We present a review of the basic ideas and techniques of the spectral density functional theory which are currently used in electronic structure calculations of strongly{correlated materials where the one{electron description breaks down. We illustrate the method with several examples where interactions play a dominant role: systems near metal{insulator transition, systems near volume collapse transition, and systems with local moments.

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Electronic structure calculations with dynamical mean-field theory

Interpretation of X-ray Powder Diffraction Patterns . By H. Lipson and H. Steeple. Pp. viii + 335 + 3 plates. (Mac-millan: London; St Martins Press: New York, May 1970.) £4.

X-Ray Diffraction

Perspectives on Titanium Science and Technology

The thermodynamic properties of 254 end-members, including 210 mineral end-members, 18 silicate liquid end-members and 26 aqueous fluid species are presented in a revised and updated internally consistent thermodynamic data set. The PVT properties of the data set phases are now based on a modified Tait equation of state (EOS) for the solids and the Pitzer & Sterner (1995) equation for gaseous components. Thermal expansion and compressibility are linked within the modified Tait EOS (TEOS) by a thermal pressure formulation using an Einstein temperature to model the temperature dependence of both the thermal expansion and bulk modulus in a consistent way. The new EOS has led to improved fitting of the phase equilibrium experiments. Many new end-members have been added, including several deep mantle phases and, for the first time, sulphur-bearing minerals. Silicate liquid end-members are in good agreement with both phase equilibrium experiments and measured heat of melting. The new dataset considerably enhances the capabilities for thermodynamic calculation on rocks, melts and aqueous fluids under crustal to deep mantle conditions. Implementations are already available in thermocalc to take advantage of the new data set and its methodologies, as illustrated by example calculations on sapphirine-bearing equilibria, sulphur-bearing equilibria and calculations to 300 kbar and 2000 °C to extend to lower mantle conditions.

An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids

Modeling hardness of polycrystalline materials and bulk metallic glasses

With the assumption that both size- and strain-broadened profiles of the pure-specimen function are described with a Voigt function, it is shown that the analysis of Fourier coefficients leads to the Warren–Averbach method of separation of size and strain contributions. The analysis of size coefficients shows that the `hook' effect occurs when the Cauchy content of the size-broadened profile is underestimated. The ratio of volume-weighted and surface-weighted domain sizes can change from ~1.31, for the minimum allowed Cauchy content, to 2, when the size-broadened profile is given solely by a Cauchy function. If the distortion Subscripts coefficient is approximated by a harmonic term, mean-square strains decrease linearly with increasing the averaging distance. The local strain is finite only in the case of purely Gaussian strain broadening, because strains are then independent of averaging distance.

Voigt-function modeling in Fourier analysis of size- and strain-broadened X-ray diffraction peaks

We propose an elastic-anisotropy measure. Zener’s familiar anisotropy index A=2C44∕(C11−C12) applies only to cubic symmetry [Elasticity and Anelasticity of Metals (University of Chicago Press, Chicago, 1948), p. 16]. Its extension to hexagonal symmetry creates ambiguities. Extension to orthorhombic (or lower) symmetries becomes meaningless because C11−C12 loses physical meaning. We define elastic anisotropy as the squared ratio of the maximum/minimum shear-wave velocity. We compute the extrema velocities from the Christoffel equations [M. Musgrave, Crystal Acoustics (Holden-Day, San Francisco, 1970), p. 84]. The measure is unambiguous, applies to all crystal symmetries (cubic-triclinic), and reduces to Zener’s definition in the cubic-symmetry limit. The measure permits comparisons between and among different crystal symmetries, say, in allotropic transformations or in a homologous series. It gives meaning to previously unanswerable questions such as the following: is zinc (hexagonal) more or less anisotropic than copper (cubic)? is alpha-uranium (orthorhombic) more or less anisotropic than delta-plutonium (cubic)? The most interesting finding is that close-packed-hexagonal elements show an anisotropy near 1.3, about half that of their close-packed-cubic counterparts. A central-force near-neighbor model supports this finding.

A general elastic-anisotropy measure

The elastic constants of a natural-quartz sphere using resonance-ultrasound spectroscopy (RUS) are measured. The measurements of the near-traction-free vibrational frequencies of the sphere are matched with the predicted frequencies from the dynamic theory of elasticity, with optimized estimates for the elastic constants driving the differences between these sets of frequencies to a minimal value. The present computational model, although based on earlier approaches, is the first application of RUS to trigonal-symmetry spheres. Quartz shows six independent elastic constants, and our estimates of these constants are close to those computed by other means. Except for C14, after a 1% mass-density correction, natural quartz and cultured quartz show the same elastic constants. Natural quartz shows higher internal frictions.

Elastic constants of natural quartz.

For a face-centered-cubic steel, new measurements of the monocrystal Voigt elastic-stiffness constants C11, C12, C44 are given. The monocrystal steel, Fe19Cr10Ni, corresponds closely to the well-known AISI-304 austenitic stainless steel. Considering seven theories for the monocrystalpolycrystal elastic constants, it is found that the Hershey-Kroner-Eshelby theory agrees best with measurements. It predicts the shear modulus within 2% of observation, where the Voigt-Beuss first-order bounds differ by 49%. Ten sets of FeCrNi Cij results are reviewed with the finding that both Zener's elastic anisotropy and the C12/C11 ratio are constant within 5%.

Hassel Ledbetter

Papers

Voigt-function modeling in Fourier analysis of size- and strain-broadened X-ray diffraction peaks

A general elastic-anisotropy measure

Elastic constants of natural quartz.

Monocrystal-Polycrystal Elastic Constants of a Stainless Steel

Internal strain (stress) in an SiCAl particle-reinforced composite: an X-ray diffraction study