmetry frozen in the structure. The program uses a mode parameterization that is as close as possible to the crystallographic conventions, expressing all quantities for the asymmetric unit of the low-symmetry structure. Distorted structures are often related to their higher-symmetry counterparts by temperature- and/or pressure- driven phase transitions, ferroic phase transitions being a particular example. The automatic symmetry-mode analysis performed by AMPLIMODES can be very useful for establishing the driving mechanisms of such structural phase transitions or the fundamental instabilities at the origin of the distorted phases.

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