Debye Theory Of Specific Heat Research Articles

The quasi-low temperature behaviour of specific heat.

A new mathematical approach to condensed matter physics, based on the finite temperature field theory, was recently proposed. The field theory is a scale-free formalism; thus, it denies absolute values of thermodynamic temperature and uses dimensionless thermal variables, which are obtained with the group velocities of sound and the interatomic distance. This formalism was previously applied to the specific heat of condensed matter and predicted its fourth power of temperature behaviour at sufficiently low temperatures, which was tested by experimental data for diamond lattice materials. The range of temperatures with the quartic law varies for different materials; therefore, it is called the quasi-low temperature regime. The quasi-low temperature behaviour of specific heat is verified here with experimental data for the fcc lattice materials, silver chloride and lithium iodide. The conjecture that the fourth order behaviour is universal for all condensed matter systems has also supported the data for glassy matter: vitreous silica. This law is long known to hold for the bcc solid helium-4. The characteristic temperatures of the threshold of the quasi-low temperature regime are found for the studied materials. The scaling in the specific heat of condensed matter is expressed by the dimensionless parameter, which is explored with the data for several glasses. The explanation of the correlation of the ‘boson peak’ temperature with the shear velocity is proposed. The critique of the Debye theory of specific heat and the Born–von Karman model of the lattice dynamics is given.

Modified Bose-Einstein and Fermi-Dirac statistics if excitations are localized on an intermediate length scale: Applications to non-Debye specific heat

Disordered systems show deviations from the standard Debye theory of specific heat at low temperatures. These deviations are often attributed to two-level systems of uncertain origin. We find that a source of excess specific heat comes from correlations between quanta of energy if excitations are localized on an intermediate length scale. We use simulations of a simplified Creutz model for a system of Ising-like spins coupled to a thermal bath of Einstein-like oscillators. One feature of this model is that energy is quantized in both the system and its bath, ensuring conservation of energy at every step. Another feature is that the exact entropies of both the system and its bath are known at every step, so that their temperatures can be determined independently. We find that there is a mismatch in canonical temperature between the system and its bath. In addition to the usual finite-size effects in the Bose-Einstein and Fermi-Dirac distributions, if excitations in the heat bath are localized on an intermediate length scale, this mismatch is independent of system size up to at least 10(6) particles. We use a model for correlations between quanta of energy to adjust the statistical distributions and yield a thermodynamically consistent temperature. The model includes a chemical potential for units of energy, as is often used for other types of particles that are quantized and conserved. Experimental evidence for this model comes from its ability to characterize the excess specific heat of imperfect crystals at low temperatures.

Analysis of thermoacoustic wave propagation in elastic media

Wave propagation in a material continuum is a thermodynamic process that is governed, not only by the mechanical character of the propagation medium, but by the thermal character as well. The dispersion relation derived on the basis of linear thermoelastic theory shows that two distinct types of dilatational waves can propagate in an isotropic elastic medium. In each such wave, mechanical and thermal effects are coupled. The behavior of these thermoacoustic waves, in particular, their speeds and absorption coefficients as functions of frequency, is described. Three models of thermoacoustic wave propagation are considered. The classic model, which incorporates the Fourier description of heat conduction, is analyzed first. The shortcomings of this model are pointed out. Two models incorporating the Vernotte hypothesis of heat conduction are examined next—a ‘‘lattice’’ model, which is based upon an idea implicit in the Debye theory of specific heats, and a ‘‘phonon-gas’’ model. Both models incorporating the Vernotte hypothesis are shown to describe linear thermoacoustic wave propagation in a more physically realistic fashion than does the classic model. Thermoacoustic wave propagation is considered from a general phenomenological (i.e., from a macroscopic) standpoint, and only the most general assumptions about the constitution of the propagation medium are made. In the analysis, only the dissipation of energy by heat conduction is considered; the medium is assumed to be elastic in that no purely mechanical mechanisms for energy dissipation (e.g., classic viscosity) are considered to be present.

