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Differential geometry provides a useful mathematical framework for describing the fundamental concepts in crystallography. The notions of point and associated vector spaces correspond to those of manifold and tangent space at a given point. A space-group operation is a one-to-one map acting on the manifold, whereas a point-group operation is a linear map acting between two tangent spaces of the manifold. Manifold theory proves particularly powerful as a unified formalism describing symmetry operations of conventional as well as modulated crystals without requiring a higher-dimensional space. We show, in particular, that a modulated structure recovers a three-dimensional periodicity in any tangent space and that its point group consists of linear applications. [ABSTRACT FROM AUTHOR]

The symmetry of a commensurately modulated crystal structure can be described in two different ways: in terms of a conventional three-dimensional space group or using the superspace concept in (3 + d) dimensions. The three-dimensional space group is obtained as a real-space section of the (3 + d) superspace group. A complete network was constructed linking (3 + 1) superspace groups and the corresponding three-dimensional space groups derived from rational sections. A database has been established and is available at http://superspace.epfl.ch/finder/. It is particularly useful for finding common superspace groups for various series of modular ('composition-flexible') structures and phase transitions. The use of the database is illustrated with examples from various fields of crystal chemistry. [ABSTRACT FROM AUTHOR]

Co-crystals of hexamethylenetetramine and resorcinol were investigated by X-ray diffraction. The structure was refined in the superspace group , . In the average structure the resorcinol molecules are disordered between two orientations. The main effect of the modulation in the structure is a harmonic modulation of the occupation probabilities of the two orientations of the resorcinol molecule. However, the modulated order is not perfect and the resorcinol molecules remain partially disordered. Below 270 K the crystal undergoes a phase transition to a commensurately modulated structure with modulation vector and superspace group . The structure of the low-temperature phase could not be determined owing to the poor quality of the crystals affected by the reconstructive phase transition. [ABSTRACT FROM AUTHOR]

After studying more than 100 different samples of calaverite Au1-pAgpTe2 (p < 0.15) three famous mineralogists declared in a well documented 1931 publication the invalidity of the law of rational indices. What could lead them to draw such an extreme conclusion from the observation law of Hauÿ dating from more than two centuries? Their observations remained unexplained for the next forty years. In the 1970''s, the room temperature structure of γ-Na2CO3 resisted any attempt for a precise structural analysis. The appearance of satellite reflections was noted on single crystal diffractograms which led to the generalisation of the concept of crystal. This generalisation consisted in using at least four integers to fully characterise each individual diffractions peaks. The theory of periodic crystals in space of higher dimension, i.e. the superspace was then developed to deal with the new experimental observations. Later, a new class of materials called composite crystals and still later, the discovery of quasicrystals only reinforced the validity of the superspace concept to describe any material requiring more than three integers to index their diffraction pattern.What is the essence of superspace to describe crystalline structures? Any crystal structure requesting more than three integers to index its diffraction pattern can be described as a periodic object in superspace with dimension equal to the number of required integer. The structure observed in our real word is a three dimensional cut of the superspace description. In general this cut is irrational and consequently the crystal is aperiodic. Calaverite and γ-Na2CO3 are examples of aperiodic crystals which includes incommensurately modulated crystals, composite and quasi-crystals. Rational cuts are also possible. In this case, the structure is periodic and is usually called a superstructure. [Copyright &y& Elsevier]

In order to perform a molecular dynamics (MD) simulation of the incommensurate crystalline structure hexamethylenetetramine suberate (C 6 H 12 N 4 )(HOOC-(CH 2 ) 6 -COOH), we present in a first step the separate simulations of the crystalline structure of each of the two pure components, hexamethylenetetramine (HMT) and suberic acid. The domain decomposition parallel MD program ddgmq is used for this purpose. A second-generation consistent force field (CFF91) is employed to describe the interactions between atoms. Starting from experimental crystal structures, both pure components were heated from low to high temperatures. Our MD results show that the HMT system can be well represented by CFF91. In the case of suberic acid the layered structure of the crystal is largely preserved although deviations in the unit cell lengths from the experimental values are ∼10%. Rather than attempt a complex re-parametrisation of CFF91 we chose to impose a fixed compensating external pressure tensor to correct for the deficiencies of the chosen force field. After optimising this compensating external pressure tensor at one temperature we find that experimental lattice constants and angles can be well reproduced over a range of temperatures. [ABSTRACT FROM AUTHOR]

