Infinitesimal Periodic Deformations and Quadrics

We describe a correspondence between the infinitesimal deformations of a periodic bar-and-joint framework and periodic arrangements of quadrics. This intrinsic correlation provides useful geometric characteristics. A direct consequence is a method for detecting auxetic deformations, identified by a pattern consisting of homothetic ellipsoids. Examples include frameworks with higher crystallographic symmetry.

Modeling and Structure Determination of Homo-Oligomeric Proteins: An Overview of Challenges and Current Approaches

Protein homo-oligomerization is a very common phenomenon, and approximately half of proteins form homo-oligomeric assemblies composed of identical subunits. The vast majority of such assemblies possess internal symmetry which can be either exploited to help or poses challenges during structure determination. Moreover, aspects of symmetry are critical in the modeling of protein homo-oligomers either by docking or by homology-based approaches. Here, we first provide a brief overview of the nature of protein homo-oligomerization. Next, we describe how the symmetry of homo-oligomers is addressed by crystallographic and non-crystallographic symmetry operations, and how biologically relevant intermolecular interactions can be deciphered from the ordered array of molecules within protein crystals. Additionally, we describe the most important aspects of protein homo-oligomerization in structure determination by NMR. Finally, we give an overview of approaches aimed at modeling homo-oligomers using computational methods that specifically address their internal symmetry and allow the incorporation of other experimental data as spatial restraints to achieve higher model reliability.

Structural Changes of Hydroxylapatite during Plasma Spraying: Raman and NMR Spectroscopy Results

Functional osseoconductive coatings based on hydroxylapatite (HAp) and applied preferentially by atmospheric plasma spraying to medical implant surfaces are a mainstay of modern implantology. During contact with the hot plasma jet, HAp particles melt incongruently and undergo complex dehydration and decomposition reactions that alter their phase composition and crystallographic symmetry, and thus, the physical and biological properties of the coatings. Surface analytical methods such as laser-Raman and nuclear magnetic resonance (NMR) spectroscopies are useful tools to assess the structural changes of HAp imposed by heat treatment during their flight along the hot plasma jet. In this contribution, the controversial information is highlighted on the existence or non-existence of oxyapatite, i.e., fully dehydrated HAp as a thermodynamically stable compound.

Space groups and crystallographic symmetry: writing a multi-featured tutorial in a new style

Knowledge of space groups and the implications of space group symmetry on the physical and chemical properties of solids are pivotal factors in all areas of crystalline solids. As Jerry Jasinski and I met to bring our ideas in teaching this subject to life, we both felt that `early and often' – teaching the concepts with textual and visual reinforcement, is the key to providing a sound basis for students in this subject. The tutorial contains > 200 PowerPoint `slides', in five modules, arranged by crystal class; a sixth module covers special topics. A `credits' module gives the direct addresses of all embedded links. Space-group diagrams appear in International Tables format. The triclinic and monoclinic groups (2 + 13) are built from `scratch', and are derived from the Hermann–Mauguin symbol. An additional section provides practice on many (but not all) of the orthorhombic groups in crystal class 222. Finally, a `Special Topics' section on enantiomorphous space groups features space groups P41 and P43. In the tutorial, lattice points build iteratively and interactively via keyclick, and the coordinates of points `pop up' as the unit cell is filled. We trust that the elements of guidance, inquiry and occasional humor will make the learning process enjoyable.

Diffraction Features from (101¯4) Calcite Twins Mimicking Crystallographic Ordering

During phase transitions the ordering of cations and/or anions along specific crystallographic directions can take place. As a result, extra reflections may occur in diffraction patterns, which can indicate cell doubling and the reduction of the crystallographic symmetry. However, similar features may also arise from twinning. Here the nanostructures of a glendonite, a calcite (CaCO3) pseudomorph after ikaite (CaCO3·6H2O), from Victoria Cave (Russia) were studied using transmission electron microscopy (TEM). This paper demonstrates the occurrence of extra reflections at positions halfway between the Bragg reflections of calcite in 0kl electron diffraction patterns and the doubling of d104 spacings (corresponding to 2∙3.03 Å) in high-resolution TEM images. Interestingly, these diffraction features match with the so-called carbonate c-type reflections, which are associated with Mg and Ca ordering, a phenomenon that cannot occur in pure calcite. TEM and crystallographic analysis suggests that, in fact, (101¯4) calcite twins and the orientation change of CO3 groups across the twin interface are responsible for the extra reflections.

Symmetry Breakdown Related Fracture in 42CrMo4 Steel

Austenite grains that underwent the f.c.c. to b.c.c. (or b.c.t.) transformation are typically composed of 24 Kurdjumov–Sachs variants that can be categorized by three axes of Bain transformations; thus, a complete transformation generally displays 3-fold symmetry in (001) pole figures. In the present work, crystallographic symmetry in 42CrMo4 steel austempered below martensite start temperature was investigated with the help of the orientation distribution function (ODF) analysis based on the FEG-SEM/EBSD technique. It is shown that, upon phase transformations, the specimens contained 6-fold symmetry in all (001), (011), and (111) pole figures of an ODF. The ODF analysis, verified by theoretical modeling, showed that under plane-strain conditions cracks prefer to propagate through areas strongly offset by the high symmetry. The origin of high symmetry was investigated, and the mechanism of the symmetry breakdown was explained.

Crystal structure of a methanol solvate of a macrocycle bearing two flexible side-arms

Di-tert-butyl N,N′-{[13,15,28,30,31,33-hexaethyl-3,10,18,25,32,34-hexaazapentacyclo[25.3.1.15,8.112,16.120,23]tetratriaconta-1(31),3,5,7,9,12(33),13,15,18,20,22,24,27,29-tetradecaene-14,29-diyl]bis(methylene)}dicarbamate methanol disolvate, C52H72N8O4·2CH3OH, was found to crystallize in the space group P21/c with one half of the macrocycle (host) and one molecule of solvent (guest) in the asymmetric unit of the cell, i.e. the host molecule is located on a crystallographic symmetry center. Within the 1:2 host–guest complex, the solvent molecules are accommodated in the host cavity and held in their positions by O—H...N and N—H...O bonds, thus forming ring synthons of graph set R 2 2(7). The connection of the 1:2 host-guest complexes is accomplished by C—H...O, C—H...N and C—H...π interactions, which create a three-dimensional supramolecular network.