Producing highly complicated materials. Nature does it better.

Through the years, mineralogical studies have produced a tremendous amount of data on the atomic arrangement and mineral properties. Quite often, structural analysis has led to elucidate the role played by minor components, giving interesting insights into the physico-chemical conditions of mineral crystallization and allowing the description of unpredictable structures that represented a body of knowledge critical for assessing their technological potentialities. Using such a rich database, containing many basic acquisitions, further steps became appropriate and possible, into the directions of more advanced knowledge frontiers. Some of these frontiers assume the name of modularity, complexity, aperiodicity, and matter organization at not conventional levels, and will be discussed in this review.

Incommensurate atomic density waves in the high-pressure IVb phase of barium.

The host-guest structures of elements at high pressure discovered a decade ago still leave many open questions due to the lack of precise models based on full exploitation of the diffraction data. This concerns in particular Ba IV, which is stable in the range 12-45 GPa. With the example of phase Ba IVb, which is characterized here for the first time, a systematic analysis is presented of possible host-guest structure models based on high-quality single-crystal diffraction data obtained with synchrotron radiation at six different pressures between 16.5 and 19.6 GPa. It is shown that a new incommensurately modulated (IM) structure model better fits the experimental data. Unlike the composite models which are commonly reported for the Ba IV phases, the IM model reveals a density wave and its pressure-dependent evolution. The crucial role played by the selected model in the interpretation of structure evolution under pressure is discussed. The findings give a new experimental basis for a better understanding of the nature of host-guest structures.

The role of the tangent bundle for symmetry operations and modulated structures.

An equivalence relation on the tangent bundle of a manifold is defined in order to extend a structure (modulated or not) onto it. This extension affords a representation of a structure in any tangent space and that in another tangent space can easily be derived. Euclidean symmetry operations associated with the tangent bundle are generalized and their usefulness for the determination of the intrinsic translation part in helicoidal axes and glide planes is illustrated. Finally, a novel representation of space groups is shown to be independent of any origin point.

Differential geometry: a natural tool for describing symmetry operations.

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.

KSm(MoO(4))(2), an incommensurately modulated and partially disordered scheelite-like structure.

The incommensurately modulated scheelite-like KSm(MoO(4))(2) structure has been refined in the monoclinic superspace group I2/b(alphabeta0)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 {KMoO(4)} to {SmMoO(4)} observed in the ab structure projection along q. The specific features of KSm(MoO(4))(2) and degree of the A-cation ordering are discussed in comparison with the previously reported structure of KNd(MoO(4))(2).

Capabilities and limitations of a (3 + d)-dimensional incommensurately modulated structure as a model for the derivation of an extended family of compounds: example of the scheelite-like structures.

The previously reported incommensurately modulated scheelite-like structure KNd(MoO(4))(2) has been exploited as a natural (3 + 1)-dimensional superspace model to generate the scheelite-like three-dimensional structure family. Although each member differs in its space-group symmetry, unit-cell parameters and compositions, in (3 + 1)-dimensional space, they share a common superspace group, a common number of building units in the basic unit cell occupying Wyckoff sites with specific coordinates (x, y, z) and specific basic unit-cell axial ratios (c/a, a/b, b/c) and angles. Variations of the modulation vector q, occupation functions and t(0) are exploited for the derivation. Eight topologically and compositionally different known structures are compared with their models derived from the KNd(MoO(4))(2) structure in order to evaluate the capabilities and limitations of the incommensurately modulated structure to act as a superspace generating model. Applications of the KNd(MoO(4))(2) structure as a starting model for the refinement and prediction of some other modulated members of the family is also illustrated. The (3 + 1)-dimensional presentation of the scheelite-like structures reveals new structural relations, which remain hidden if only conventional three-dimensional structure descriptions are applied.

Hexagonal ferrites: a unified model of the (TS)nT series in superspace.

Hexagonal ferrites represent an extensive family of mixed-layer magnetic materials with periods up to 1500 A along the stacking direction, probably constituting the largest unit cells in the inorganic realm. The (TS)(n)T subfamily includes P3m1 and R3m structures that can be derived from Y ferrite Ba(2)M(2)Fe(12)O(22) (M = Zn, Fe, Co, Mg, Mn) by introducing stacking faults. A unified (3 + 1)-dimensional superspace model is proposed for all members of the (TS)(n)T family. The model belongs to the superspace group X3m1(00gamma) with X = {(1/3, 2/3, 0, 1/3), (2/3, 1/3, 0, 2/3)}, has a unit cell of the basic structure with a = 5.88, c = 4.84 A and modulation vector q = 4n+3/9n+6 c*, where n is rational for periodic structures and irrational for the aperiodic ones. The model was tested on calculated data of one of the principal members of the (TS)(n)T family, the Y ferrite (n = infinity). The fit obtained with the superspace model was excellent. The model allowed a reduction of refinable parameters by 19% with respect to the ordinary refinement without a significant increase of the refinement R values.

Potassium gallium hydrogenphosphate fluoride, K(2)Ga[H(HPO(4))(2)]F(2).

Hydrothermally synthesized dipotassium gallium {hydrogen bis[hydrogenphosphate(V)]} difluoride, K(2)Ga[H(HPO(4))(2)]F(2), is isotypic with K(2)Fe[H(HPO(4))(2)]F(2). The main features of the structure are ([Ga{H(HPO(4))(2)}F(2)](2-))(n) columns consisting of centrosymmetric Ga(F(2)O(4)) octahedra [average Ga-O = 1.966 (3) Angstrom and Ga-F = 1.9076 (6) Angstrom] stacked above two HPO(4) tetrahedra [average P-O = 1.54 (2) Angstrom] sharing two O-atom vertices. The charge-balancing seven-coordinate K(+) cations [average K-O,F = 2.76 (2) Angstrom] lie in the intercolumn space, stabilizing a three-dimensional structure. Strong [O...O = 2.4184 (11) Angstrom] and medium [O...F = 2.6151 (10) Angstrom] hydrogen bonds further reinforce the connections between adjacent columns.

