Showing papers on "Debye model published in 1969"

Linewidths and Thermal Shifts of Spectral Lines in Neodymium-Doped Yttrium Aluminum Garnet and Calcium Fluorophosphate

Abstract: Measurements in the 4.2-500\ifmmode^\circ\else\textdegree\fi{}K temperature range are reported of the widths, shapes, and thermal shifts of the 1.06- and 0.9-\ensuremath{\mu} lines of the ${\mathrm{Nd}}^{3+}$ ions in yttrium aluminum garnet (YAlG) and calcium fluorophosphate. The results are interpreted in terms of phonon-impurity interactions and static crystal strains. The obtained effective Debye temperature of YAlG is compared with the phonon density spectrum deduced from vibronic sidebands of the $R$ lines in YAlG: ${\mathrm{Cr}}^{3+}$. The $R\ensuremath{\rightarrow}{Z}_{5}$ and $R\ensuremath{\rightarrow}{Y}_{6}$ transitions of YAlG: Nd are observed to shift exceptionally to the higher energies with an increase of temperature. This is tentatively ascribed to the pushing effect of nearby higher-lying manifolds. Comparison of widths of several lines in YAlG: Nd at low temperatures reveals that the fluctuation of the second-degree crystal-field parameters contributes to the inhomogeneous linebroadenings. Discussions are also presented on the suitability of hard crystals with high phonon cutoff frequencies for the laser host materials.

Elastic properties of polycrystalline TiB2, ZrB2 and HfB2 from room temperature to 1300 °K☆

Abstract: The Young's, shear, and bulk moduli of polycrystalline TiB2, ZrB2, and HfB2 were determined from 298 ° to 1300 °K by the sonic resonance technique. All three of these moduli decreased by about 9% over the temperature range considered. The Young's and shear moduli were fitted to Wachtman's empirical equation. The theoretical Soga-Anderson equation was applied to the bulk modulus-temperature data and found to fit the data within 1% despite the variation of the Soga-Anderson constant, δ, with temperature. The Gruneisen constant, γ, was found to remain invariant with temperature while the Poisson's ratio increased with increasing temperature. Debye temperatures were computed from the measured Young's and shear moduli.

Elastic Properties of Polycrystalline Yttrium Oxide, Dysprosium Oxide, Holmium Oxide, and Erbium Oxide: Room Temperature Measurements

Abstract: The Young's and shear moduli of polycrystalline yttrium oxide, dysprosium oxide, holmium oxide, and erbium oxide were determined at room temperature as a function of volume fraction porosity using the sonic resonance technique. Linear relations empirically described the data for Y2O3, Hr2O3, and Er2O3. The Young's and shear moduli data for Dy2O3 were empirically described by Hasselman's and Spriggs’equations, respectively. The empirical curves which best fit the data were compared to theoretical expressions and agreed closely with a modified form of Mackenzie's equation. Values for bulk modulus, Poisson's ratio, and the Debye temperature were computed for each oxide, and the bulk modulus-volume relation was determined and compared to that of other oxides.

