Occupational modulation in the (3+1)-dimensional incommensurate structure of (2S,3S)-2-amino-3-hydroxy-3-methyl-4-phenoxybutanoic acid dihydrate.

The incommensurately modulated structure of (2S,3S)-2-amino-3-hydroxy-3-methyl-4-phenoxybutanoic acid dihydrate (C11H15NO4·2H2O or I·2H2O) is described in the (3+1)-dimensional superspace group P212121(0ß0)000 (ß = 0.357). The loss of the three-dimensional periodicity is ascribed to the occupational modulation of one positionally disordered solvent water molecule, where the two positions are related by a small translation [ca 0.666 (9) Å] and ∼168 (5)° rotation about one of its O-H bonds, with an average 0.624 (3):0.376 (3) occupancy ratio. The occupational modulation of this molecule arises due to the competition between the different hydrogen-bonding motifs associated with each position. The structure can be very well refined in the average approximation (all satellite reflections disregarded) in the space group P212121, with the water molecule refined as disordered over two positions in a 0.625 (16):0.375 (16) ratio. The refinement in the commensurate threefold supercell approximation in the space group P1121 is also of high quality, with the six corresponding water molecules exhibiting three different occupancy ratios averaging 0.635:0.365.

A modulated structure derived from the XA-type Mn2RuSn Heusler compound.

A modulated structure derived from the inverse Heusler phase (the XA-type and the disordered variant L21B-type) has been observed in rapidly quenched Mn2RuSn ribbons. The powder X-ray diffraction pattern of the quenched ribbons can be indexed as an L21B-type structure. Electron diffraction patterns of the new structure mostly resemble those of the XA-type (and the disordered variant L21B-type) structure and additional reflections with denser spacing indicate a long periodicity. Orthogonal domains of the modulated structure were revealed by a selected-area electron diffraction pattern and the corresponding dark-field transmission electron microscopy images. The structure was further studied by the crystallographic analysis of high-resolution transmission electron microscopy images. A model for the modulated structure has been proposed to interpret the experimental results.

Decoding the Double/Multiple Hysteresis Loops in Antiferroelectric Materials.

Antiferroelectric materials has become one of the most promising candidates for pulsed power capacitors. The polarization versus electric-field hysteresis loop is the key electrical property for evaluating their energy-storage performance. Here, we applied in situ biasing transmission electron microscopy to decode two representative energy-storage behaviors-namely, multiple and double hysteresis loops-in (Pb,La)(Zr,Sn,Ti)O3 system. Simultaneous structural examination and domain/defects observation establish a direct relationship between phase transitions and hysteresis loops. Sustaining a smaller period of modulated structure to a certain applied electric field and then undergoing additional transitions among varying antiferroelectric phases with increasing modulation periods before the final antiferroelectric-ferroelectric transition leads to the favorable multiple-loop configuration that realizes a high energy-storage performance. Such phenomenon is described by a model in terms of defect-driven phase transition. The distinctive mechanisms of multiple phase transition will inspire future composition-design for high switch-fielding antiferroelectric materials.

Incommensurately modulated crystal structure of α' (O'3)-type sodium cobalt oxide NaxCoO2 (x ∼ 0.78).

A single-phase sample of α' (O'3)-type layered sodium cobalt oxide NaxCoO2 (x ∼ 0.78) was prepared and its incommensurately modulated crystal structure was analyzed using the (3+1)-dimensional superspace approach to the powder neutron diffraction data. The crystal structure of the cobaltate is accurately described based on the superspace group C2/m(α0γ)00, wherein the positions of Na atoms are most significantly modulated in the monoclinic a direction to form an ordered arrangement. Such a displacive modulation causes a quasi-periodic shift of Na atoms from the centers of the NaO6 polyhedra between undulated CoO2 sheets, changing the form of the NaO6 polyhedron from an octahedral coordination (O) to a trigonal prismatic (P) one, via an intermediate capped trigonal prismatic NaO7 coordination (C). At the positions where the Na atoms are most significantly shifted, the neighboring Na atoms are located at almost touching distances. However, the occupation factor of Na atoms becomes zero at such positions, yielding Na-deficient sites VNa, sandwiched either between C and P, or C and C-type polyhedra.

The first bismuth borate oxyiodide, Bi4BO7I: commensurate or incommensurate?

