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Publication# Phase-transition-induced twinning in the 1 : 1 adduct of hexamethylenetetramine and azelaic acid

Résumé

The title compound, C6H12N4. C9H16O4, undergoes several thermotropic phase transitions. The crystalline structure is layered, with sheets of azelaic acid linked to sheets of hexamethylenetetramine by hydrogen bonds. In the room-temperature phase, the azelaic acid molecules are disordered. By lowering the temperature, this disorder partially disappears. The ordering is clearly observed in reciprocal space where on the rods of diffuse scattering, present in the room-temperature phase, a series of superstructure reflections emerges. This phase transition leads to twin-lattice quasisymmetry (TLQS) twinning. The structure of this twinned phase is explored in this paper. There are two orientational domains linked by a mirror plane which relates disordered orientations of the acid molecules above the phase transition. A single domain has space group P2(1)/c. The structure has been solved and refined on the complete set of data to R-1 = 0.0469. The chains remain partially disordered, showing two acid groups with unequal population: the major form corresponding to a carboxylic acid and the minor to a carboxylate. The ordering of the structure, when going through the phase transition, is interpreted in terms of stabilization by C-H ... O hydrogen bonding. A least-squares estimator of the twinning volume ratio is developed that gives an expression for the twinning ratio in terms of the intensities of nonoverlapping reflections. The twinning ratio obtained in the structure refinement compares very well with that obtained from this estimator. [References: 23]

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Urotropin (U) and azelaic acid (AA) form 1:1 co-crystals (UA) that give rise to a rather complex diffraction pattern, the main features of which are diffuse rods and bands in addition to the Bragg reflections. UA is characterized by solvent inclusions, parasite phases, and high vacancy and dislocation densities. These defects compounded with the pronounced tendency of U to escape from the crystal edifice lead to at least seven exotic phase transitions (many of which barely manifest themselves in a differential scanning calorimetry trace). These involve different incommensurate phases and a peritectoid reaction in the recrystallization regime (T-h >0.6). The system may be understood as an OD (order-disorder) structure based on a layer with layer group P(c)c2 and cell a(o) similar or equal to 4.7, b similar or equal to 26.1 and c similar or equal to 14.4 Angstrom. At 338 K the layer stacking is random, but with decreasing temperature the build-up of an orthorhombic MDO (maximal degree of order) structure with cell a(1) = 2a(o), b(1) = b, c(1) = c and space group Pcc2 is begun (at similar to 301 K). The superposition structure of the OD system at T = 286 (1) K with space group Bmmb and cell (a) over cap = 2a(o), (b) over cap = b and (c) over cap = c/2 owes its cohesion to van der Waals interactions between the AA chains and to three types of hydrogen bonds of varied strength between U-U and U-AA. Before reaching completion, this MDO structure is transformed, at 282 K, into a monoclinic one with cell a(m) = a(o) + c/4, b(m) = b, c(m) = -2(a(o) + c/2), space group P2(1)/c, spontaneous deformation similar to2degrees, and ferroelastic domains. This transformation is achieved in two steps: first a furtive triggering transition, which is not yet fully understood, and second an improper ferroelastic transition. At similar to 233 K, the system reaches its ground state (cell a(M) = a(m), bM = b, c(M) = c(m) and space group P2(1)/c) via an irreversible transition. The phase transitions below 338 K are described by a model based on the interaction of two thermally activated slip systems. The OD structure is described in terms of a three-dimensional Monte Carlo model that involves first- and second-neighbour interactions along the a axis and first-neighbour interactions along the b and c axes. This model includes random shifts of the chains along their axes and satisfactorily accounts for most features that are seen in the observed diffraction pattern.

2003Jean-Philippe Ansermet, Helmuth Berger, Simon Granville

We present magnetization and Se-77 Nuclear Magnetic Resonance (NMR) measurements in single crystals of the magnetoelectric compound Cu2OSeO3. The temperature and field dependence of the magnetization suggest a ferrimagnetic ordering at T-c similar or equal to 60 K in a 3up-1down configuration. The easy axis of the magnetization is along the [100] crystallographic direction. The Se-77 NMR line shape data collected at 14.09 T are consistent with the symmetries imposed by the cubic P2(1)3 space group in the temperature range 20-290 K and confirm that the magnetic transition is not accompanied by any lowering of the crystal symmetry as has recently been proposed by Bos et al. [Phys. Rev. B 78 094416 (2008)].

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.