Grüneisen constant and thermal properties of crystalline and glassy polymers

Abstract: Polymer crystals are characterized by strong anisotropy of binding forces among units, i.e., the intrachain force constant f is much larger than the interchain force constant g. The anisotropic lattice is reduced to an isotropic one, in which each lattice point represents N* units (segment) along the chain axis of the anisotropic lattice [N = 2(f/g)1/2]. Vibrational modes of the reduced lattice correspond to interchain modes of the original lattice, i.e., modes whose frequencies are governed by interchain potential. Anharmonicity of crystalline force field is assumed to be related predominantly with interchain force alone. Thermodynamic and transport equations for a simple lattice are applied to the reduced, isotropic lattice, and numerical results are obtained for high-density polyethylene. The Gruneisen constant γ was obtained from the pressure dependence of sound velocity. The heat capacity of the reduced lattice, Cinter (interchain specific heat), was calculated from Gruneisen's equation, α = γβCinter (where α = thermal expansion coefficient, β = compressibility), and the mass of a segment m* was estimated from Dulong-Petit's equation, Cinter = 3ρk/m* (where ρ = density, k = Boltzmann constant). The value of m* is consistent with N* from force constants, m* = N*m (where m = mass of a unit in the original lattice). m*θ3 (where θ denotes the Debye temperature of the reduced lattice) is calculated from low temperature specific heat. The value of m* calculated from m*θ3 and θ from other sources agrees with that from the estimate by Dulong-Petit's equation. The high-temperature thermal conductivity K was calculated through Leibfried-Schloemann's equation by employing γ and m*θ3 as estimated as described above; satisfactory agreement was obtained with experiment. Poly(methyl methacrylate) and polystyrene were also studied by similar methods.

Pressure dependence of the superconducting transition temperature.

Abstract: A theory of the superconducting transition temperature using a simple model for the electron-phonon interaction has been used to calculate the pressure dependence of the transition temperature for nontransition metals. The pressure dependence of the Debye temperature and phonon frequencies is included by using a Gr\"uneisen model. The theory is compared to experiment for aluminum, lead, zinc, cadmium, indium, and tin, and the agreement is generally quite good. The results have been used to predict the critical pressures at which ${T}_{c}=0$; and for the cases of zinc, cadmium, and possibly aluminum, it should be possible to carry out experiments where ${T}_{c}$ is pushed below presently measurable temperatures. The empirical ${T}_{c}=f(P)$ relations published in the literature are discussed and compared with the present results.

Optical spectra of orientationally disordered crystals. iii. infrared spectra of the sound waves

Abstract: The theory of the absorption of light by the translational vibrations of orientationally disordered crystals has been extended to low frequencies where the vibrations are short‐wavelength sound waves whose density of states can be represented by the Debye model. Both the integrated intensity of a normal vibration and the density of states are proportional to the frequency squared, so the absorptivity is proportional to the fourth power of the frequency. The proportionality constant can be related to the difference between the limiting low‐ and high‐frequency permittivities of the translation band. The absorptivity of ice has been measured below 45 cm−1 in order to test the theory. As predicted, it is approximately proportional to the fourth power of the wavenumber. At low frequencies the absorptivity becomes approximately proportional to the square of the wavenumber and is probably caused by two‐phonon difference bands involving high‐frequency acoustic phonons. The microwave absorptivity reported by others appears to be consistent, at least approximately, with this square line extrapolated, which suggests that it also is due mainly to two‐phonon difference bands. The absorptivity is less than predicted due to two approximations in the theory, namely, the neglect of the strong orientational correlation in ice and the assumption that R defined by Eq. (20) is independent of the type of internal coordinate.

Lattice dynamics of terbium

Abstract: The frequency-wave-vector dispersion relation for the normal modes of vibration of terbium at room temperature has been measured by means of slow-neutron inelastic scattering techniques. The triple-axis spectrometer at the Oak Ridge high flux isotope reactor was used, mostly in the constant-$Q$ mode of operation. Phonon frequencies for wave vectors along the principal symmetry directions have been determined and, in addition, measurements of phonon frequencies along the boundaries of the Brillouin zone and along a more general direction are reported. The data have been fitted with a Born-von K\'arm\'an force model which includes interactions out to the eighth nearest neighbor. The interactions have been assumed to be general (tensor) out to the fourth neighbor and axially symmetric beyond. The model has been used to calculate a frequency distribution function $g(\ensuremath{ u})$ and related quantities such as the lattice specific heat and Debye temperature.

