Efficient Debye Function Interpolation Formulae: Sample Applications to Diamond

The well-known classical heat capacity model developed by Debye proposed an approximate description of the temperature-dependence of heat capacities of solids in terms of a characteristic integral, the T-dependent values of which are parameterized by the Debye temperature, Θ D . However, numerous tests of this simple model have shown that within Debye’s original supposition of approximately constant, material-specific Debye temperature, it has little chance to be applicable to a larger variety of non-metals, except for a few wide-band-gap materials such as diamond or cubic boron nitride, which are characterized by an unusually low degree of phonon dispersion. In this study, we present a variety of structurally simple, unprecedented algebraic expressions for the high-temperature behavior of Debye’s conventional heat-capacity integral, which provide fine numerical descriptions of the isochoric (harmonic) heat capacity dependences parameterized by a fixed Debye temperature. The present sample application of an appropriate high-to-low temperature interpolation formula to the isobaric heat capacity data for diamond measured by Desnoyers and Morrison [17], Victor [24], and Dinsdale [25] provided a fine numerical simulation of data within a range of 200 to 600 K, involving a fixed Debye temperature of about 1855 K. Representing the monotonically increasing difference of the isobaric versus isochoric heat capacities by two associated anharmonicity coefficients, we were able to extend the accurate fit of the given heat capacity ( C p ( T ) ) data up to 5000 K. Furthermore, we have performed a high-accuracy fit of the whole C p ( T ) dataset, from approximately 20 K to 5000 K, on the basis of a previously developed hybrid model, which is based on two continuous low-T curve sections in combination with three discrete (Einstein) phonon energy peaks. The two theoretical alternative curves for the C p ( T ) dependence of diamond were found to be almost indistinguishable throughout the interval from 200 K to 5000 K.

In situ high-temperature X-ray diffraction study of Sc-doped titanium oxide nanocrystallites

Titanium dioxide is an inexpensive wide-gap highly ionic semiconductor with striking photocatalytic capabilities in several heterogeneous photoredox reactions. A small crystal size is desirable to maximize the surface area, since photocatalytic reactions occur at the surface of a photocatalyst. Presented here are the synthesis and microstructural characterization of 4 at.% Sc-doped TiO2 (4SDT) prepared by water-based co-precipitation. The crystal structure of 4SDT was examined via in situ high-temperature powder X-ray diffraction experiments from 25 to 1200°C. Rietveld analysis revealed single-phase anatase up to 875°C, while at 900°C the anatase-to-rutile phase transformation occurred and at higher temperatures additional reflections of Sc-rich phases (Sc2TiO5 from 975°C and Ti3Sc4O12 or Sc2O3 at 1200°C) were observed. Debye function analysis (DFA) was applied to model the total scattering pattern directly in reciprocal space, allowing the reconstruction of Ti vacancies. Both Rietveld and DFA methods were applied to estimate the nanocrystallite size and shape with consistent growth in crystallite size with temperature: an ellipsoid shape with equatorial ∼4.7 nm / axial (001) ∼6.9 nm at 25°C to equatorial ∼27.9 nm / axial (001) ∼39.6 nm at 900°C refined by Rietveld analysis, versus a cylinder shape with D a,b = 4.3 nm and size dispersion σ ab = 1.5 nm, L c = 4.9 nm and σ c = 2.3 nm at 25°C to D a,b = 21.4 nm, σ ab = 8.3 nm, L c = 23.9 and σ c = 10.9 nm at 900°C estimated by DFA. The microstructural changes obtained by Rietveld and DFA methods were supported by high-resolution transmission electron microscopy image analysis, as well as by the less direct nitrogen sorption techniques that provide information on the size of non-agglomerated and dense particles. The Ti site-occupancy factor showed a linear increase from 0.6–0.8 at 25°C to unity at 900°C for anatase, and from ∼0.7 at 900°C to unity at 1200°C for rutile, via Rietveld analysis and DFA.

A real-space approach to the analysis of stacking faults in close-packed metals: G(r) modelling and Q-space feedback

An R-space approach to the simulation and fitting of a structural model to the experimental pair distribution function is described, to investigate the structural disorder (distance distribution and stacking faults) in close-packed metals. This is carried out by transferring the Debye function analysis into R space and simulating the low-angle and high-angle truncation for the evaluation of the relevant Fourier transform. The strengths and weaknesses of the R-space approach with respect to the usual Q-space approach are discussed.

Analytic approximation of the Debye function

An expression in a closed form is proposed for the approximation of the Debyefunction used in thermodynamic models of solids. This expression defines an analytic functionthat has the same limiting behavior as the Debye function at low and high temperatures. Theapproximation gives the maximum relative deviation from the value of the Debye function lessthan 0.001. The proposed expression can be useful in the equations of state of solids in a widetemperature range.

Comparación entre la función de cole-cole y la función de Debye para modelar el tejido biológico

The study of the biological tissue varies its behavior, in function of the frequency to which it is subjected. The functions of Cole-Cole, Debye and Lorentz are considered the most used to study the parameters of conductivity and permittivity of biological tissue such as: liver, stomach, gray matter, pancreas, blood, breast tissue to name a few. Depending on the frequency, there may be three different relaxation times. These depend on the nature and physical characteristics of the tissue and its effect in the presence of an electromagnetic signal. The Cole-Cole function is most appropriate when considering a study in the frequency domain, and the Debye function is established to perform the study in the time domain. It is of particular interest to make a comparison between the methods of Cole-Cole and Debye, to define the advantages and disadvantages of each of them. The Lorentz function is used for frequencies of the Tera-Hertz order. Comparative graphs are obtained between both for the biological tissues mentioned above. For each of them the electrical parameters are calculated in a frequency range between 100 Hz and 10 GHz.

Structure factors for generalized grey Browinian motion

In this paper we investigate the form factors of paths for a class of non Gaussian processes. These processes are characterized in terms of the Mittag-Leffler function. In particular, we obtain a closed analytic form for the form factors, the Debye function, and can study their asymptotic decay.

Synthesis of cerium oxide (CeO2) by co-precipitation for application as a reference material for X-ray powder diffraction peak widths

Several methods can be used to obtain, from powder diffraction patterns, crystallite size and lattice strain of polycrystalline samples. Some examples are the Scherrer equation, Williamson–Hall plots, Warren/Averbach Fourier decomposition, Whole Powder Pattern Modeling, and Debye function analysis. To apply some of these methods, it is necessary to remove the contribution of the instrument to the widths of the diffraction peaks. Nowadays, one of the main samples used for this purpose is the LaB6 SRM660b commercialized by the National Institute of Standard Technology; the width of the diffraction peak of this sample is caused only by the instrumental apparatus. However, this sample can be expensive for researchers in developing countries. In this work, the authors present a simple route to obtain micron-sized polycrystalline CeO2 that have a full width at half maximum comparable with the SRM660b and therefore it can be used to remove instrumental broadening.

Whole-nanoparticle atomistic modeling of the schwertmannite structure from total scattering data

Schwertmannite is a poorly crystalline nanometric iron sulfate oxyhydroxide. This mineral shows a structural variability under different environments. Because of that, the determination of its structure and, consequently, of its physical–chemical properties is quite challenging. This article presents a detailed structural investigation of the structure of schwertmannite conducted under different approaches: X-ray absorption spectroscopy, Rietveld refinement, and a combined reverse Monte Carlo and Debye function analysis of the whole nanoparticle structure. The schwertmannite model presented here is, to the auhors' knowledge, the most complete model so far reported.