Roderick J. Hill

Quantitative phase analysis from neutron powder diffraction data using the Rietveld method

The weight of a phase in a mixture is proportional to the product of the scale factor, as derived in a multi-component Rietveld analysis of the powder diffraction pattern, with the mass and volume of the unit cell. If all phases are identified and crystalline, the weight fraction W of phase p is given by W_{p} = S_{p}(ZMV)_{p}/\sum_{i}S_{i}(ZMV)_{i},where S, Z, M and V are, respectively, the Rietveld scale factor, the number of formula units per unit cell, the mass of the formula unit and the unit-cell volume. This is the basis of a method providing accurate phase analyses without the need for standards or for laborious experimental calibration procedures. The method is demonstrated by measurements on binary mixtures of rutile, corundum, silicon and quartz.

Systematics of the spinel structure type

Systematic trends in the geometry of 149 oxide and 80 sulfide binary and ternary spinels have been examined from the standpoint of ionic radius and electronegativity. The mean ionic radii of the octahedral and tetrahedral cations, taken together, account for 96.9 and 90.5% of the variation in the unit cell parameter, a, of the oxides and sulfides, respectively, with the octahedral cation exerting by far the dominant influence in sulfides. The mean electronegativity of the octahedral cation exerts an additional, but small, influence on the cell edge of the sulfides. The equation a=(8/3√d)dtet+(8/3)doct, where dtet and doct are the tetrahedral and octahedral bond lengths obained from the sum of the ionic radii, accounts for 96.7 and 83.2% of the variation in a in the oxides and sulfides, respectively, again testifying to the applicability of the hard-sphere ionic model in the case of the spinel structure. Comparison of observed and calculated u values for 94 spinels indicates that up to 40% of the experimentally measured anion coordinates may be significantly in error. In addition to these compounds, u values are given for 52 spinels for which no data have previously been determined. Diagrams are presented for the rapid interpretation of the internal consistency of published data and the prediction of the structural parameters of hypothetical or partially studied spinels.

Peak shape variation in fixed-wavelength neutron powder diffraction and its effect on structural parameters obtained by Rietveld analysis

Analyses of high-resolution neutron powder diffraction data, using both the Pearson VII and pseudo-Voigt peak shape functions, have revealed a range of peak shapes from essentially Gaussian to Lorentzian and beyond. Moreover, the refinements show that the Lorentzian character of the peaks in each pattern increases with increasing diffraction angle. Both kinds of shape change are associated with varying relative contributions to the peak profiles of the instrumental resolution, isotropic crystallite strain and crystallite size effects. Rietveld analysis of powder data with the standard Gaussian form when the peaks have significant Lorentzian character has little effect on atomic positional parameters, but it leads to an overestimation of the thermal vibration coefficients and higher least-squares residuals.

Crystal structure refinement and electron density distribution in diaspore

Diaspore from Dilln, Hungary, AlOOH, is orthorhombic with space group Pbnm, a=4.4007(6), b=9.4253(13), c=2.8452(3) Å, and Z=4. The crystal structure and electron distribution have been refined from 791 graphite-monochromatized MoKα data (maximum 2θ=130°) to R=0.035 (Rw=0.029). Difference maps show substantial electron density ascribed to covalent bonding in the hydroxyl group, O(2)-H, but no residual density is observed along the Al-O(1,2) bonds. An analysis of the charge distribution implies net charges of +1.47(26), −1.08(16), −0.59(13) and +0.20(5) for Al, O(1), O(2) and H respectively. Semi-empirical molecular orbital calculations of the Hückel type agree with the experimentally determined atomic charge distribution and also allow a rationalization of the observed bond length variations.

THE POLYMORPHS OF ZIRCONIA : PHASE ABUNDANCE AND CRYSTAL STRUCTURE BY RIETVELD ANALYSIS OF NEUTRON AND X-RAY DIFFRACTION DATA

Rietveld analysis of neutron and X-ray powder diffraction data has been used to obtain the crystallinity and relative abundances of cubic (22 at% Y), tetragonal (5.8 at% Y), and monoclinic zirconia in synthetic binary and ternary mixtures of the pure phases. The stabilized cubic and tetragonal forms are shown to be 94 (2) and 96 (2)% crystalline, respectively, but no amorphous material was detected in the chemically pure monoclinic phase. In both stabilized polymorphs, the yttrium atoms randomly substitute into the zirconium site, with a charge-compensating proportion of vacancies on the oxygen atom site. In this near worst-case situation of very similar cubic and tetragonal unit cell dimensions, the phase abundances determined by Rietveld analysis of neutron data are very accurate and superior to those determined by integrated-intensity methods of analysis. For X-ray data, the accuracy is diminished by the presence of extinction, and by uncertainty in the definition of the peak shape of the cubic phase due to the absence of intense high-angle reflections and the resultant dependence on strongly overlapping low-angle data.

