Debye model

About: Debye model is a research topic. Over the lifetime, 7462 publications have been published within this topic receiving 133987 citations.

Thermodynamics and lattice vibrations of minerals: 5. Applications to phase equilibria, isotopic fractionation, and high‐pressure thermodynamic properties

Abstract: In previous papers in this series, a model that uses available elastic, structural, and spectroscopic data on minerals has been used to predict the thermodynamic functions Cv (heat capacity), S (entropy), E (internal energy), and F (Helmholtz free energy). In this paper, four applications to problems of current geochemical and geophysical interest are presented: (1) interpretation of complex trends of calorimetric data; (2) calculation of phase equilibria; (3) calculation of oxygen-isotopic fractionation factors; and (4) estimation of the effect of pressure on thermodynamic functions. The model demonstrates that trends in high-temperature thermodynamic properties of silicates are determined by the position and relative numbers of high-frequency modes, generally antisymmetric (Si, Al)-O stretching modes. The position of these modes varies systematically with degree of polymerization of tetrahedra, and therefore high-temperature calorimetric behavior is relatively systematic as a function of crystal structure and mineral composition. Trends at low frequency are much more complex because the low-frequency optic modes that most strongly influence the low-temperature thermodynamic functions depend in a complex way on the size, coordination, and mass of cations and various polyhedra in the minerals. The heat capacity curves of kyanite, andalusite, and sillimanite and of quartz, coesite, and stishovite show crossovers that cannot be explained by Debye theory, which accounts only for acoustic mode behavior, but can be explained by the model spectra proposed because proper account is taken of the changing low- and high-frequency optic modes upon polymorphic transformations. The proposed model is sufficiently accurate that phase equilibrium problems can be addressed: the quartz-coesite-stishovite equilibrium curves, the kyanite-andalusite-sillimanite triple point, and the breakdown of albite to jadeite-plus-quartz are cited as specific examples. For each example, predicted slopes of equilibrium curves agree moderately well to excellently with slopes determined experimentally. The calculated slopes are sensitive to spectroscopic parameters, particularly to the distribution of optic modes in the far infrared; this sensitivity is discussed in detail for the albite breakdown reaction. The model can be used for prediction of isotopic fractionation factors if spectra of the isotopic forms of the mineral are known or postulated. A simple set of ‘rules’ for generating hypothetical spectra of 18O minerals from measured spectra of the 16O forms is given. Reduced partition functions are calculated for 13 minerals. At 298°K the model values of reduced partition function, 1000 ln α, of these minerals decrease in the order quartz > calcite ≳ albite > muscovite > clinoenstatite ≈ anorthite > diopside > pyrope > grossular > zircon > forsterite > andradite > rutile, in good agreement with experimental data. At 1000°K the first six minerals show small crossovers so that the order becomes calcite, muscovite ≈ albite, quartz, anorthite, and clinoenstatite; the differences in 1000 ln α at high temperature for these minerals are so small that the model probably cannot address the deviations from experimental trends. The model clearly defines the region in which the fractionation factors do not follow a 1/T² trend and should be useful for extrapolation of experimental data to low temperatures. Finally, a modified Gruneisen parameter model is proposed for shift of the lattice vibrational frequencies under compression, and thermodynamic properties to 1000 kbar, 1000°K, are given for nine minerals. At 1 Mbar, the predicted decrease in entropy at 298°K ranges from 54% (of the 1-bar value) for periclase to 25% for stishovite.

Lattice dynamics of ionic and covalent crystals

Abstract: Lattice dynamics has some claim to being the oldest branch of solid state physics. Planck's Theory of Radiation and the Theory of Specific Heat was published by Einstein in 1907, On Vibrations in Space Lattices by Born and von Karman in 1912, and On the Theory of Specific Heat by Debye in the same year. Other early papers on lattice dynamics included those by Debye and by Waller on the effect of temperature op the scattering of x-rays by a crystal. In the 1920's Peierls made fundamental contributions to the theory of thermal conductivity and of electrical conductivity involving lattice dynamics, and in the 1930's Blackman brought to an end the acceptance of the complete validity of the Debye theory of specific heat and also contributed the first papers on anharmonic effects in the absorption of infrared radiation by ionic crystals. The work we have mentioned used the formal theory of lattice dynamics to account for the thermal and other properties of crystals. It was not until 1940 that Kellerman...

Equation of State of Polycrystalline and Single‐Crystal MgO to 8 Kilobars and 800°K

Abstract: MgO has been measured again. Most measurements of the elastic properties of materials, interesting to geophysics, have been made as a function of pressure at room temperature or as a function of temperature at atmospheric pressure. A lapped seal between a buffer rod and sample has made it possible to use ultrasonic interferometry to 1000°K and 10 kb. The elastic constants of poly crystalline and single-crystal MgO were measured in a gas high-pressure system over a temperature range from 300°K to the Debye temperature of MgO. Data from the polycrystalline specimen indicated large effects of temperature on the pressure derivatives. These data did not agree with the results obtained from single-crystal measurements. Upon remeasuring the ceramic sample it becomes apparent that the data are not reproducible after the sample has been cycled to high temperature and pressure. Additional sintering, deformation of the individual grains, and recrystallization take place, which change the properties of the sample. These problems and the problems of sintering isotropic aggregates of theoretical density limit the usefulness of this widely used procedure. Order from the American Geophysical Union, Suite 435, 2100 Pennsylvania Ave., N.W., Washington D.C 20037. Document J70-001; $1.00.

Lattice Dynamics of Gold

Abstract: The complete phonon dispersion relations for gold in the high-symmetry directions have been measured at room temperature by the coherent inelastic scattering of neutrons. It is found that the forces in gold are not homologous with the other noble metals, the frequencies of gold lying appreciably higher than those "scaled" from copper and silver. An analysis of the data in terms of different force-constant models reveals that a general tensor force is required for the first-neighbor interaction, whereas for neighbors beyond the first either general tensor or axially symmetric forces give an excellent fit to the data. The axially symmetric model alone does not adequately describe the data even when forces extending to ninth-nearest neighbors are included in the fit. In addition, simple screened-pseudopoential models were fit to the data and these results also indicate the need for the first-neighbor interaction to be general. Frequency distribution functions and related thermodynamic quantities were calculated from the various force-constant models. The Debye temperature ${\ensuremath{\Theta}}_{C}$ versus temperature curves obtained show an anomaly at low temperatures consistent with the ${\ensuremath{\Theta}}_{C}(T)$ obtained from specific-heat measurements. The relation between this anomaly and the character of the dispersion curves is given.

Lattice Dynamics of Rutile

Abstract: The phonon dispersion relation for rutile (Ti${\mathrm{O}}_{2}$) has been measured by the coherent inelastic scattering of thermal neutrons along principal symmetry directions of the Brillouin zone. Theoretical models based on rigid-ion and shell models, with either axially symmetric or tensor first- and second-neighbor forces, have been fitted to the measured dispersion relation. Only the shell model with tensor forces for all interactions except the second-neighbor oxygen-oxygen interaction was able to give acceptable qualitative agreement with the data, and that agreement is good for only some modes. A frequency distribution and Debye temperature spectrum are presented for that model. The temperature dependence of the frequency of the ${\ensuremath{\Gamma}}_{1}^{\ensuremath{-}}({A}_{2u})$ transverse optic mode is measured from 4 to 300\ifmmode^\circ\else\textdegree\fi{}K, and the behavior of the square of the frequency is in good agreement with that predicted by the static dielectric constant measurements of Parker.