This article reviews the current status of lattice-dynamical calculations in crystals, using density-functional perturbation theory, with emphasis on the plane-wave pseudopotential method. Several specialized topics are treated, including the implementation for metals, the calculation of the response to macroscopic electric fields and their relevance to long-wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodology is demonstrated with a number of applications existing in the literature.

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Phonons and related crystal properties from density-functional perturbation theory

The thermodynamic properties of 154 mineral end-members, 13 silicate liquid end-members and 22 aqueous fluid species are presented in a revised and updated data set. The use of a temperature-dependent thermal expansion and bulk modulus, and the use of high-pressure equations of state for solids and fluids, allows calculation of mineral–fluid equilibria to 100 kbar pressure or higher. A pressure-dependent Landau model for order–disorder permits extension of disordering transitions to high pressures, and, in particular, allows the alpha–beta quartz transition to be handled more satisfactorily. Several melt end-members have been included to enable calculation of simple phase equilibria and as a first stage in developing melt mixing models in NCKFMASH. The simple aqueous species density model has been extended to enable speciation calculations and mineral solubility determination involving minerals and aqueous species at high temperatures and pressures. The data set has also been improved by incorporation of many new phase equilibrium constraints, calorimetric studies and new measurements of molar volume, thermal expansion and compressibility. This has led to a significant improvement in the level of agreement with the available experimental phase equilibria, and to greater flexibility in calculation of complex mineral equilibria. It is also shown that there is very good agreement between the data set and the most recent available calorimetric data.

An internally consistent thermodynamic data set for phases of petrological interest

The newly developed GSAS-II software is a general purpose package for data reduction, structure solution and structure refinement that can be used with both single-crystal and powder diffraction data from both neutron and X-ray sources, including laboratory and synchrotron sources, collected on both two- and one-dimensional detectors. It is intended that GSAS-II will eventually replace both the GSAS and the EXPGUI packages, as well as many other utilities. GSAS-II is open source and is written largely in object-oriented Python but offers speeds comparable to compiled code because of its reliance on the Python NumPy and SciPy packages for computation. It runs on all common computer platforms and offers highly integrated graphics, both for a user interface and for interpretation of parameters. The package can be applied to all stages of crystallographic analysis for constant-wavelength X-ray and neutron data. Plans for considerable additional development are discussed.

GSAS-II: the genesis of a modern open-source all purpose crystallography software package

Strong Closed-Shell Interactions in Inorganic Chemistry.

With very few exceptions the numerous polymorphs of SiO2 all consist of SiO4 tetrahedra linked through their corners, thus forming three-dimensional framework structures. The topology of the tetrahedral linkage and the efficiency of space filling are different for the polymorphs. For a given type of framework, for example, that of quartz or cristobalite, space filling can be improved by so-called displacive transformations from a more open high-temperature form (e.g., “high”, “h”, or β “quartz”) to a denser form stable at lower temperatures (“low” or “α quartz”). These transformations do not change the topology of the framework, i.e., chemical bonds in a crystal can be deformed, but are not broken and rearranged.

The Scientific Basis

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.

A correction for powder diffraction peak asymmetry due to axial divergence

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.

https://science.sciencemag.org/content/sci/163/3870/926.full.pdf

Estimating Proportions in Petrographic Mixing Equations by Least-Squares Approximation

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.

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

Comparative crystal chemistry

The unit-cell parameters of quartz, SiO2, have been determined by single-crystal diffraction at 22 pressures to a maximum pressure of 8.9 GPa (at room temperature) with an average precision of 1 part in 9000. Pressure was determined by the measurement of the unit-cell volume of CaF2 fluorite included in the diamond-anvil pressure cell. The variation of quartz unit-cell parameters with pressure is described by: a −4.91300 (11) = −0.0468 (2) P + 0.00256 (7) P2 − 0.000094 (6) P3, c − 5.40482 (17) = − 0.03851 (2) P + 0.00305 (7) P2 − 0.000121 (6) P3, where P is in GPa and the cell parameters are in angstroms. The volume–pressure data of quartz are described by a Birch–Murnaghan third-order equation of state with parameters V0 = 112.981 (2) a3, KT0 = 37.12 (9) GPa and K′ = 5.99 (4). Refinement of K′′ in a fourth-order equation of state yielded a value not significantly different from the value implied by the third-order equation. The use of oriented quartz single crystals is proposed as an improved internal pressure standard for high-pressure single-crystal diffraction experiments in diamond-anvil cells. A measurement precision of 1 part in 10 000 in the volume of quartz leads to a precision in pressure measurement of 0.009 GPa at 9 GPa.

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Larry W. Finger

Papers

A correction for powder diffraction peak asymmetry due to axial divergence

Estimating Proportions in Petrographic Mixing Equations by Least-Squares Approximation

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

Comparative crystal chemistry

The Use of Quartz as an Internal Pressure Standard in High-Pressure Crystallography