The first bismuth borate oxyiodide, Bi4BO7I, has been prepared by solid-state reaction in evacuated silica ampoules. Its crystal structure [space group Immm(00γ)000] comprises litharge-related layers of edge-sharing OBi4 tetrahedra; the interlayer space is filled by I- and [BO3]3- anions. The wavevector, q = 0.242 (3)c*, is very close to the rational value of c*/4, yet refinement based on commensurate modulation faces serious problems indicating the incommensurate nature of the modulation. The I-/[BO3]3- anions are ordered in a complex sequence along [001], i.e. -<-BO3-BO3-I-I->n = 28-I-I-I-<-BO3-BO3-I-I->n = 28-BO3-BO3-BO3-, leading to a structural modulation. The principal feature of the latter is the presence of -I-I-I- and -BO3-BO3-BO3- sequences that cannot be accounted for in the a × b × 4c supercell. The thermal expansion of Bi4BO7I is weakly anisotropic (αa = 8, αb = 15 and αc = 17 × 10-6 K-1 at 500 K) which is caused by preferential orientation of the borate groups.

The incommensurately modulated structures of low-temperature labradorite feldspars: a single-crystal X-ray and neutron diffraction study.

Labradorite feldspars of the plagioclase solid solution series have been known for their complicated subsolidus phase relations and enigmatic incommensurately modulated structures. Characterized by the irrationally indexed e-reflections in the diffraction pattern, e-labradorite shows the largest variation in the incommensurate ordering states among the e-plagioclase structures. The strongly ordered low-temperature e-labradorite is one of the last missing pieces of the e-plagioclase puzzle. Nine plutonic and metamorphic labradorite feldspar samples from Canada, Ukraine, Minnesota (USA), Tanzania and Greenland with compositions ranging from An52.5 to An68 were studied with single-crystal X-ray diffraction. Two crystals from Labrador, Canada, and Duluth, MN, USA, with wide enough twin lamellae were analyzed with single-crystal neutron diffraction. The incommensurately modulated structures of e-plagioclase are refined for the first time with neutron diffraction data, which confirmed that the T-O distance modulation in the low-temperature e-plagioclase results from the Al-Si ordering in the framework. Detailed configurations of the M site are also observed in the structures refined from neutron diffraction data, which were not possible to see with X-ray diffraction data. The relation between the q-vectors and the mole% An composition is revealed for the entire compositional range of e-plagioclase, from An25 to An75. The previously proposed two-trend relation depending on the cooling rate and phase transition path is confirmed. A new classification of e-plagioclase (eα, eß and eγ) is proposed based on the q-vector of the structure, which makes it an independent character from the presence/absence of density modulation. New parameters are proposed to quantify the ordering states of these complicated aperiodic structures of e-plagioclases, such as the difference between 〈T1o-O〉 and 〈T1m-O〉 at phase t = 0.2 or the normalized intensity of the (071\bar 1) reflection.

Compositely modulated structures of phosphor materials SrxLi2+xAl2-xO4:Eu2.

Composite crystals SrxLi2+xAl2-xO4:Eu2+ were synthesized and their structures were determined using single-crystal X-ray diffraction. The commensurate structure with a modulation wavevector q = 5c*/6 was analyzed in a conventional manner in 3D space, while a structure model in (3+1)-dimensional superspace was used for the other two crystals with modulation wavevectors slightly differing from 5c*/6. The superstructure of the commensurate phase was described using the space group P4/n and a common superspace group I4/m(00γ)00 was used for the (3+1)D structures of all three crystals. The whole structure of each crystal consists of two substructures. Basis vectors a and b are common, but c is different for the two substructures. The first substructure is a host framework constructed by (Li/Al)O4 tetrahedra sharing edges. A linear connection of cavities is seen to be channel-like, in which Sr ions locate as guest cations forming the second substructure. The crystal of q = 5c*/6 contains five Sr ions per six cavities in a channel. Sr ions are distributed at seven sites, some of which are partially occupied. Statistical disorder of local structure models for the location of Sr ions in the channel was assumed to explain the results. Such a partially disordered character was also seen in the incommensurate phases and properly embodied by a (3+1)D model containing an atomic domain of the Sr ion with occupational modulation. Plots of the occupation factor, interatomic distances and the bond valence sum at each metal site as functions of t (= x4 - q·r) are roughly identical in the three crystals, which are considered as members of the same series of composite crystals.

