Debye, characteristic temperature

Thermal Properties at Low Temperatures For sohds, the Debye model developed with the aid of statistical mechanics and quantum theoiy gives a satisfactoiy representation of the specific heat with temperature. Procedures for calculating values of d, ihe Debye characteristic temperature, using either elastic constants, the compressibility, the melting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ed., Oxford University Press, 1985, pp 24-29). 

In this equation v is a phonon frequency, such that hv is approximately k, with the Debye characteristic temperature of the metal. The quantity p is the product of the density of electrons in energy at the Fermi surface, N(0), and the electron-phonon interaction energy, V. 

Recent work has shown 103) that for alkali halides the Debye characteristic temperature is not very sensitive to volume changes produced by thermal expansion. This probably indicates that, in general, volume changes of an adsorbent will not markedly affect its bulk thermodynamic properties. 

The measured values of the thermal expansion coefficient were used for calculation of the Debye characteristic temperature (0o) and root-mean-square... 

Problem Calculate the value of Cv for any element when its temperature is equal to the Debye characteristic temperature B. 

It may be mentioned that the Debye characteristic temperature can be derived from other properties of the element, particularly from the compressibility and Poisson s (elasticity) ratio. Where such data are available it is thus possible to obtain reasonably accurate heat capacities, at moderate and high temperatures, from elasticity measurements. 

Using the Debye curve obtained in the preceding exercise, and the values of Cv for aluminum given below, determine the (mean) Debye characteristic temperature for this element. 

The Debye characteristic temperature of silver is 212. Calculate the atomic heat capacity Cv of this metal at 20.0° K and 300° K. 

Show that if the Debye heat capacity equation were applicable, the entropy of a perfect solid at very low temperatures should be equal to JCp, where Cp is the heat capacity at the given temperature. What would be the value in terms of the Debye characteristic temperature ... 

Debye characteristic temperature. A parameter relating to the lattice specific heat of a solid. The temperature at which the specific heat of a simple specific cubic crystal equals 5.67 calories per degree per mole. 

Whilst it is clear that any simple Debye-like treatment of lattice vibrations in ice is inadequate, it is useful for some semi-quantitative purposes to describe the state of the crystal in terms of an equivalent Debye characteristic temperature which will itself vary with the temperature of the crystal. If intramolecular motions and librations are neglected, then the crystal has 3iV degrees of... 

The Debye characteristic temperatures 6jy (K) of R2C3 have been given, based on the neutron scattering temperature factor coefficients of R2C3 at 296 K, to be 247 K... 

In addition, the connection between the characteristic temperature and the elastic constants and the heat of atomization can be established directly by the Debye method. We note, for example, that the Debye characteristic temperature is directly proportional to the average velocity of elastic waves in a crystal (0= kc) and it is proportional to the square root of the atomization energy per unit atomic weight, at least in the first approximation (0= kVU/A). 

The Debye characteristic temperature of isotropic bodies is directly associated with the bulk modulus and therefore with all the other properties of matter dependent, to a certain degree, on the atomization energy, the surface energy u, the elasticity moduli, the expansion coefficients, the width of the forbidden band in semiconductors, etc. 

As we have previously shown [5], there is also a definite relationship between the Debye characteristic temperature and the heat of activation of diffusion U, etc. However, an analysis of the experimental curves of the specific heat as a function of temperature shows that, for all bodies, including those whose specific heat is more or less satisfactorily described by the Debye equation, the characteristic temperature of the whole crystal is essentially a function of temperature (Fig.3). In addition, it is observed that the greater the deviation of the curve of from the horizontal straight line, the more the temperature dependence of the specific heat deviates from the Debye law. Despite the fact that the temperature dependence of the specific heat is comparatively insensitive to the form of the phonon spectrum, an evaluation of the trend of the specific heat curves already indicates that the vibration spectrum of ion vibrations in real solids differs essentially from the Debye law. 

The Debye characteristic temperature may be the most useful single parameter for characterizing various solid-state phenomena. It can be computed simply and accurately from elastic constants [i... 

Total thermal conductivity is a sum of the lattice and electronic parts, K = Ki + Ke- The lattice part of the thermal conductivity describes the scattering of phonons on the vibrations of atoms, whereas the electronic part describes thermal conductivity appearing due to conduction electrons and is related to the electrical conductivity Wiedemann-Franz equation, = a T Lo, where T is the absolute temperature and Lq is the ideal Lorenz number, 2.45 X 10 Wf2K [64]. The electronic part of the thermal conductivity is typically low for low-gap semiconductors. For the tin-based cationic clathrates it was calculated to contribute less than 1% to the total thermal conductivity. The lattice part of the thermal conductivity can be estimated based on the Debye equation /Cl = 1 /3(CvAvj), where C is the volumetric heat capacity, X is the mean free path of phonons and is the velocity of sound [64]. The latter is related to the Debye characteristic temperature 6 as Vs = [67t (7V/F)] . Extracting the... 

The quantity 2) the dimensions of a temperature and is called the Debye characteristic temperature of the particular substance. According to (13 59) Cyis the same function of the ratio Tjdj for all substances, and the heat capacity per mole may therefore be written... 

TABLE 11 Debye characteristic temperature, Einstein oscillator frequencies, and a for YCI3, YbCl3, and LUCI3... 

The example of experimental data processing for LuCla from (Brunetti et al., 2005) considered above substantiates our views on the optimum selection of the Debye characteristic temperature and Einstein characteristic frequencies used to calculate the temperature dependences of heat capacities. 

Tables 1.8 and 1.9 and Fig. 1.21 give some reference data on the values of the thermal coefficient of linear expansion for oxides, refractory, and ceramic materials [100-102]. Crystals with a cubic lattice (CaO, MgO) have equal values of linear coefficients of expansion along aU axes. The typical linear coefficients of thermal expansion for such materials are 6-8 x 10 and increase with the temperature up to 10-15 X 10 K . Anisotropic crystals with low symmetry have different values of linear coefficients of thermal expansion along different axes, but with a temperature increase, this difference becomes smaller. Materials with strong chemical bonds (silicon carbide, titanium diboride, diamond) have low values of linear coefficients of thermal expansion. However, these materials have high values of Debye characteristic temperature (values of the linear coefficients of thermal expansion grow below the Debye temperature and are almost constant above it).

Debye cut-off frequency Maximum possible frequency for a vibrational wave in a crystal the Debye characteristic temperature is the Debye cut-off frequency multiplied by Planck s constant and divided by Boltzmann s constant. 

FIGURE 5 The molar specific heat Cy of an insulating solid, according to Debye theory, plotted as a function of T/Oq where T is the absolute temperature and 6o is the Debye characteristic temperature for any given material (see Table I). 

TABLE I Debye Characteristic Temperatures 6 for Some Common Eiements ... 

Table 3.3. Debye Characteristic Temperatures of Some Selected Elements and Compounds (in Kelvins at T 0 /2)... 

Debye characteristic temperatures of polytypes, 0d, obtained from the a temperature dependences, are shown in Table 1. 

---