X-ray Debye Temperature Research Articles

Anharmonic effects in cubic crystals

The anharmonic effects due to isochoric self-energy and isothermal volume expansion of cubic crystals have been considered to study the temperature dependence of frequency shifts, low-temperature specific heat and X-ray Debye temperature. The calculated results are in good agreement with the experimental data.

The anomaly of the X-ray Debye temperature of chromium in the temperature range 35–51°C

The integrated intensities of the 550 and 651 diffraction lines from a chromium single crystal were measured in the temperature range 35-51°C by steps of 2°C. A plot of the intensities of these lines versus temperature showed an abrupt change between 43 and 45°C corresponding to a discontinuity of about (3.6 ± 1.1)°C for the Debye characteristic temperature. This value is in good agreement with the one calculated from the discontinuity of the elastic constants.

A determination of the Debye–Waller temperature factor and the X-ray Debye temperature for Ni, Cr, Fe, Mo and W

The integrated intensities of high-order reflexions for Ni, Cr, Fe, Mo, and W powder samples were measured at 296 and 77°K with Mo Kα radiation. The values 397 ± 6, 510 ± 6, 435 ± 6, 407 ± 6, and 333 ± 6°K respectively were obtained for the Debye temperatures after correction for thermal diffuse scattering. These values are in good agreement with those calculated from force-constant models determined by inelastic neutron scattering.

The electron distribution in silicon - II. Theoretical interpretation

The absolute structure factors of silicon measured in Part I are examined in detail using Dawson’s structure factor formalism with recent theoretical estimates of free atom Hartree-Fock wavefunctions (Clementi 1965). It is not possible to describe the structure of silicon with only Kubic Harmonic expansions having physically reasonable coefficients. If the valence-shell form factor is radially compressed by 6.8% then we can obtain an excellent fit using only the third- and fourth-order Kubic Harmonics with amplitudes F 3 ═ + 1.012 and G 4 ═ — 0.206. There is some evidence that the anomalous dispersion corrections for silicon which are available in the literature are in error by 0.02 electron units. From the high-order reflexions we have measured the X-ray Debye temperatures of silicon as ⊝ M ═ 536.3 ± 5.0 K at 92.2 K and ⊝ M ═ 532.5 ± 2.0 K at 293.2K . The increase in ⊝ M at low temperature is smaller than had been expected theoretically. Our measurements of the 555 structure factor permit us to confirm Nunes’s (1971) upper-bound estimate of the anharmonic force constant for silicon. Finally, we have compared our results with the self-consistent orthogonalized-plane-wave band structure calculations of Stukel & Euwema (1970) for silicon atoms in a crystal. We find that the Kohn-Sham-Gaspar model of exchange results in structure factors which are in much better agreement with those measured than are structure factors calculated using Slater’s exchange term. As in the interpretation based on Dawson’s formalism there is evidence here that the theoretical anomalous dispersion corrections may be inadequate.

Debye-Waller factor and Debye temperature for Fe, Mo, Ta, and W

The Debye-Waller parameter and the Debye temperature for iron, molybdenum, tantalum, and tungsten have been calculated from general force constant models. Comparison of the x-ray Debye temperatures with the heat capacity Debye temperatures is made and discussed.

X-Ray Debye Temperatures of Zinc Sulfide and Zinc Telluride

The x-ray Debye temperatures ΘM of zinc sulfide and zinc telluride have been determined to be 277 ± 7°K and 180 ± 6°K, respectively, by measurement of the temperature variation of the integrated intensities of Bragg reflections from powder specimens (method 1) in the temperature range 93–298°K. Only at the low end of the temperature range was Θm found to vary with temperature and Θm (93°K) appeared to be about 10% larger than the average values. The room-temperature (298°K) Debye temperature for ZnTe was determined to be 175 ± 8°K from integrated intensities of selected Bragg peaks (method 2). Measured absolute structure amplitudes were used to calculate the root-mean-square vibrational amplitudes of the individual ions in ZnTe, which were found to be 0.094 ± 0.025 Å and 0.135 ± 0.030 Å for the Zn and Te ions, respectively. These results are discussed in relation to previous work on other materials.