LaTe1.82(1): modulated crystal structure and chemical bonding of a chalcogen-deficient rare earth metal polytelluride.

Crystals of the rare earth metal polytelluride LaTe1.82(1), namely, lanthanum telluride (1/1.8), have been grown by molten alkali halide flux reactions and vapour-assisted crystallization with iodine. The two-dimensionally incommensurately modulated crystal structure has been investigated by X-ray diffraction experiments. In contrast to the tetragonal average structure with unit-cell dimensions of a = 4.4996 (5) and c = 9.179 (1) Šat 296 (1) K, which was solved and refined in the space group P4/nmm (No. 129), the satellite reflections are not compatible with a tetragonal symmetry but enforce a symmetry reduction. Possible space groups have been derived by group-subgroup relationships and by consideration of previous reports on similar rare earth metal polychalcogenide structures. Two structural models in the orthorhombic superspace group, i.e. Pmmn(α,ß,1/2)000(-α,ß,1/2)000 (No. 59.2.51.39) and Pm21n(α,ß,1/2)000(-α,ß,1/2)000 (No. 31.2.51.35), with modulation wave vectors q1 = αa* + ßb* + 1/2c* and q2 = -αa* + ßb* + 1/2c* [α = 0.272 (1) and ß = 0.314 (1)], have been established and evaluated against each other. The modulation describes the distribution of defects in the planar [Te] layer, coupled to a displacive modulation due to the formation of different Te anions. The bonding situation in the planar [Te] layer and the different Te anion species have been investigated by density functional theory (DFT) methods and an electron localizability indicator (ELI-D)-based bonding analysis on three different approximants. The temperature-dependent electrical resistance revealed a semiconducting behaviour with an estimated band gap of 0.17 eV.

Sr2Pt8-x As: a layered incommensurately modulated metal with saturated resistivity.

The high-pressure synthesis and incommensurately modulated structure are reported for the new compound Sr2Pt8-x As, with x = 0.715 (5). The structure consists of Sr2Pt3As layers alternating with Pt-only corrugated grids. Ab initio calculations predict a metallic character with a dominant role of the Pt d electrons. The electrical resistivity (ρ) and Seebeck coefficient confirm the metallic character, but surprisingly, ρ showed a near-flat temperature dependence. This observation fits the description of the Mooij correlation for electrical resistivity in disordered metals, originally developed for statistically distributed point defects. The discussed material has a long-range crystallographic order, but the high concentration of Pt vacancies, incommensurately ordered, strongly influences the electronic conduction properties. This result extends the range of validity of the Mooij correlation to long-range ordered incommensurately modulated vacancies. Motivated by the layered structure, the resistivity anisotropy was measured in a focused-ion-beam micro-fabricated well oriented single crystal. A low resistivity anisotropy indicates that the layers are electrically coupled and conduction channels along different directions are intermixed.

Phase transitions in ferroelectric 4-aminopyridinium tetrachloroantimonate(III) - revisited.

New X-ray diffraction studies on the crystal structure of ferroelectric [4-NH2C5H4NH][SbCl4] indicate that in the broad temperature range from 240 to 304 K covering the three intermediate phases, the crystal structure is modulated. Phase II is incommensurately modulated with modulation vector q = ßb*, ß varying from 0.60 to 0.66 and monoclinic C2/c(0ß0)s0 superspace group. Ferroelectric phase III is commensurate with q = 2\over 3b* and Cc(0ß0)0 symmetry. Polar phase IV is incommensurately modulated with ß varying from 0.66 to 0.70 and Cc(0ß0)0 superspace group. In all phases only first-order satellites are observed along the b* direction. Two types of periodic deformation are present in the structure of modulated phases. The 4-aminopyridinium cations are subjected to occupation modulation whereas [SbCl4]-n chains are displacively modulated. The paraelectric-ferroelectric phase transition is an example of the incommensurate-commensurate transition of the lock-in type. A new mechanism for this transformation is proposed.

Incommensurately modulated structure of morpholinium tetrafluoroborate and configurational versus chemical entropies at the incommensurate and lock-in phase transitions.

