D. E. Cox

Rietveld refinement guidelines

A set of general guidelines for structure refinement using the Rietveld (whole-profile) method has been formulated by the International Union of Crystallography Commission on Powder Diffraction. The practical rather than the theoretical aspects of each step in a typical Rietveld refinement are discussed with a view to guiding newcomers in the field. The focus is on X-ray powder diffraction data collected on a laboratory instrument, but features specific to data from neutron (both constant-wavelength and time-of-flight) and synchrotron radiation sources are also addressed. The topics covered include (i) data collection, (ii) background contribution, (iii) peak-shape function, (iv) refinement of profile parameters, (v) Fourier analysis with powder diffraction data, (vi) refinement of structural parameters, (vii) use of geometric restraints, (viii) calculation of e.s.d.s, (ix) interpretation of R values and (x) some common problems and possible solutions.

A Correction for Powder Diffraction Peak Asymmetry due to Axial Divergence

Analysis of a crystal structure using the Rietveld profile technique requires a suitable description of the shape of the peaks. In general, modern refinement codes include accurate formulations for most effects; however, the functions used for peak asymmetry are semi-empirical and take very little account of diffraction optics. The deficiencies in these methods are most obvious for high-resolution instruments. This study describes the implementation of powder diffraction peak profile formulations devised by van Laar & Yelon [J. Appl. Cryst. (1984), 17, 47-54]. This formalism, which describes the asymmetry due to axial divergence in terms of finite sample and detector sizes, does not require any free parameters and contains intrinsic corrections for the angular dependence of the peak shape. The method results in an accurate description of the observed profiles for a variety of geometries, including conventional X-ray diffractometers, synchrotron instruments with or without crystal analyzers and neutron diffractometers. other more recent applications, such as auto-indexing and accurate lattice-parameter determination in diamondanvil cells and other constrained environments, it is essential that the effects of axial divergence are properly described in the peak-shape function. In a diffraction experiment with a nondivergent point source and a randomly oriented powdered sample, the radiation scattered by a given diffraction line will lie on the surface of a cone with semi-angle 20. The entrance slit of a detector lies on the surface of a right cylinder with axis parallel to the 20 axis of the diffractometer. The intersection of the diffraction cone with the detector cylinder is an ellipse. As 120-901 increases, the ellipticity decreases, as can be seen in a Debye-Scherrer film. The center of the ellipse is at zero for 20 90 ° . As shown in Fig. 1, this curvature leads to a peak asymmetry because intensity from the ends of the intercepted piece of the diffraction cone will intersect the detector slit on the side of the peak closer to the center of the

Crystal structure of Ba2Bi3+Bi5+O6

Abstract The structure of BaBiO3 at 25°C has been solved and refined with the use of neutron diffraction powder data. The valence situation is found to be BaBi3+ 1 2 Bi5+ 1 2 O3 rather than BaBi4+O3. The Bi3+ and Bi5+ cations take on an ordered arrangement and the resultant structure has monoclinic symmetry 12/m ( a = 6.181 A , b = 6.136 A , c = 8.670 A , β = 90.17°). This is the first example of an ordered perovskite in which the ordered cations are the same element, and also represents the first case of Bi3+ octahedrally coordinated to oxygen. The semiconducting properties of BaBiO3 are readily understood on the basis of the ordered structure. The structure becomes rhombohedral at about 130°C, and cubic in the region of 450°C.

Crystal growth and structure of BiVO4

Abstract Large crystals of BiVO 4 were grown. Extensive twinning can be related to the ferroelastic transition at 528K. The structure of BiVO 4 was refined at 4.5, 295 and 566K from powder neutron diffraction data and at 295K from powder X-ray diffraction data by the Rietveld profile technique. Space groups of I 4 1 a above and I 2 b below 528K were confirmed. It is suggested that the transition in BiVO 4 is driven by the lone-pair cation Bi 3+ . The Bi-O polyhedron is regular above the transition but becomes significantly distorted below the transition. This lone-pair distortion increases on cooling from 295 to 4.5K. The V-O tetrahedron remains regular at all temperatures.

Orientational disorder in solvent-free solid c70.

The high-temperature structure of solvent-free C70 has been determined with high-resolution x-ray powder difraction and electron microscopy. Samples crystallized from solution form hexagonal close-packed crystals that retain an appreciable amount of residual toluene, even after prolonged heating. Samples prepared by sublimation, which contain no detectable solvent, are primarily face-centered cubic with some admixture of a hexagonal phase. The relative volume of the hexagonal phase can be further reduced by annealing. The structures of both phases are described by a model of complete orientational disorder. The cubic phase contains an appreciable density of stacking faults along the [111] direction.