The X-ray Debye temperature of aluminum

Accurate photographic X-ray intensities have been obtained for nearly all reflexions within the Mo Kα limiting sphere for two single crystals of aluminum. These intensities have been used to give the X-ray Debye temperature, Θd, in the first part of an experimental determination of the electron-density distribution in solid aluminum. The results of this work show that Θd = 393 ± 1°K at 293°K, varying with temperature to Θd = 362 ± 9°K at 559°K. The Debye parameter is found to be 0.849 ± 0.005 Å2 at 293°K.

The discontinuity of the X-ray Debye temperature of chromium near the Néel temperature

The integrated intensity of the lines 550 and 712 from a chromium powder sample was measured at 310.2 and 313.2°K, that is below and above the Néel temperature. From the difference in the integrated intensity of each line and after correction for thermal diffuse scattering, it was found that in this interval there exists a discontinuity in the Debye temperature of approx. 8°K. This discontinuity is 2.8 times larger than the one calculated from the discontinuity of the elastic constants data.

Lattice Dynamics and Thermodynamic Properties of β-ZnS, GaP and β-SiC

The lattice dynamics of β-ZnS, GaP and β-SiC has been investigated on the basis of a rigid ion model with seven parameters. Results are obtained for the dispersion curves and effective calorimetric and X-ray Debye temperatures and are shown to compare favourably with the available experimental data. Calculated values of equivalent Debye frequencies are also reported.

The X-ray Debye Temperatures of V, Ni, Cu, Nb and Ag from the measured integrated intensities at 300 K and 4 K

Abstract The relative integrated intensities of five high-order reflections for V, Ni, Cu, Nb and Ag powders have been measured at 300 K and 4 K by MoKα radiation. The data were corrected for thermal diffuse scattering. At room temperatures the values 372 ± 6, 417 ± 12, 307 ± 3, 284 ± 6 and 200 ± 3 K were obtained for the Debye Temperatures. At absolute zero the Debye-Waller coefficients were found to be 0·152 ± 0·006, 0·117 ± 0·005, 0·147 ± 0·005, 0·109 ± 0·004 and 0·133 ± 0·004 A2, respectively.

The X-ray debye temperatures of some II—VI compounds

Abstract The x-ray Debye temperatures of ZnSe and CdTe have been measured by two methods. The integrated intensities of Bragg peaks were measured in the temperature range 96–300°k. It was estimated that the ω M values were constant to within 20% and average values for the temperature range were calculated to be 218 ± 6°k for ZnSe and 141 ±5°k for CdTe. The calculation of the slope of the graph of In (F 2/f 2) versus sin2 θ/Δ2 yielded θM = 198 ± 8°k for ZnSe and θ M = 139 ± 10°k for CdTe. The present results are discussed in relation to work on other II—VI compounds.

Influence of the static electron gas screening function on the lattice dynamics of sodium

The static electron gas screening function calculated by Geldart and Taylor has been used in conjunction with the theory of Vosko, Taylor, and Keech to calculate the phonon dispersion curves, the effective calorimetric and X-ray Debye temperatures, and moments of the phonon frequency distribution function of sodium. The results which compare very favorably with those obtained using other screening functions give good agreement with experiment, indicating that the new screening function is reasonably adequate for calculations in sodium.

Mean-square atomic displacement and associated Debye temperature of holmium

The two principal components of the mean-square atomic displacement resulting from thermal vibrations in holmium, 〈uz2〉 and 〈ux2〉, are determined from single-crystal, X-ray diffraction intensities, corrected for thermal diffuse scattering and for extinction effects. The room temperature values of 〈uz2〉 and 〈ux2〉 are found to be 0.01225 ± 0.00063 and 0.0122 ± 0.0021 Å2 respectively. These two components of the average displacement combine to yield an X-ray Debye temperature for holmium of 147 ± 9°K.