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[sub 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[sub o] ... 4.7, b ... 26.1 and c ... 14.4 Å. 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[sub 1] = 2a[sub o], b[sub 1] = b, c[sub 1] = c and space group Pcc2 is begun (at ∼301 K). The superposition structure of the OD system, at T = 286 (1)K with space group Bmmb and cell â = 2a[sub o], &bcirc; = b and ĉ = 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[sub m] = -a[sub o] +c/4, b[sub m] = b, c[sub m] = -2(a[sub o] + c/2), space group P2[sub 1]/c, spontaneous deformation ~2°, 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 ∼233 K, the system reaches its ground state (cell a[sub M] = a[sub m], b[sub M] = b, c[sub M] = c[sub m] and space group P2[sub 1]/c) via an irreversible transition. The phase... [ABSTRACT FROM AUTHOR]

Using electrodeposition from a bath of molten fluorides single crystals of tetragonal β-tantalum have been obtained for the first time at normal pressure; The unit-cell parameters a = 10.211 (3), c = 5.3064 (10) Å, space group P&macr4;21m. The β-Ta structure belongs to the σ-type Frank-Kasper structures which are typical for binary intermetalliccompounds and β-U. In comparison to the σ-type, additional intercalated Ta atoms (population factor ∼0.01) have been detected between the atoms located in the channels of the structure. The shorter interatomic distances observed between the channel atoms in comparison with the atoms of the framework justify the 'self-hosting' characteristic. β-Ta exhibits common features with the complex tetragonal structures of the high-pressure phases for the elements Rb, Ba, Sr, Bi and Sb. [ABSTRACT FROM AUTHOR]

An La2Co1.7 crystals was investigated by single-crystal neutron and X-ray diffraction. The neutron measurement was performed with a Laue white-beam technique at 15 K and room temperature, using a large position-sensitive detector. The X-ray measurements were obtained at room temperature from a CCD detector. The average structure of La2Co1.7 is hexagonal with cell parameters a = 4.885 (1), c = 4.273 (2) Å and space group P63/mmc. The satellites are located at the vertices of small hexagons perpendicular to the c axis. The modulated crystal was indexed assuming a sixfold twinned 3 + 1 dimensional structure with q = (α, 0, γ). The structure was solved in the pseudoorthorhombic cell, with a = 8.461 (1), b = 4.885 (1), c = 4.273 (2) Å, in the superspace group C2/m(α, 0, γ). Owing to space requirements, the Co atoms cannot fit precisely into the octahedral sites of the La h.c.p. (hexagonal close packing). Instead, the Co atoms adopt a different periodicity, which is not commensurate with the periodicity of the La atoms. Two structure models have been refined in order to describe this behavior, one using the sawtooth function for the positional modulation of cobalt and the other describing the structure as a composite system. The chemical composition calculated from the composite model is La2O1.8 (1) with the estimated standard deviation arising from the variation of q for different samples. In both models lanthanum is incommensurately modulated, while the position of cobalt seems not to be affected by any relative periodic displacement. [ABSTRACT FROM AUTHOR]

The structural phases observed in LiKSO4 crystals due to thermotropic transitions have been studied for more than a century. Nowadays many different phases are referenced, but some of the results are still controversial. Structural studies by single-crystal X-ray diffraction from room temperature to 803 K are presented here. Phase II (708 < T < 943 K) is extensively discussed on the basis of ordered and disordered models, using harmonic and anharmonic atomic displacements, and considering a twinned crystal composed of three orthorhombic domains Analyses of the same phase at different temperatures determine the best structure model. [ABSTRACT FROM AUTHOR]

The incommensurately modulated structure of the mineral natrite has been refined for the first time. Two single-crystal grains, Lv and Kh, from two different occurrences [Mt. Karnasurt, Lovozero massif (Lv), Kola peninsula, Russia, and the pegmatite of Mt. Koashva, Khibiny massif (Kh)], have been investigated at 293 and 120 K using synchrotron X-ray data. The average structures of both minerals are identical and the basic features of the structural modulations are similar to the synthetic γ-Na2CO3 phase previously published. The γ (incommensurate) → δ (lock-in) phase transition reported at low temperature for the synthetic compound was not observed down to 120 K in natural natrite. Crystal-chemical aspects, especially about the second coordination sphere for the carbon atoms, are examined to explain the different structural behaviors observed at low temperature. The possible role played by the minor isomorphous substitutions in natural natrite specimens is also discussed. [ABSTRACT FROM AUTHOR]

The structure of room-temperature ZrP2O7 is shown to be orthorhombic by a combination of high-resolution synchrotron powder diffraction and single-crystal synchrotron diffraction data. Small nontwinned single crystals were obtained by synthesizing the compound using solvothermal methods at temperatures below the cubic to orthorhombic phase transition. The average P-O-P angle is 146°. DFT calculations (B3LYP/AUG-cc-pVDZ) on the isolated P2O74- anion yield a P-O-P angle of 153.42° and indicate that the barrier to inversion is of the order 3.6 kJ mol-1. [ABSTRACT FROM AUTHOR]