Neutron irradiation of superconducting compounds

Abstract The effects of neutron irradiation on the superconducting and normal state properties of alloys and compounds are presented. Particular emphasis is placed on the A-15 compounds where the effects of neutron irradiation on T c , H c 2 , long range order parameter and lattice parameter are described. Large depressions (up to 80%) in T c are observed for all the A-15 compounds studied with the exception of Mo 3 Os where much smaller decreases in T c are seen. Along with the decrease in T c and increase in lattice parameter, the degree of long range order, as measured by X-ray and neutron diffraction, decreases. Also presented are the results of isothermal and isochronal anneals up to 900°C. The unirradiated value of T c can be restored by annealing, and for those systems where measurements have been made, recovery of the lattice parameter and order parameter also takes place. The effects observed in irradiated material, together with those observed in unirradiated but compositionally variant A-15 compounds, lead us to conclude that the principal defects associated with the depression of T c in both cases are anti-site defects. Other models are also discussed.

Thermal expansion and structural distortion of perovskite — data for NaMgF3 perovskite. Part I

Abstract The crystal structure of NaMgF3 perovskite (Neighborite) has been studied at high temperature by X-ray powder diffraction. Data were collected using a position sensitive detector with a monochromatic synchroton radiation source. Changes in unit cell and atomic positions of the perovskite structure were defined using the Rietveld refinement technique. The linear and volumetric thermal expansion coefficients are observed to be αa = 4.04 × 10−5 K−1, αb = 1.53 × 10−5 K−1, αc = 3.06 × 10−5 K−1, αv = 8.80 × 10−5 K−1 for the orthorhombic Pbnm phase, and αa0 = 3.16 × 10−5 K−1, αv0 = 9.49 × 10−5 K−1 for the cubic Pm3m phase of NaMgF3 perovskite, respectively. The temperature-induced linear and volumetric changes of the centrosymmetrically distorted ABX3 perovskite structure can be empirically expressed as a combination of the change of the (BX) bond length and the change of tilting of the BX6 octahedral framework. The considerable anisotropy of linear thermal expansion, αa > αc > αb, for the orthorhombic Pbnm phase reflects the progressive decrease of structural distortion and the development of the phase transition of the NaMgF3 perovskite. The tilting angle of the MgF6 octahedral framework is observed to decrease rapidly toward zero as the temperature approaches Tc = 765°C in the manner expected for a ferroelastic phase transition. More interestingly, the apparent (MgF) bond lengths of the MgF6 octahedra shrink dramatically throughout a temperature interval of about 100°C before the phase transition. The volumetric thermal expansion increases drastically in a critical manner as the temperature approaches Tc.

Structures of C60 intercalation compounds

Abstract In the past two years, many intercalation compounds of C 60 , have been prepared and characterized by X-ray powder diffraction techniques. The M-C 60 systems with M = Li, Na, K, Rb, Cs and Ca have been studied, and the structures of several compounds have been determined in considerable detail, including the face-centered cubic superconductors K 3 C 60 and Rb 3 C 60 , body-centered tetragonal K 4 C 60 , body-centered cubic M 6 C 60 with M = K, Rb and Cs, simple hexagonal C 60 I 4 , and the monoclinic molecular ferromagnet tetrakis-dimethylamino-ethylene (TDAE) C 2 N 4 (CH 3 ) 8 C 60 . The Rietveld profile technique for structure refinement coupled with high resolution synchrotron X-ray powder data has played an important role in providing accurate structural data for many of these compounds. In this article, a detailed review of the literature dealing with structural studies on the intercalated compounds of C 60 will be given.

Synchrotron X-ray Diffraction Measurements of Single-Crystal Hydrogen to 26.5 Gigapascals.

The crystal structure and equation of state of solid hydrogen have been determined directly to 26.5 gigapascals at room temperature by new synchrotron x-ray diffraction techniques. Solid hydrogen remains in the hexagonal close-packed structure under these pressure-temperature conditions and exhibits increasing structural anisotropy with pressure. The pressure-volume curve determined from the x-ray data represents the most accurate experimental measurement of the equation of state to date in this pressure range. The results remove the discrepancy between earlier indirect determinations and provide a new experimental constraint on the molecular-to-atomic transition predicted at higher pressures.

Critical phenomena and phase transition of perovskite — data for NaMgF3 perovskite. Part II

Abstract The crystal structure of NaMgF 3 perovskite is observed to transform directly from orthorhombic ( Pbnm ) to cubic ( Pm 3 m ) at a temperature of T c = 765°C. Superlattice diffractions associated with in-phase and anti-phase octahedral tilts vanish simultaneously at the transition temperature. The intensities of the superlattice diffractions, the atomic displacements, and the octahedral tilts follow a Landau type of critical behavior as the temperature approaches T c . The structural phase transition in NaMgF 3 perovskite can be modelled as a tricritical ferroelastic phase transition. The octahedral tilts θ and φ represent the primary order parameters of the phase transition, and the coupling between these two order parameters are observed to be bi-quadratic. The spontaneous strain for the m 3 m F mmm ferroelastic species is derived in terms of lattice parameters and its relation to the ferroelastic species m 3 m F4/ mmm and 4/ mmm F mmm are discussed. It is demonstrated experimentally that the coupling between the spontaneous strain and the octahedral tilts of perovskites is in a linear-quadratic form. Excess physical properties (thermal expansion and heat capacity, etc.) are observed to have a λ-anomaly during the structural phase transition in NaMgF 3 perovskite, and are considered to be directly associated with the excess Gibbs free energy.