Lattice dynamics of III–V compounds

The lattice dynamics of III–V compounds has been investigated on a 'rigid ion' type of model. The interactions are assumed to consist of Coulomb forces with an effective charge and a general force constant field up to and including second neighbors. Results are presented for the dispersion curves, frequency distribution functions, and effective calorimetric and X-ray Debye temperatures for AlSb, GaP, GaAs, and InSb. The agreement with the experimental data, where available, is reasonable.

The X-ray debye temperature of tungsten

The x-ray Debye temperature ( θ M ) of tungsten has been measured to be 377 ± 11°K. The method used was the measurement of the integrated intensity of the {310} and {321} reflections at two temperatures.

X-ray measurement of the Debye temperature for diamond at low temperatures

The X-ray Debye temperature, θ M , was measured at liq. He, liq. N 2 and room temperature for two diamond powders of industrial quality, having a mean particle size of 6μ and 780Å respectively. The results are compatible with θ M ⋍ 1880°K for the fine and θ M ⋍ 1500°K for the coarse powder. A slight decrease of θ M at low temperatures is possible within the experimental errors. Because no use is made of the atomic scattering factor, the claim is not substantiated that previous results around 1500°K necessitate a revision of the scattering factor curve for carbon.

The x-ray debye temperatures of tellurium and cadmium

The Debye temperatures of tellurium and cadmium have been determined to be 139 ± 6°K and 134 ± 4°K respectively by measurement of the x-ray integrated intensity of Bragg reflections from powder specimens in the temperature range 77 to 293°K. A thermal diffuse scattering correction was made. No significant anisotropic effects could be detected even though tellurium and cadmium have non-cubic unit cells.

X-ray Debye temperatures for Al, Nb and Pb

Data from neutron scattering experiments are used to calculate X-ray Debye temperatures for Al, Nb and Pb.

Examination of the Thermal Variation of the Mean Square Atomic Displacements in Zinc and Evaluation of the Associated Debye Temperature

The two principal components of the mean square atomic displacement differences due to thermal vibrations in zinc have been measured from single-crystal x-ray diffraction intensities over a thermal range from 4.85 to 600\ifmmode^\circ\else\textdegree\fi{}K. The absolute value of the components of the mean square atomic displacements parallel to the hexagonal axis and lying in the basal plane have been evaluated from the data by means of a self-consistent technique; these mean square atomic displacements are compared with existing theoretical and experimental values. The x-ray Debye temperatures ${\ensuremath{\Theta}}^{M}(T)$ have been evaluated; ${\ensuremath{\Theta}}^{M}(0)$ is found to be 219.6\ifmmode^\circ\else\textdegree\fi{}K, whereas ${\ensuremath{\Theta}}^{M}(600)$ drops to 187\ifmmode^\circ\else\textdegree\fi{}K. The values of ${\ensuremath{\Theta}}^{M}(T)$ are compared with theoretical calculations, as well as with other experimental results. The thermal variation of ${\ensuremath{\Theta}}^{M}(T)$ appears to be explained by volume-expansion effects over a limited thermal range ($100\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}<T<400\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}$). At higher temperatures, additional anharmonic effects appear to be present. An analysis of measured and calculated values of the temperature derivative of the mean square displacement has led to an evaluation of constant-volume anharmonic effects. Comparison is also made between ${\ensuremath{\Theta}}^{M}(T)$ and Debye temperatures relating to other averages over the frequency-distribution function.

Angular Forces in the Lattice Dynamics of Face-Centered Cubic Metals

The lattice dynamics of the fcc lattice has been investigated using a model in which, in addition to central forces, interatomic forces include angular forces of the type employed by Clark, Gazis, and Wallis. The model has been applied to copper, and results are presented for dispersion curves, vibration spectra, and effective calorimetric and x-ray Debye temperatures.