Morpholinium tetrafluoroborate, [C4H10NO]+[BF4]-, belongs to a class of ferroelectric compounds ABX4. However, [C4H10NO]+[BF4]- does not develop ferroelectric properties because the incommensurate phase below Tc,I = 153 K is centrosymmetric with superspace group Pnam(σ100)00s and σ1 = 0.42193 (12) at T = 130 K; the threefold superstructure below Tc,II = 117-118 K possesses the acentric but non-ferroelectric space group P212121. At ambient conditions, [C4H10NO]+[BF4]- comprises orientationally disordered [BF4]- anions accommodated in cavities between four morpholinium cations. A structure model for the incommensurately modulated phase, which involves modulated orientational ordering of [BF4]- together with modulated distortions and displacements of the morpholinium ions is reported. A mechanism is proposed for the phase transitions, whereby at low temperatures morpholinium cations are shaped around the tetrafluoroborate anion in order to optimize the interactions with one orientation of this anion and, thus, forcing [BF4]- into this orientation. This mechanism is essentially different from a pure order-disorder phase transition. It is supported by consideration of the transition entropy. The difference in configurational entropy between the disordered and incommensurate phases has been computed from the structure models. It is shown to be much smaller than the experimental transition entropy reported by Owczarek et al. [Chem. Phys. (2011), 381, 11-20]. These features show that the order-disorder contribution is only a minor contribution to the transition entropy and that other factors, such as conformational changes, play a larger role in the phase transitions.

A commensurately modulated structure of parabutlerite, FeIIISO4(OH)·2H2O.

Parabutlerite, orthorhombic FeIIISO4(OH)·2H2O, has been reinvestigated using single-crystal X-ray diffraction. The structure of parabutlerite is commensurately modulated, with a = 20.0789 (8), b = 7.4024 (7), c = 7.2294 (15) Šand q = 0.4b*. The superstructure has been determined, using a superspace approach, as having the superspace group Pnma(0ß0)s0s and t0 = 1/20, and refined to R = 0.0295 for 2392 main reflections with I > 3σ(I). The structure consists of infinite chains of Fe octahedra that are linked via vertices (OH groups); these chains are encased from both sides by SO4 tetrahedra. The displacive modulation of atoms in parabutlerite is connected with a tilt of the chains around the b axis towards the adjacent chains due to the accommodation of an energetically more favorable hydrogen-bond geometry.

Low-temperature behaviour of K2Sc[Si2O6]F: determination of the lock-in phase and its relationships with fresnoite- and melilite-type compounds.

K2Sc[Si2O6]F exhibits, at room temperature, a (3 + 2)-dimensional incommensurately modulated structure [a = 8.9878 (1), c = 8.2694 (2) Å, V = 668.01 (2) Å3; superspace group P42/mnm(α,α,0)000s(-α,α,0)0000] with modulation wavevectors q1 = 0.2982 (4)(a* + b*) and q2 = 0.2982 (4)(-a* + b*). Its low-temperature behaviour has been studied by single-crystal X-ray diffraction. Down to 45 K, the irrational component α of the modulation wavevectors is quite constant varying from 0.2982 (4) (RT), through 0.2955 (8) (120 K), 0.297 (1) (90 K), 0.298 (1) (75 K), to 0.299 (1) (45 K). At 25 K it approaches the commensurate value of one-third [i.e. 0.332 (3)]: thus indicating that the incommensurate-commensurate phase transition takes place between 45 K and 25 K. The commensurate lock-in phase of K2Sc[Si2O6]F has been solved and refined with a 3 × 3 × 1 supercell compared with the tetragonal incommensurately modulated structure stable at room temperature. This corresponds to a 3 × 1 × 3 supercell in the pseudo-orthorhombic monoclinic setting of the low-temperature structure, space group P2/m, with lattice parameters a = 26.786 (3), b = 8.245 (2) c = 26.824 (3) Å, ß = 90.00 (1)°. The structure is a mixed tetrahedral-octahedral framework composed of chains of [ScO4F2] octahedra that are interconnected by [Si4O12] rings with K atoms in fourfold to ninefold coordination. Distorted [ScO4F2] octahedra are connected to distorted Si tetrahedra to form octagonal arrangements closely resembling those observed in the incommensurate structure of fresnoite- and melilite-type compounds.

Crystal structures of η''-Cu3+xSi and η'''-Cu3+xSi.

