The determination of the elastic properties of composite materials (multiphase aggregates, polycrystals, and porous or cracked solids) from the elastic properties of the components may be approached in several ways. The problem may be treated statistically, via scattering theory, through variational principles, or by the assumption of specific geometries for the material under consideration. Each of these methods is reviewed in turn. The widely used Voigt-Reuss-Hill average can be a poor approximation for both two-phase composites and polycrystals, and its replacement by the two Hashin-Shtrikman bounds is recommended. For pore-containing or crack-containing media, specific geometry models must be considered if useful results are to be obtained. If aggregate theory is used to estimate the moduli of individual components of a composite whose bulk properties are known, the shear moduli of the component phases must be matched (within a factor of 2 or 3) for the method to be useful. Results for nonlinear composites (which allow calculation of the pressure variation of aggregate moduli) have been obtained for only a few special cases.

The Elastic Properties of Composite Materials

We report on extensive experimental studies on thin film, single crystal, and ceramics of multiferroic bismuth ferrite $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_{3}$ using differential thermal analysis, high-temperature polarized light microscopy, high-temperature and polarized Raman spectroscopy, high-temperature x-ray diffraction, dc conductivity, optical absorption and reflectivity, and domain imaging, and show that epitaxial (001) thin films of $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_{3}$ are clearly monoclinic at room temperature, in agreement with recent synchrotron studies but in disagreement with all other earlier reported results. We report an orthorhombic order-disorder $\ensuremath{\beta}$ phase between 820 and 925 $(\ifmmode\pm\else\textpm\fi{}5)\phantom{\rule{0.2em}{0ex}}\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$, and establish the existence range of the cubic $\ensuremath{\gamma}$ phase between 925 $(\ifmmode\pm\else\textpm\fi{}5)$ and 933 $(\ifmmode\pm\else\textpm\fi{}5)\phantom{\rule{0.2em}{0ex}}\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$, contrary to all recent reports. We also report the refined ${\mathrm{Bi}}_{2}{\mathrm{O}}_{3}\text{\ensuremath{-}}{\mathrm{Fe}}_{2}{\mathrm{O}}_{3}$ phase diagram. The phase transition sequence rhombohedral-orthorhombic-cubic in bulk [monoclinic-orthorhombic-cubic in $(001)\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_{3}$ thin film] differs distinctly from that of $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$. The transition to the cubic $\ensuremath{\gamma}$ phase causes an abrupt collapse of the band gap toward zero (insulator-metal transition) at the orthorhombic-cubic $\ensuremath{\beta}\text{\ensuremath{-}}\ensuremath{\gamma}$ transition around $930\phantom{\rule{0.2em}{0ex}}\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$. Our band structure models, high-temperature dc resistivity, and light absorption and reflectivity measurements are consistent with this metal-insulator transition.

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βphase andγ−βmetal-insulator transition in multiferroicBiFeO3

The Velocity of Compressional Waves in Rocks to 10 Kilobars, Part 2

We have measured and analyzed the Extended X-ray Absorption Fine Structure (EXAFS) of BaTiO3 at the barium k edge and the X-ray Absorption Near Edge Structure (XANES) at the titanium K edge. Our structural data show that the sequence of phase transitions in this material as the temperature increases is explained by a disordering of domains wherein the local structural environment remains approximately rhombohedrally distorted at all temperatures around both metal sites. As the temperature is raised, the long range correlations between these local distortions change, resulting in the observed sequence of phase transitions. Our measurements confirm the model of eight-site disorder used to explain the phase diagram of BaTiO3. We show that EXAFS and XANES are sensitive probes of both the magnitude and direction of the local structural distortions which accompany ferroelectricity and therefore are sensitive probes of the microscopic mechanism of ferroelectricity.

Local structure and the phase transitions of BaTiO3

The elastic properties of candidate mantle phases are used to test the viability of olivine-rich (pyrolitic) and CaO + Al_2O_3-rich (eclogitic) assemblages for the mantle. High temperature adiabats for each phase of interest are constructed and compared to mantle seismic properties. Both pyrolitic and eclogitic assemblages satisfy the seismic properties between ∼ 200 and 400 km. Between 400 and 670 km depth an eclogitic assemblage yields a superior match to velocities and velocity gradients. The 400 km seismic discontinuity may represent a chemical boundary between pyrolite and picritic eclogite (“piclogite”) or phase transformations in the olivine + orthopyroxene components of a piclogitic assemblage containing about 16% olivine. High velocity gradients in the transition zone may be explained by the transformation of Ca-rich cpx to majorite garnet. Seismic properties at the top of the lower mantle are consistent with pyrolite, piclogite or perovskite, implying that the 670 km discontinuity may be a chemical boundary.