Dispersion curves for thermomechanical waves

When mechanical deformation and heat conduction are both accounted for, two coupled partial differential equations (PDEs) describe wave propagation in a medium. For one‐dimensional wave propagation in a medium without mechanical dissipation, these PDEs yield a dispersion relation of complex quadratic form, which shows that two kinds of waves can propagate. Each wave has a mechanical and a thermal component. The speeds and the attenuations of such waves are illustrated in two cases. First, it is shown that the “classical” form of the thermomechanical PDEs provides an interesting but nonphysical description of wave propagation. Second, a Vernotte thermal relaxation time is introduced into the PDEs. When the value of the Vernotte parameter is set by applying the Debye theory of specific heats in a rudimentary way, it is shown that a physically plausible description of wave propagation can be obtained. In this latter connection, the consequences of using a “phonon‐gas” model of a thermomechanical medium is considered briefly. [Work supported by ONR.]When mechanical deformation and heat conduction are both accounted for, two coupled partial differential equations (PDEs) describe wave propagation in a medium. For one‐dimensional wave propagation in a medium without mechanical dissipation, these PDEs yield a dispersion relation of complex quadratic form, which shows that two kinds of waves can propagate. Each wave has a mechanical and a thermal component. The speeds and the attenuations of such waves are illustrated in two cases. First, it is shown that the “classical” form of the thermomechanical PDEs provides an interesting but nonphysical description of wave propagation. Second, a Vernotte thermal relaxation time is introduced into the PDEs. When the value of the Vernotte parameter is set by applying the Debye theory of specific heats in a rudimentary way, it is shown that a physically plausible description of wave propagation can be obtained. In this latter connection, the consequences of using a “phonon‐gas” model of a thermomechanical medium is co...

Thermal properties of spinel Nickel-Zinc ferrite in the temperature range 400 to 1000 k

Specific heat capacity (cp), thermal conductivity (λ), and thermal diffusivity (a) coefficients are measured for spinel ferrite system prepared according to the following composition: Ni1.67−xZnxFeSb0.33O4 where x = 0.00, 0.33, and 0.67. The specific heat capacity measurements indicate the presence of a magnetic phase transition from ferrimagnetic to paramagnetic. The Curie temperature decreases as the concentration of Zn ions increases. Below and above the magnetic transition region cp obey's Debye theory of specific heat. For the thermal conductivity coefficient, it is found that the main mechanism of heat transfer is due to phonons in the region below and above the magnetic phase transition, while in the magnetic transition the main mechanism is due to both, phonons and magnons. The role of electrons and phonons is negligible in the measured temperature range.

THE DETERMINATION OF DEBYE CHARACTERISTIC TEMPERATURES OF CRYSTALS FROM X-RAY DIFFRACTION INTENSITIES

The methods of determining Debye characteristic temperatures from X-ray diffraction intensities for the case of homogeneous and isotropic crystals have been fully discussed.It is proposed that if the common logarithms of the ration of the calculated intensities to observed intensities log (Icalc/Iobs) of all diffraction lines are plotted against sin2θ, a straight line should be obtained, the slope of which gives 2Bloge/λ2, where B is a physical quantity to be determined contained in the Debye factor e(-2Bsin2θ/λ2) in the intensity expression, λ being the wave length of the radiation used. In the Debye theory of specific heats, B may be expressed as (6h2T/MkΘD2){Φ(x) + x/4}, where h and k represent Planck constant and Boltzmann constant respectively, M is the mass of the atom or of the group of atoms situated at the lattice points, T is the absolute temperature at the time of taking Debye-Scherrer photographs, and ΘD is the Debye characteristic temperature. X = ΘD/T, and φ(x) is a function of x, given in the original Debye theory. It is seen that if we let G=BMkT/6h2, then φ(x)+x/4=Gx2 Having obtained B, G in this equation is a measurable number, and solution of the equation may be performed graphically. By making Y1=Gx2 and Y2=φ(x)+x/4, the plotting of these two equations should give two curves, the intersection of which should give x which determines the characteristic temperature at that temperature.It is pointed out that owing to the fact that ΘD itself is a function of temperature, the method proposed affords a possibility of determining Debye temperatures at required temperatures.

