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Specific heat of iron-vanadium alloys between 125 and 625°K
Abstract
The specific heat of iron-vanadium alloys with 8–45 at.% iron was measured between 125°K and 625°K in an adiabatic calorimeter. The total specific heat was separated into a lattice contribution, a dilatation correction, and a term linear in temperature. The coefficient of the linear term was associated with the electronic specific heat coefficient, γ, and compared with results of specific heat measurements at liquid helium temperatures. These agree for the alloys with up to 33 at.% iron, but differ considerably for the alloys with higher iron concentration.A similar analysis of enthalpy measurements made on these alloys from 800 to 1200°K yields γ values which agree with the liquid helium results for all alloy compositions.It is concluded that the interpretation of the liquid helium measurements is essentially correct and that between 100°K and 700°K there are magnetic contributions to the specific heat of the alloys of 35 and 45 at.% iron.
The impurity induced infrared absorption in CsCl and CsI is calculated and compared with experiment. The perfect crystal is described by a breathing shell model the density of states and the Green functions of the perfect crystal are calculated using 27000 points in the Brillouin zone. The impurities are described by models involving changed central and noncentral force constants between the defect and its nearest neighbours, and between the nearest neighbours themselves. Fitting the experimental data yields not only the parameters necessary to describe the defects but also suggests improved parameters for describing the perfect crystals.
The multiphonon absorption spectra of MgO and CaO single crystals have been recorded, and analysed in terms of the infrared active sum modes of lattice vibrations. A critical point analysis was carried out using the dispersion curves obtained from previous neutron scattering studies and the 'breathing shell' model, and it produced similar mode combination patterns for both substances. The two main peaks I and II (846.5 cm-1 and 981 cm-1 for MgO; 628 cm-1 and 738 cm-1 for CaO) in the two- phonon region are interpreted as the sum modes W'2+W"3 at W and Sigma 1LO+ Sigma 3TO near X respectively, in both crystals. Three- phonon bands were observed for both MgO and CaO and a four- phonon band in MgO, all at wavenumbers in agreement with theoretical predictions.
The quasiharmonic and lowest order anharmonic contributions to the Helmholtz free energy of sodium chloride have been calculated. A breathing shell model was used in the quasiharmonic calculation, and Coulomb and polarization contributions to the anharmonic matrix elements were included more completely than has been done before. The thermodynamic properties derived from the free energy are in good agreement with experimental results. The inclusion of Coulomb anharmonicity was particularly important, reducing the magnitude of the second order anharmonic contribution by 40%. The quasiharmonic calculation gives good agreement with experiment up to one quarter of the melting temperature, and the lowest order anharmonic theory appears to be adequate up to at least two thirds of the melting temperature.
The phonon dispersion curves of strontium fluoride in the (001), (110) and (111) directions were observed by inelastic neutron scattering on a triple axis spectrometer. These observations, and all other data relevant to the vibrational properties, were fitted by the 'shell' model for lattice dynamics. Some harmonic vibrational properties were calculated.
A consistent set of shell parameters for alkali halide crystals of both rocksalt and CsCl structures is reported. The fitting procedure, values of the parameters and their sensitivity are discussed. Short range interionic potentials are determined and are shown to give a good description of both static and dynamic properties of the crystals.
Phonon frequencies in RbBr at 80 K have been obtained by inelastic scattering of thermal neutrons. The measurements were performed in the symmetry directions Delta , Sigma and Lambda and along the line Z. Some zone boundary phonons were also measured at 300 K. Four versions of the shell model were fitted to the experimental frequencies, and the elastic and dielectric constants were calculated from the model parameters. From an eleven parameter model the authors have also calculated the phonon frequency spectrum and the temperature variation of the Debye temperature.
The lowest order anharmonic contributions to the one-phonon Green functions for long wavelength phonons, and to the dielectric properties, have been calculated for sodium chloride. A breathing shell model was used to provide frequencies and eigenvectors. Coulomb contributions to the anharmonic matrix elements were included, and the volume dependence of all contributions was treated exactly. The phonon shifts and the static dielectric constant are in good agreement with experiment up to about three quarters of the melting temperature. The lowest order contribution to the width is about two thirds of the experimental value and, when the volume dependence is properly incorporated, varies with temperature as T1.6 in good agreement with experiment.
Some new experimental data (optical measurement and elastic constant determination) have shown discrepancies with model calculations, in LiH and LiD crystals. Using the classical shell model, the author has shown that second-neighbour H-H interactions are very important in this crystal. In order to correct the frequencies, particularly on the TO branches in the (1, 1, 1) direction, shell deformation effects have been introduced into the model and it has been shown that there exist no short-range forces able to improve the calculation. It is then necessary to introduce long-range three-body forces to explain all the dynamic properties of these two crystals.