The binary phase diagram of Cu-Si is unexpectedly complex in the vicinity of Cu3+xSi. The low-temperature region contains three closely related incommensurately modulated phases denoted, in order of increasing temperature of stability, η''', η'' and η'. The structure analysis of η' has been reported previously [Palatinus et al. (2011). Inorg. Chem. 50, 3743]. Here the structure model for the phases η'' and η''' is reported. The structures could be solved in superspace, but no superspace structure model could be constructed due to the complexity of the modulation functions. Therefore, the structures were described in a supercell approximation, which involved a 4 × 4 × 3 supercell for the η'' phase and a 14 × 14 × 3 supercell for the η''' phase. Both structures are very similar and differ only by a subtle symmetry lowering from η'' to η'''. A comparison of the structure models of η'' and η''' with the reported structure of η' suggests that the reported structure model of η' contains an incorrect assignment of atomic types.

Communication between cation environments in aluminosilicate frameworks: incommensurately modulated crystal structure of an e-plagioclase.

Despite being one of the most common minerals in the earth's crust the crystal structure of intermediate e-plagioclase remains only partially understood, due in a large part to its complex diffraction patterns including satellite reflections. In this article we present a detailed analysis of the structure of e-plagioclase (An44) using single-crystal X-ray diffraction measured at ambient and low temperature (T = 100 K), in which the full modulated structure is successfully refined. As in earlier studies, the diffraction pattern exhibits strong main a-reflections and weak e-satellite reflections. The average structure could be solved in terms of an albite-like basic cell with the triclinic centrosymmetric and non-centrosymmetric space groups P \bar 1 and P1 (treated in its C \bar 1 and C1 setting, respectively, to follow conventions in the literature), while the incommensurately modulated structure was modeled in (3 + 1)D superspace, employing both the centro- and non-centrosymmetric superspace groups X \bar 1(αßγ)0 and X1(αßγ)0, where X refers to a special (3 + 1)D lattice centering with centering vectors (0 0 ½ ½), (½ ½ 0 ½), and (½ ½ ½ 0). Individual positional and occupational modulations for Ca/Na were refined with deeper insights being revealed in the non-centrosymmetric structure model. Through the structural details emerging from this model, the origin of the modulation can be traced to the communication between Ca/Na site positions through their bridging aluminosilicate (Si/Al)O4 tetrahedra.

Modulated crystal structure of InMo4O6.

The (3 + 1)-dimensional modulated crystal structure of the metal-rich cluster compound InMo4O6 was solved and refined from single-crystal data in the superspace group P4/mbm(00γ)00ss [q = 0, 0, 0.1536 (4); a = 9.6664 (9), c = 2.8645 (3) Å; R1(all) = 0.046, wR(all) = 0.076]. The crystal structure is closely related to the NaMo4O6 structure type. It is built from rods of Mo6 clusters condensed via trans edges. These form channels parallel to [001], in which In6 and In7 oligomers alternate. Weak diffuse planes parallel to (001)* interconnect the satellite reflections; they occur due to two-dimensional rod disorder of the In oligomer chains.

The Z' = 12 superstructure of Λ-cobalt(III) sepulchrate trinitrate governed by C-H...O hydrogen bonds.

Λ-Cobalt(III) sepulchrate trinitrate crystallizes in P6322 with Z = 2 (Z' = 1/6) at room temperature. Slabs perpendicular to the hexagonal axis comprise molecules Co(sepulchrate) alternating with nitrate groups A and B. Coordinated by six sepulchrate molecules, highly disordered nitrate groups C are accommodated between the slabs. Here we report the fully ordered, low-temperature crystal structure of Co(sep)(NO3)3. It is found to be a high-Z' structure with Z' = 12 of the 12-fold 6a_{h}\times\sqrt{3}b_{h}\times c_{h} superstructure with monoclinic symmetry P21 (c unique). Correlations between structural parameters are effectively removed by refinements within the superspace approach. Superstructure formation is governed by a densification of the packing in conjunction with ordering of nitrate group C, the latter assuming different orientations for each of the Z' = 12 independent copies in the superstructure. The Co(sep) moiety exhibits small structural variations over its 12 independent copies, while orientations of nitrate groups A and B vary less than the orientations of the nitrate group C do. Molecular packing in the superstructure is found to be determined by short C-H...H-C contacts, with H...H distances of 2.2-2.3 Å, and by short C-H...O contacts, with H...O distances down to 2.2 Å. These contacts presumably represent weak C-H...O hydrogen bonds, but in any case they prevent further densification of the structure and strengthening of weak N-H...O hydrogen bonds with observed H...O distances of 2.4-2.6 Å.