Surface Thermodynamic Functions for Noble‐Gas Crystals

Abstract: Surface thermodynamic quantities have been calculated as functions of temperature between 0°K and the melting temperature for the (111), (100), and (110) surfaces of the noble‐gas solids Ne, Ar, Kr, and Xe. The method of calculation differs from the methods used in previous treatments of surface thermodynamic functions in that the atomic vibrations are taken into account, and the vibrational frequencies are properly calculated rather than obtained from a Debye model or an Einstein model. The quantities calculated are the static surface energy, vibrational surface energy, surface entropy, vibrational surface free energy, and surface specific heat. In addition, a surface frequency‐distribution function f8(ω) has been calculated; f8(ω) is positive at low frequencies, because of the presence of surface modes of vibration, and negative at higher frequencies. This behavior of f8(ω) produces a narrow peak in the graph of the surface specific heat as a function of temperature.

The lattice dynamics of niobium I. Measurements of the phonon frequencies

Abstract: Measurements of the frequencies of the normal modes of vibration of niobium at 296 °K have been made by inelastic neutron scattering techniques. Results are presented for several hundred phonons with wave vectors at general positions in the [11, macron0] reciprocal lattice plane. The phonon dispersion relations in symmetry directions agree well with the previous work of Nakagawa and Woods. The measured frequencies cannot be fitted well by tensor force-constant models even when the range of the forces extends to tenth-nearest neighbours, but the critical points in the phonon density of states derived from the eighth-neighbour model agree with those observed in superconducting tunnelling experiments, and the calculated Debye temperature also agrees quite well with measured values.

The specific heat of solid oxygen

Abstract: Measurements have been made in the temperature range from 1.3° K to about 71°K with particular attention to the behavior at very low temperatures, and in the neighborhood of the α-β transition at about 23.89° K. Assuming that near 0° K each molecule moves as a single mass point with the passage of lattice waves, the effective Debye temperature at 0° K is extrapolated to be 104±2° K. As the temperature rises above 10° K, the specific heat rises more rapidly than a reasonable Debye model would predict, suggesting the appearance of additional degrees of freedom, which are thought to be the superposition of a librational motion of the molecules superposed on the longitudinal and transverse lattice waves controlling the motion of a molecule's center of mass. The specific heat shows a very sharp high “spike” at the α-β transition with an entropy change of aboutR ln 1.65; there is no evidence for a latent heat associated with this transition.

Dielectric response of the alkaline earth fluorides

Abstract: The temperature dependence of the low-frequency dielectric response of CaF2, SrF2 and BaF2 over the range 2-350 °k is reported, together with the pressure dependence, for hydrostatic pressures up to 5 kbar, at selected temperatures in the range 60 to 350 °k. The low-temperature data are used to assess the nature and extent of the electronic structure and interionic forces of the alkaline earth fluorides. The temperature dependence of the dielectric response is found to be due principally to intrinsic volume changes in the crystals rather than to intrinsic temperature changes in the anharmonic contribution. The temperature dependence under constant volume of the anharmonic contribution to the dielectric constant of all three salts exhibits a maximum value near 0·25 θD, where θD is the characteristic Debye temperature. The quasi-harmonic contributions are analysed to determine values for the κ similar, equals 0 longitudinal and infra-red active transverse optic phonon Gruneisen constants.

Debye–Waller factor and anomalous absorption (Ge: 293–5°K)

Abstract: Experiments measuring the intensity of three reflexions from a `very thick' Ge crystal (Laue case) were made, while the temperature of the crystal was lowered from 293 to 5°K. The variation of the intensity agreed with the dynamic theory, when the Debye temperature Θ, used for the evaluation of the Debye–Waller factor, was assumed to be 290°K throughout the temperature range. This result may suffer some slight correction in the future, but there is no variation of Θ at 20°K. Θ = constant constrasts with the specific heat data of Θ but essentially agrees with Batterman & Chipmans's prediction.