I1line shape as a temperature probe: The thermal relaxation of highly excited CdS

We demonstrate that analysis of the line shape of the acoustic-phonon wings of the ${I}_{1}$ bound-exciton emission in CdS can provide a means to determine crystal temperature $T$ and use this result to investigate the thermal relaxation of CdS following photoexcitation by an intense ${\mathrm{N}}_{2}$-laser pulse. We find $T\ensuremath{\sim}15$ \ifmmode^\circ\else\textdegree\fi{}K for \ensuremath{\sim}1-kW incident power and $T$ varies as the fourth root of the absorbed energy in accord with the Debye theory of specific heat. In addition the crystal-temperature rise is found to persist for long times (\ensuremath{\sim}1-100 \ensuremath{\mu}sec). From the variation of the crystal temperature with time, we deduce a constant value of heat flow from the sample to the liquid-helium bath (\ensuremath{\sim}4 ${\mathrm{W}/\mathrm{c}\mathrm{m}}^{2}$).

Effect of Dislocation Strain Field on Lattice Specific Heat

The change of the lattice specific heat caused from the strain fields of dislocations has been evaluated on the basis of the higher order elasticity theory. The variations of the lattice vibrational frequencies due to strains were calculated using the quasi-harmonic approximation, and the usual and the higher order mode Gruneisen parameters were expressed as functions of the second-, third-, and fourth-order elastic constants. The displacement fields of an edge and a screw dislocation were also calculated with taking into account the elastic anisotropy of the crystal. These results were combined with the Debye theory of specific heat to obtain the expression for the specific heat change from dislocations. Numerical evaluation has been done for the case of copper, and it was found that the specific heat was increased several per cent at liquid helium temperature by dislocations with density of 10 12 cm -2 and the effect was larger for edge dislocations.

Relationship between Sonic Velocity and Entropy in Molten Salts

Sonic velocity in molten salts can be estimated from absolute molar entropy by means of an equation developed from the Debye theory of specific heats. When tested on 29 molten halide compounds, for which the required data are available, the equation reliably predicts sonic velocity and its negative temperature dependence. When applied to molten alumina, the equation predicts the magnitude of the sonic velocity, but does not yield the correct temperature dependence. For more complex molten salt compounds (such as nitrates and sulfates), a substantial reduction in the molar entropy is necessary for the predicted sonic velocity to agree with the experimental. When entropy contributions, associated with interatomic vibrations and free rotation of the complex ion, are subtracted from the total molar entropy, the equation yields excellent results for these complex molten salt compounds. The nature of the subtracted entropy terms implies that ``gaslike'' degrees of freedom, if present in the liquid, have very little effect on sonic volocities in salt melts containing complex ion species.

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...

Phonon dispersion in rare-gas crystals

A central-force, rigid-atom model has been used to study the lattice dynamics of frozen rare gases. The model assumes central, short-range, two-body forces derivable from a modified Buckingham (exp, 6) potential. The intermolecular potential parameters of rare-gas solids have been calculated using crystal data of the sublimation energy and lattice spacing extrapolated to 0°K. The model takes account of zero-point energy and its volume derivatives by the Debye-theory of specific heats. The contribution of cubic and quartic terms of potential energy expansion to lattice free energy at 0°K has been taken into account by successive approximations. The theoretical frequency distribution and phonon dispersion curves computed on the basis of Born-Karman theory, using the above potential parameters are found to compare reasonably well with the available experimental data.

Dispersion of Phonons in Ideal Crystals

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.

Theoretical Phonon Spectra and Heat Capacities of Solidified Argon and Krypton

AbstractThe phonon spectra of solidified Argon and Krypton have been calculated on the basis of the Born‐Karman theory using the interatomic potential ϕ(r) = ‐Ar−6 + AB × exp (‐r/Q), and taking into account the zero‐point energy and interactions of the four nearest neighbours. The evaluation of the parameters A, B, and Q requires a knowledge of the vibration spectrum of the crystal which is itself a function of these constants. A method of iteration is therefore employed to evaluate the potential parameters, after estimating the frequency‐dependent terms using the Debye theory of specific heats. The static lattice energy is obtained by a method of successive approximations. The specific heats and (θ,T) curves show a fair agreement with the experimental results.