Exploration of structure and bonding in stishovite with fourier and pseudoatom refinement methods using single crystal and powder X-ray diffraction data

The structure and bonding in stishovite, SiO2, is explored with Fourier summation and pseudoatom refinement of merged x-ray single crystal and powder diffraction data. Replacement of the 25 lowest-angle, highly extinction-affected, single crystal reflections with structure factors obtained from low-extinction powder diffraction data has resulted in a significant improvement in the analysis compared with earlier studies. The deformation electron density, total electrostatic potential and total and valence electron densities are mapped. Accumulations of electron density are observed in both SiO bonds, together with non-bonding features displayed about the oxygen on both sides of a plane formed by three bonds with Si. Deficits of electron density between O atoms across the shared-edges are rationalized in terms of the Pauli exclusion principle. There is no evidence for strong repulsion of Si atoms across the same ring. The total electrostatic potential has a continuous low value for the vacant channels in the structure along c with localized minima between O atoms on opposite sides of the channel. The sizes of Si and O are related to the electron density and to the electrostatic potential.

The crystal structures of lead dioxides from the positive plate of the lead/acid battery

Abstract Full profile neutron powder diffraction structure refinements have been performed on electrolytically- and chemically-prepared varieties of the dimorphs of lead dioxide, the primary constituents in the charged positive plate of the lead/acid battery. Electrolytic samples of α-PbO 2 (columbite related structure) contain significant amounts of noncrystalline material, have severe cation disorder, and display evidence of severe lattice strain and unit cell incoherency in the [010] direction. On the other hand, chemically-prepared varieties of α-PbO 2 are more highly crystalline, are essentially stoichiometric, and are well ordered. Electrolytic samples of β-PbO 2 (rutile structure) from both new and used battery plates also contain more noncrystalline material than chemically-prepared forms, but all varieties are near-stoichiometric, well ordered, and structurally indistinguishable insofar as coherent neutron diffraction is concerned. Given that positive plates cycled under normal conditions for long periods of time consist primarily of the beta form of lead dioxide, the onset of battery failure is therefore not manifest in the long range crystal structure of this dioxide. In view of the high proportion of noncrystalline material present in electrolytic samples of PbO 2 , property changes previously associated with loss of crystalline dioxide activity may relate instead to the “amorphous” component of the positive plate mass.

The thermal expansion of ScAlO3 — A silicate perovskite analogue

The crystal structure of ScAlO3 has been refined at temperatures up to 1100° C on the basis of x-ray powder diffraction data. The thermal expansion is adequately described by a Grüneisen-Debye model with the elastic Debye temperature and an effective Grüneisen parameter of 1.6. The volumetric thermal expansion of 3.0% between 10 and 1100° C, corresponding to a mean thermal expansion coefficient of 2.7 × 10−5 K−1, is entirely attributable to the expansion of the AlO6 octahedra. The interoctahedral angles, though not fixed by symmetry, do not vary significantly with temperature —indicating that the expansivities of the constituent AlO6 and distorted ScO8 polyhedra are well matched. Similar considerations of polyhedral expansivity suggest thermal expansion coefficients of ∼2 × 10−5K−1 for cubic CaSiO3 perovskite and a value between 2 × 10−5 K−1 and 4 × 10−5 K−1 for MgSiO3 perovskite. The lower value is consistent with the reconnaissance measurements for Mg0.9Fe0.1SiO3 (Knittle et al. 1986) below 350° C, with low-temperature measurements of single-crystal MgSiO3 (Ross and Hazen 1989), and with the results of some recent calculations. The markedly greater expansivity ∼4 × 10−5 K−1 measured at higher temperatures (350–570° C) by Knittle et al. is inconsistent with the simple Grüneisen-Debye quasiharmonic model and may reflect the marginal metastability of the orthorhombic perovskite phase. Under these circumstances, extrapolation of the measured expansivity is hazardous and may result in the under-estimation of lower mantle densities and the drawing of inappropriate inferences concerning the need for chemical stratification of the Earths mantle.

Structure of PbSb2O6 and its relationship to the crystal chemistry of PbO2 in antimonial lead-acid batteries

Abstract The crystal structure of PbSb2O6, space group P 3 1m, a = 5.3006(1), c = 5.3792(1) , A, V = 130.89(2), A3, Z = 1, Dx = 6.936 g cm−3, has been refined at 295 K by Rietveld analysis of step-scan neutron powder diffraction data (λ = 1.500 A, μ = 0.179 cm−1): Rwp = 0.0614 for 2898 step intensities, RB = 0.0141 for 150 reflections. The structure consists of gibbsitelike sheets of edge-sharing SbO6 octahedra oriented perpendicular to the c-axis. These sheets alternate with layers of isolated PbO6 octahedra positioned above and below the vacant sites in the adjacent (Sb2O6)2− layers. The SbO6 octahedra have symmetry 32 and a bond length of 1.9904(4) A; the PbO6 octahedra have symmetry 3 m and a bond length of 2.5633(6) A. The properties of this structure and its relationship to the polymorphs of PbO2 provide possible explanations for the well-known beneficial effect of antimony on the cycle life of lead-acid batteries.

The electron distribution in corundum. A study of the utility of merging single-crystal and powder diffraction data

Powder X-ray diffraction data for corundum were collected by a variety of methods and reduced to structure amplitudes by two profile-fitting techniques. The resulting averaged powder-data set was merged with three different single-crystal data sets to assess the improvements possible over least-squares modelling of extinction for accurate electron density analysis of minerals. With reference to the deformation electron density derived from multipole refinements, it is concluded that this strategy offers advantages over the post facto modelling of severe extinction effects commonly observed in such systems. The deformation electron density is found to be in quantitative agreement with the results of recent ab initio calculations on clusters and the bulk.