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Composition of the upper mantle: Geophysical tests of two petrological models

Phonon dispersion curves of wustite (FeO) have been measured at room temperature by inelastic neutron scattering in the three high-symmetry directions [001], [110], and [111] of reciprocal space. The experimental curves are interpreted by various lattice-dynamical models including rigid-ion and shell models. The models show that both ions are highly polarizable and that overlap effects seem to play an important role on the shell charge ${Y}_{1}$ of the iron ion, and on the second-neighbor interaction constants. The best-fit model has been applied in calculating structure factors, optical and elastic constants, and the frequency-distribution function.

Lattice dynamics of wustite (FeO)

We have calculated the phonon dispersion curves of mixed crystals of ${\mathrm{KTaO}}_{3}$ and ${\mathrm{KNbO}}_{3}$ for various niobium concentrations. In our calculations, a nonlinear shell model is used, in which the fourth-order anharmonic core-shell interaction of the oxygen ions is taken into account in the self-consistent phonon approximation. The measured phonon dispersion curves, their temperature dependence, and, in particular, the temperature dependence of the ferroelectric soft-mode frequency for different niobium concentrations between 0 and 100 at. % are well reproduced by our calculations in which only two coupling parameters depend on the niobium concentration in a systematic way. These two parameters are temperature independent and characterize the linear and nonlinear polarizability of the oxygen ions. All other coupling parameters are temperature and concentration independent.

Lattice dynamics of KTa 1 − x Nb x O 3 solid solutions in the cubic phase

Raman scattering measurements are made in the tetragonal phase of KNbO3. The results are in good agreement with the previous corresponding neutron data, and can be assigned to symmetry species. Comparison of these results to those available in the other phases enables to observe the effect of the phase transformations on the LO–TO splitting and on the degeneracy lifting of each initial cubic mode. It is shown that the lowest frequency F1u cubic mode is the most affected one by the different cell distortions; at each transition, it splits into “soft” and “stiffened” components; it is the so-called “ferroelectric” mode which drives the successive phase changes. On the other hand, it is remarkable that the phonons derived from the F2u “silent” mode are nearly degenerate, even in the ferroelectric phases; it is also the case for the phonons issued from the second F1u cubic mode. Finally, it is emphasized that the phase transitions in KNbO3 can be described by a displacive-type mechanism.



Nous reportons des mesures de diffusion Raman obtenues dans la phase quadratique de KNbO3. En accord avec les mesures neutroniques anterieures, ces resultats sont associes aux types de symetrie correspondants. La comparaison de ces mesures avec celles disponibles dans les autres phases permet d'observer l'effet des changements de phase sur le splitting LO–TO et sur la levee de degenerescence de chaque mode initial de la phase cubique. Nous montrons que le mode cubique F1u le plus bas est le plus sensible aux differentes distorsions du reseau cristallin; a chaque transition, il se separe en composantes „molle” et „dure”; c'est le mode „ferroelectrique”, ainsi nomme parce qu'il induit les changements de phase successifs. D'autre part, nous remarquons que les phonons qui proviennent du mode „silencieux” F2u sont presque degeneres, měme dans les phases ferroelectriques, c'est aussi le cas des phonons issus du second mode cubique F1u. Finalement, nous mettons en evidence que les transitions de phase dans KNbO3 peuvent ětre decrites par un mecanisme de type displacif.

Long-Wavelength Phonons in the Different Phases of KNbO3

Low-energy magnetic excitations in wustite (Fe 1-x O)

Inelastic neutron scattering experiments in tetragonal KNb${\mathrm{O}}_{3}$ show a high anisotropy in some regions of the phonon dispersion relation and the existence of strong one-dimensional intercell correlations between atomic motions along the [100] and [010] directions, in agreement with x-ray diffuse-scattering measurements in the same phase. The transverse phonon modes propagating normally to these directions of strong correlations, that is, with polarization vectors parallel to them, are relatively soft, not only near the zone center but all the way to the zone boundary. Comparison of these results with the Raman, neutron, and x-ray data obtained in the other phases suggests a description of the ferroelectric phase transition mechanism in KNb${\mathrm{O}}_{3}$ in terms of soft modes and dynamic correlations in each phase.

G. E. Kugel

Papers

Lattice dynamics of wustite (FeO)

Lattice dynamics of KTa 1 − x Nb x O 3 solid solutions in the cubic phase

Long-Wavelength Phonons in the Different Phases of KNbO3

Low-energy magnetic excitations in wustite (Fe 1-x O)

Inelastic neutron scattering in tetragonal KNb O 3