Lattice Vibrations in Graphite

In both natural and artificial graphites the crystallites are reported to be agglomerates of smaller units, each containing perhaps not more than 12 to 17 well-ordered planes. For atomic displacements perpendicular to a plane the restoring forces are much weaker than for similar displacements in the plane itself. For the latter the restoring forces are so strong that these lattice vibrations will not be excited at low temperatures. In this paper the requisite adaptation of the Debye theory of specific heat is given; it leads to a ${T}^{2}$ law instead of the familiar ${T}^{3}$ law at low temperatures. Unpublished measurements of specific heat between 25\ifmmode^\circ\else\textdegree\fi{} and 60\ifmmode^\circ\else\textdegree\fi{}K by Estermann and Kirkland are cited and compared with the theory; they are found to lead to a value ${\ensuremath{\theta}}_{D}={614}^{\ensuremath{\circ}}$ for the Debye temperature for the vibrations with weak restoring forces. Combining this with a value ${\ensuremath{\theta}}_{D}={2100}^{\ensuremath{\circ}}$ for the remaining modes of vibration is found to represent the specific heat over the whole range of temperature.

The Thermal Expansion of Pure Metals: Copper, Gold, Aluminum, Nickel, and Iron

Extremely accurate determinations of the linear thermal expansions have been made interferometrically from --- 196\ifmmode^\circ\else\textdegree\fi{}C to temperatures about +400\ifmmode^\circ\else\textdegree\fi{}C for Al and +700\ifmmode^\circ\else\textdegree\fi{}C for Fe, Ni, Cu and Au. The relationship between true thermal coefficient of expansion and temperature conforms very well to the Grueneisen-Debye theory when values are chosen for the Debye characteristic temperatures which turn out to agree well with those chosen to achieve agreement with the Debye theory of specific heats. Our values for these characteristic temperatures are: 410\ifmmode^\circ\else\textdegree\fi{}K for Ni, 420\ifmmode^\circ\else\textdegree\fi{}K for Fe, 400\ifmmode^\circ\else\textdegree\fi{}K for Al, 325\ifmmode^\circ\else\textdegree\fi{}K for Cu, and 190\ifmmode^\circ\else\textdegree\fi{}K for Au. The magnetic Curie temperature for Ni is found to be 352\ifmmode^\circ\else\textdegree\fi{}C. In plotting true coefficient of thermal expansion versus temperature Simon and Bergmann found a horizontal plateau at about 175\ifmmode^\circ\else\textdegree\fi{} to 235\ifmmode^\circ\else\textdegree\fi{}K for Ni and Fe; but we do not confirm this.

Effect of Temperature on the Intensity of X-ray Reflections from Copper

IN a recent letter in NATURE1, Prof. E. A. Owen and Mr. R. W. Williams have given results for the effect of temperature on the intensity of X-ray reflections from copper. They find that throughout the range of temperature from 290° to 840° A., the decline of intensity with temperature is greater than predicted by the usual Debye-Waller formula, exp.—2M, where M has the well-known expression involving the characteristic temperature ⊝ which occurs in the Debye theory of specific heats, and that their results agree closely with a temperature factor of the form exp.—3M.

Contributions to the theory of the specific heat of crystals II—On the vibrational spectrum of cubical lattices and its application to the specific heat of crystals

1—An investigation of the features of the vibrational spectrum of a cubical crystal has been made by means of a geometrical method described below; the two-dimensional case has been treated in detail, the three-dimensional case in outline. The main result has been the discovery of a number of maxima of the density of the vibrations. The significance of the results is discussed especially in relation to the Debye Theory of specific heats. 2—One-Dimensional Theory The density of the vibrations for a linear lattice containing two types of particles has been worked out in Part I. The density curves are shown in fig. 1, the mass difference causing the appearance of two additional maxima; it is as well to emphasize here that it is really the number of vibrations in the immediate neighbourhood of the maximum—what one might term the "weight" of the maximum—which is important, rather than the fact that the density becomes infinite. For instance, one obtains the first two maxima for very small mass differences, where one can hardly expect a perceptible difference in any physical property as compared with the case of equal masses.