Dispersion of Phonons in Ideal Crystals

Abstract: A quasiharmonic central force rigid-atom model has been used to study the lattice vibrations of frozen rare gases. The model takes care of interactions up to fourth neighbour and estimates zero-point energy and its volume derivatives by the Debye theory of specific heats. The theoretical frequency distribution and phonon dispersion curves are found to compare reasonably well with the available experimental data. Various causes of the discrepancies and possibilities of improvement of the results are discussed.

Temperature Dependence of the Elastic Constants of Tin Telluride

Abstract: The temperature dependence of the elastic constants of SnTe has been measured between 1.2 and 300 K. A change was found in the slope of C44 around 135 K in crystals with a carrier concentration of 6×1020 carriers/cm3. No indication of a low temperature phase transition was seen. The 0°K Debye temperature for this carrier concentration is 192.2±0.4 K.

Scattering of long-wavelength phonons by point defects in cubic lattices

Abstract: The phase-shift method in solid state scattering theory is applied to the study of the scattering of long-wavelength phonons by point defects in the cubic lattice model of Montroll and Potts. Expressions for the mean free path and Debye temperature are derived for a solid containing a small concentration of point defects.

Thermal Effects on the Fluorescence Lines of Sm2+ in Crystals

Abstract: The effects of temperature on the fluorescence lines of Sm2+ in CaF2, SrF2, and BaClF were studied. The results for the first two samples were obtained in the region 10°–100°K. For CaF2 a strong dependence of the linewidth on temperature was found; this width is ∼ 1.7 cm−1 at 15°K and ∼ 40 cm−1 at 95°K. For SrF2, the linewidth is constant and equal to ∼ 2.15 cm−1 up to ∼ 77°K. This difference in behavior is explained qualitatively as due to the fact that in CaF2 the metastable state belongs to the 4f55d configuration and therefore, the electron charge is strongly affected by the thermal perturbations of the environment. We also measured the widths and positions of several fluorescence lines of BaClF:Sm2+ in the 30°–600°K region. The thermal broadening of most of these lines is explained as due to microscopic strains and Raman scattering of phonons, the thermal shifts as due to emission, and absorption of virtual phonons. A Debye model of phonon distribution was used with the same TD for both the lineshift...

Single‐Crystalline Elastic Constants of ZrCo2 and HfCo2 in the Temperature Range 4.2°–300°K

Abstract: The elastic constants of single crystals of the cubic Laves phases ZrCo2 and HfCo2 were determined by the pulse‐echo technique over the temperature range 4.2°–300°K. Debye temperatures for the two materials were computed from extrapolations of the elastic constants to 0°K. The ratio of these Debye temperatures is such as to indicate that the primary difference between the two materials is a difference in vibrational amplitudes resulting from the mass difference between zirconium and hafnium. Further substantiation of a close similarity of interatomic force interactions in the two materials is obtained by comparison of appropriate combinations of lattice parameters and force constants.

An empirical relation between debye temperature and formation energy of vacancies in alkali halides

Abstract: An empirical relation, θD = 32·37 (Ws/MV 2/3)1/2, connecting the Debye temperature (θD) and the formation energy of Schottky defects in alkali halides (Ws ), is found and this relation is used for predicting the formation energy of Schottky defects in various alkali halides. The predicted values for Ws are compared with the experimental values wherever possible and in many cases the agreement between them is found to be well within 10%. It is believed that the simple relation found here can be useful for a rapid and reasonably accurate calculation of the formation energies of vacancies in alkali halides.

Heat Capacity and Elastic Constants of CsI at Low Temperatures

Abstract: Measurement of the heat capacity from 2.13° to 10°K and of the elastic constants at 4.2°K have been made on a single crystal of CsI. The Debye temperature at 0°K, θ0, as calculated from the heat capacity is θ0=127.67°K. Values of the elastic constants at 4.2°K are c11=2.725±0.6%, c44=0.873±0.6%, and c12=0.767±3% in units of 1011 dyn/cm2. θ0 calculated from the elastic constants at 4.2°K is equal to 129.4°K.