The authors have developed new potentials describing the short-range interactions between ions in the alkali halides. They have used the measured elastic constants to fix the near-neighbour overlap repulsion and study the second-neighbour potentials. The description of the next-neighbour interactions requires the inclusion of large van der Waals attractions, in general accord with latest estimates of these forces. They have used theoretical estimates of the repulsive part of the second-neighbour interaction and added a variable attractive part to fit the crystal data. The authors have first used a conventional Buckingham potential, but a more flexible form is required to provide a completely coherent picture of these interactions in the whole series of alkali halides.
Pair interaction potentials for NaI are obtained by fitting to solid state data at 0K and to some properties of the diatomic molecule. In order to examine the effects of polarization on properties of molten NaI, molecular dynamics simulations are performed with rigid ion and shell model potentials the shell model potentials being fitted to the same data as are used for the rigid ion potential with the additional parameters being determined from dielectric constant data. Two shell models are considered; one is fitted to diatomic molecule properties and the other is more in line with empirical ionic radii. Simulations with 216 ions have been carried out for these potentials and the following results are presented: radial distribution functions velocity and force autocorrelation functions and mean square displacements.
Measurements of the inelastic neutron scattering by longitudinal optic phonons in MgO are reported. These results and those previously obtained for other branches are used in an investigation of various lattice dynamical models, including shell and breathing shell models. It is shown that the agreement with the measured data is improved by the inclusion of breathing effects.
The authors discuss the nature of deformation of the electron charge cloud of an ion implied in the deformable shell model and considers an application of the model to the case of LiF crystals. It is found that the properties of the crystal are fairly well reproduced by the present model and certain discrepancies noted by earlier authors are removed. The results of other theoretical investigations are also included for comparison.
Phonon dispersion relationships have been determined for the acoustic branches in MnO along the [100], [011] and [111] directions by neutron spectrometry. The measurements have been interpreted in terms of the rigid-ion and the shell model. The rigid-ion model does not fit the data adequately and the shell model, while producing a much better fit to the data, cannot be said to give a satisfactory description.
Extensive neutron inelastic scattering measurements of the frequencies of normal modes in strontium titanate propagating in the crystallographic directions (0, 0, zeta ), ( zeta , zeta , 0), ( zeta , zeta , zeta ), (1/2, 1/2, zeta ) and ( zeta , zeta , 1/2) are reported. The temperature dependence of the 'soft' modes at points Gamma and R have been investigated with particular emphasis on the mode of symmetry Delta 2, propagating in the direction (1/2, 1/2, zeta ); quantitative estimates of the isotropy of the dispersion surface at Gamma and R are given. Rigid ion, rigid shell, and deformable shell models of the crystal dynamics have been developed. Based on these, the frequency distribution, specific heat, one-phonon X ray scattering intensity. Debye-Waller factor, and two-phonon Raman spectrum have been calculated; where possible, these are compared with the available experimental measurements.
For pt.I see ibid., vol.11, p.1523 (1978). A set of interionic potentials of the Born-Mayer form is derived for the alkali halides. The specification is completely in terms of parameters of two types, (a) parameters which relate to the constituent ions and (b) parameters which apply to all crystals. Thus the parametrisation scheme contains no crystal dependent terms and is of a form which may readily be applied to systems of mixed alkali halides. Each of the nine ions is described by five parameters, its polarisability, its shell charge and three parameters which relate to the ion size. The van der Waals coefficients are calculated consistently from the ion polarisabilities. The model system, which has 47 free parameters, reproduces the lattice constants, high and low frequency dielectric constants and transverse optic frequencies for the 20 crystals to high accuracy. Of crystal properties which are not included in the fitting procedure, the cohesive energies are given in most cases to better than 1%, the NaCl/CsCl structural relative stability is correctly described in all cases except CsF, and reasonable phonon dispersion relations and elastic constants are obtained.
The two phonon Raman spectrum of KBr is calculated. The calculation is based on the shell model and assumes a nonlinear coupling between the shell and the core of the negative ions. It is found that the results are considerably improved by using a breathing shell model rather that a rigid shell model. The theory then contains three parameters which may be chosen to give a reasonable description of the experimental results. An investigation of the importance of inter-ionic nonlinearities suggests that they are far less important for describing Raman scattering than are the intra-ionic terms.
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