Larry W. Finger

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

Estimating Proportions in Petrographic Mixing Equations by Least-Squares Approximation

Petrogenetic hypotheses involving fractional crystallization, assimilation, or mixing of magmas may be expressed and tested as problems in leastsquares approximation. The calculation uses all of the data and yields a unique solution for each model, thus avoiding the ambiguity inherent in graphical or trial-and-error procedures. The compositional change in the 1960 lavas of Kilauea Volcano, Hawaii, is used to illustrate the method of calculation.

Crystal structure and isothermal compression of Fe2O3, Cr2O3, and V2O3 to 50 kbars

Crystal structures of several of the corundum‐type oxides have been determined at pressures to 50 kbars. All materials have linear compression within the pressure range and precision of the techniques used. Compression of Cr2O3 and Al2O3 is essentially isotropic (c/a remains constant), Fe2O3 has a slightly anisotropic compression, with c/a decreasing slightly with pressure, and V2O3 is very anisotropic, with the a axis nearly three times more compressible than c. Similar differences are observed in the structural parameters. Aluminum, iron, and chromium sesquioxides simply scale, whereas atomic positions in V2O3 approach an ideal HCP arrangement with increasing pressure. The differences in structural variation with pressure for these ’’isostructural’’ compounds emphasize the difficulty in using simple bonding parameters to predict details of crystal structures under nonambient conditions.

High-Pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): Comparisons with silicate spinels

High-pressure crystal structures and compressibilities have been determined by x-ray methods for MgAl2O4 spinel and its isomorph magnetite, Fe3O4. The measured bulk moduli, K, of spinel and magnetite (assuming K′=4) are 1.94±0.06 and 1.86±0.05 Mbar, respectively, in accord with previous ultrasonic determinations. The oxygen u parameter, the only variable atomic position coordinate in the spinel structure (Fd3m, Z=8), decreases with pressure in MgAl2O4, thus indicating that the magnesium tetrahedron is more compressible than the aluminum octahedron. In magnetite the u parameter is unchanged, and both tetrahedron and octahedron display the 1.9 Mbar bulk modulus characteristic of the entire crystal. This behavior contrasts with that of nickel silicate spinel (γ-Ni2SiO4), in which the u parameter increases with pressure because the silicon tetrahedron is relatively incompressible compared to the nickel octahedron.

High-pressure crystal chemistry and amorphization of α-quartz

Single-crystal X-ray diffraction experiments on α-quartz at pressures to 15 GPa reveal structural instabilities that result in its gradual transition to an amorphous state. With increasing pressure the average SiO distance (≈ 1.61 ± 0.01 A) and SiO4 tetrahedral volume (≈ 2.14 ± 0.02 A3 remain constant. Compression of α-quartz results from a dramatic decrease in SiOSi angle and corresponding decrease in inter-tetrahedral (i.e., OO) distances. The onset of amorphization coincides with bending of all SiOSi angles to less than 120° and severe distortion of SiO4 tetrahedra, as oxygens approach a close-packed configuration.

Bulk moduli and high-pressure crystal structures of rutile-type compounds

Abstract Unit-cell parameters and crystal structures of five rutile-type compounds, including TiO 2 ; SnO 2 , GeO 2 , RuO 2 ; and MnF 2 ;, have been determined at pressures to 50 kbar at 20°C. All five compounds compress anisotropically with the a axis approximately twice as compressible as c . The one variable positional parameter, x of oxygen or fluorine, changes little with pressure. The uniform behavior of these RX 2 compounds at high pressure contrasts with their highly variable structural changes at high-temperature. Rutile-type oxides are, therefore, unlike most oxides and silicates, in which structural variations at high pressure mirror those at high temperature.

Crystal structure and compression of ruby to 46 kbar

Crystal structures and lattice constants have been determined for ruby at hydrostatic pressures up to 46 kbar using a gasketed opposed‐anvil diamond cell on a four‐circle diffractometer. The measured compressibility is slightly anisotropic, having a value of 1.36±0.03×10−4 kbar−1 parallel to c and 1.22±0.03×10−4 kbar−1 perpendicular to c. If a Birch‐Murnaghan equation of state is used and K′0 is assumed to be 4, the isothermal bulk modulus is 2.57±0.06 Mbar. Refined atomic coordinates do not change with pressure; therefore, the structure compresses in a uniform manner. This study demonstrates that crystal structures may be determined at high pressure on single crystals with a precision approaching that of room‐pressure results.

Polyhedral tilting: A common type of pure displacive phase transition and its relationship to analcite at high pressure

Abstract Polyhedral tilt transformations are a subgroup of pure displacive solid-solid phase transitions that occur in many ionic compounds and have all the following characteristics: (1) the transitions occur in compounds with structures composed of corner-linked, rigid polyhedral elements; (2) transitions are between a higher-symmetry or less-distorted form (stable at higher temperature or lower pressure) and a lower-symmetry or more-distorted form (stable at lower temperature or higher pressure); (3) transitions are nonquenchable, and single crystals are preserved through the transition; (4) twinning is common in the low-symmetry form, with the twin law governed by a symmetry operator lost in the high-to-low transition; (5) the value of dP/dT is always positive and is similar to the ratio of large-site thermal expansivity to compressibility. Analcite, (NaAl)x Si2-x O6 · H2O with 0.9 > x < 1.0, is a zeolite mineral that undergoes several polyhedral tilt transitions. Under room conditions analcites are p...

SINGLE: a program to control single-crystal diffractometers

SINGLE, a program that runs under Windows operating systems, can be used to control a variety of four-circle Eulerian-cradle single-crystal diffractometers including Huber instruments equipped with Huber motor controllers, the Stoe Stadi-4 and the Siemens P4. The software can be configured, via variables set by the user, to operate any of these diffractometers with a variety of environmental devices including furnaces and diamond-anvil pressure cells. It is specifically designed to determine unit-cell parameters to a precision of 1 part in 30 000.

High‐pressure and high‐temperature crystal chemistry of beryllium oxide

The crystal structure of synthetic BeO, bromellite, has been determined at several pressures to 5.0 GPa and several temperatures to 1183 K. The single variable atomic‐positional parameter, z of oxygen, does not vary significantly with pressure, but does undergo a small increase with temperature. The hexagonal axial ratio c/a is constant with pressure, but decreases slightly from 1.624±0.001 to 1.623±0.001 between 300 and 1183 K. The observed crystal bulk modulus is 212±3 GPa, if K’ is assumed to be 4. The bulk modulus of the beryllium tetrahedron in BeO is 210 GPa, identical to that of the crystal, and close to the value observed for beryllium tetrahedra in other beryllium minerals. The average volume thermal expansivity of the BeO beryllium tetrahedron between 298 and 1183 K is 2.5±0.2×10−5 K−1, compared to 2.66±0.10×10−5 K−1 for the crystal.