Growth-induced incommensurability observed in the organic co-crystal hexamethylenetetramine resorcinol.

Co-crystals of hexamethylenetetramine and resorcinol were investigated by X-ray diffraction. The structure was refined in the superspace group Xmcm(0beta0)s0s, X=(1/2 1/2 0 1/2). 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 q=(0 1/2 1/4) and superspace group X2/m(0betagamma)0s. 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.

The room-temperature superstructure of ZrP2O7 is orthorhombic: there are no unusual 180 degrees P-O-P bond angles.

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 degrees. DFT calculations (B3LYP/AUG-cc-pVDZ) on the isolated P2O7(4-) anion yield a P-O-P angle of 153.42 degrees and indicate that the barrier to inversion is of the order 3.6 kJ mol(-1).

The incommensurately modulated structure of a tricyclic natural-product-like compound of empirical formula C22H20O3.

3,4-Diphenyl-2a,5a,6,7,8,8a,8b-heptahydro-furo[4,3,2-de]chromen-2-one (1) was prepared as part of a project aimed at the synthesis of polycyclic natural-product-like scaffolds. X-ray analysis of crystals grown from ethanol revealed an incommensurately modulated structure. Data include main and satellite reflections up to second order. The modulation vector was refined using the NADA program. The modulated character of the structure of the organic compound C(22)H(20)O(3) is interpreted in terms of the intermolecular C-H...O hydrogen bonds and close-contact approximation.

Getting more out of an incommensurately modulated structure: the example of K5Yb(MoO4)4.

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 = betab* 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 beta can be represented as a polytypic modification with 1/2b* < or = q < or = 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.

A reinterpretation of the phase transitions in Na2CO3.

Based on the structural data of phases alpha (hexagonal; 756-972 K), beta (monoclinic; 605-751 K), gamma (incommensurate, monoclinic; 295 K) and delta (lock-in, monoclinic; 110 K) of sodium carbonate, Na2CO3, we could draw a parallel between the phase transitions and the evolution of the second coordination sphere of the C atoms. The temperature-dependent structures observed in the beta phase are reproduced in the incommensurate gamma phase as a modulation wave, which relates to the content of the symmetrically equivalent {110} lattice planes in the alpha phase. By decreasing the temperature, the phase transitions are associated with a stepwise increase in the number of Na ions participating in the second coordination sphere of the C atoms. Over the full temperature range, this number increases from 3 to 7. The C-O distances and the mobility of the O atoms depends on the number of Na ions in the vicinity of the C atoms.

The role of second coordination-sphere interactions in incommensurately modulated structures, using beta-K5Yb(MoO4)4 as an example.

The incommensurate palmierite-like structure of beta-K5Yb(MoO4)4, potassium ytterbium tetramolydate, has been refined in the (3 + 1)-dimensional monoclinic superspace group X2/m(0rho0)00, with X = [0 0 0 0; (1/2) (1/2) 0 0; 0 0 (1/2) (1/2); (1/2) (1/2) (1/2) (1/2)] and the unit-cell parameters a = 10.4054 (16), b = 6.1157 (12), c = 19.7751 (18) A, beta = 136.625 (10) degrees ; q = 0.6354 (30)b*. The occupations of the K and Yb atomic positions are described by crenel functions. The structure model reveals a balanced interaction between the atoms of the first and second coordination spheres. It is shown that the third coordination sphere should not be neglected in studies of modulated structures. The ordering of the K and Yb atoms appears to be the driving force for the modulation of all the other atoms.

The alpha2-polymorph of salicylideneaniline.

N-Salicylideneaniline (SA), C13H11NO, belongs to the large family of aromatic Schiff bases. It is of particular importance owing to its reversible photoreactivity. SA forms two photochromic polymorphs, both with two non-coplanar benzene rings. In addition, we have recently discovered a planar polymorph, named the beta-polymorph, which will be discussed in a subsequent paper. We report here the structure of the alpha2-polymorph in the orthorhombic crystal system. This compound exhibits a strong intramolecular O-H...N hydrogen bond and the dihedral angle between the two rings varies with temperature.

Molecular dynamics investigations of modulated phases in organic materials.

In order to apply the molecular dynamics (MD) method to simulate modulated phases in organic materials, a compensating external pressure tensor is proposed to compensate for the deficiencies of the force field applied in the simulation. MD can well reproduce modulated phases that have been measured by diffraction. Mechanisms of incommensurate modulation are revealed from the simulations. Details of the structures relating to the origins and mechanisms giving rise to the formation of modulated phases are presented.

Quininium (R)-mandelate, a structure with large Z' described as an incommensurately modulated structure in (3+1)-dimensional superspace.

Quininium (R)-mandelate, C(20)H(25)N(2)O(2)(+).C(8)H(7)O(3)(-), is an organic salt with an incommensurately modulated structure. The superspace approach is required for a precise description of the structure at room temperature. In addition to the main reflections, the diffraction pattern also exhibits higher-order satellite reflections. The large number of first- and second-order satellite reflections with relatively strong intensities indicates a significant modulation. No average structure solution could be obtained on the basis of the main reflections only and hence the structure solution was performed in two suitable superstructure approximations, including satellite intensities. Positional modulation functions and modulation functions for the anisotropic atomic displacement parameters were introduced. The modulation originates from a competition between intramolecular and intermolecular forces, which is reflected, in the superspace formalism, in the modification of hydrogen bonds along the internal space variable t.