New insight on bismuth cuprates with incommensurate modulated structures.

The incommensurate modulated crystal structure of Bi2.27Sr1.73CuO6 + δ (2201) phase [a = 5.3874 (5), b = 5.3869 (4), c = 24.579 (3) Å; ß = 90.01 (1)°, q = 0.2105 (3)a(*) + 0.538 (4)c(*), Z = 4, the (3 + 1)-dimensional monoclinic A2/a(α0γ) group] has been refined with R = 0.041, wR = 0.052 from X-ray single-crystal data including up to third-order satellite reflections. The same structure has also been considered as incommensurate composite with a2 = 2.437, b2 = 5.387, c2 = 24.614, ß2 = 93.06, q2 = 0.4524a2(*)-0.243c2(*) and the (3 + 1)-dimensional A2/m(α0γ)0s group for the second component. Both approaches give quite similar results. The structure possesses oxygen disorder in the oxygen-rich region of the BiO layer. An extra O atom is determined in the bridging position shifted ∼ 0.6 Šfrom BiO towards the SrO layer. Its presence is the cause of the tremendous increase of the bismuth U(11) atomic displacement parameter in ∼ 20% of the unit cells (t = -0.05-0.15). Vacancies are determined in the oxygen site of the SrO layer, which may result in the oxygen content variation with annealing at different oxygen pressures. The total refined oxygen content 6.18 (1) corresponds to the results of chemical analysis.

Crystal pathologies in macromolecular crystallography.

Macromolecules, such as proteins or nucleic acids, form crystals with a large volume fraction of water, ~50% on average. Apart from typical physical defects and rather trivial poor quality problems, macromolecular crystals, as essentially any crystals, can also suffer from several kinds of pathologies, in which everything seems to be perfect, except that from the structural point of view the interpretation may be very difficult, sometimes even impossible. A frequent nuisance is pseudosymmetry, or non-crystallographic symmetry (NCS), which is particularly nasty when it has translational character. Lattice-translocation defects, also called order-disorder twinning (OD-twinning), occur when molecules are packed regularly in layers but the layers are stacked (without rotation) in two (or more) discrete modes, with a unique translocation vector. Crystal twinning arises when twin domains have different orientations, incompatible with the symmetry of the crystal structure. There are also crystals in which the periodic (lattice) order is broken or absent altogether. When the strict short-range translational order from one unit cell to the next is lost but the long-range order is restored by a periodic modulation, we have a modulated crystal structure. In quasicrystals (not observed for macromolecules yet), the periodic order (in 3D space) is lost completely and the diffraction pattern (which is still discrete) cannot be even indexed using three hkl indices. In addition, there are other physical defects and phenomena (such as high mosaicity, diffraction anisotropy, diffuse scattering, etc.) which make diffraction data processing and structure solution difficult or even impossible.

Incommensurate modulation and thermal expansion of Sr3B(2†+†x)Si(1†-†x)O(8†-†x/2) solid solutions.

Crystal structures of Sr3B(2 + x)Si(1 - x)O(8 - x/2) solid solutions with nominal compositions x = 0.28, 0.53, 0.78 in the Sr3B2SiO8-Sr2B2O5 section of the SrO-B2O3-SiO2 system are refined using single-crystal X-ray diffraction data. Incommensurate structure modulations are mainly associated with various orientations of corner-sharing (B,Si)-polyhedra. Preference is given to the (3 + 2)-dimensional symmetry group Pnma(0ßγ)000(0ßγ)000 for a single crystal compared with an alternate model of a twin formed by monoclinic components, each of them corresponding to the (3 + 1)-dimensional symmetry group P2(1)/n(0ßγ). Single-phase polycrystalline samples of solid solutions are investigated by high-temperature X-ray powder diffraction in air. Orientation preferences of the BO3 units lead to a strong anisotropy of thermal expansion. Negative expansion is observed along the a axis over the temperature range 303-753 K. Anisotropy decreases both on heating and decreasing of the boron content.