Lattice vibrations Debye approximation

Dulong-Petit limit can be reproduced. In the Debye model, lattice vibrations are approximated as a continuous elastic body considering only acoustic modes, and therefore, phonon spectrum is treated linearly. In... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

As for graphite, its zero-point energy, ZPE = R6 + jR0 , is most conveniently deduced from Debye s theory [197,198] by separating the lattice vibrations into two approximately independent parts, with Debye temperatures (in plane) and 6j (perpendicular). A balanced evaluation gives ZPE 3.68 kcal/mol [199]. 

Comment on the choice of representative values of Vj for the 12 vibrational modes of the crystal. How much would reasonable changes (say, 10 to 20 percent) in these values affect the results of the calculations If possible, conunent on the effect of using the Debye approximation for the acoustic lattice modes instead of the Einstein approximation. 

The free energy due to harmonic lattice vibrations (or equivalently the Debye temperature) is approximately the same for bcc, fee, and hep structures but with a significant tendency for the bcc value to be a few percent lower. The more open bcc structure has a transverse phonon mode with a particularly low frequency which causes a more rapid decrease in the free energy with temperature. On cooling, sodium and lithium transform partially from bcc to hep at very low temperatures (0.1-0.2 Tm). Calcium, strontium, beryllium, and thallium transform to a bcc phase at high temperatures (0.66-0.98 Tm) when there is a considerable anharmonic contribution to the free energy. 

The heat capacity and transition enthalpy data required to evaluate Sm T ) using Eq. 6.2.2 come from calorimetry. The calorimeter can be cooled to about 10 K with liquid hydrogen, but it is difficult to make measurements below this temperature. Statistical mechanical theory may be used to approximate the part of the integral in Eq. 6.2.2 between zero kelvins and the lowest temperature at which a value of Cp,m can be measured. The appropriate formula for nonmagnetic nonmetals comes from the Debye theory for the lattice vibration of a monatomic crystal. This theory predicts that at low temperatures (from 0 K to about 30 K), the molar heat capacity at constant volume is proportional to Cv,m = aT, ... 

The Stockmayer-Hecht-model deserves to be valued in consideration of the fact that the approximation methods of Debye and Tarasov (Section II.4.4) to estimate the lattice vibration ectrum and the heat capacity of polymers are useful only to a limited extent. The validity of the Debye approximation for pdl3nners is confined to a temperature interval of about 0 10° K. This is due to the fact that, at... 

As a good approximation it is assumed, that the adsorbed species are vibrating in resonance with the lattice phonon vibrations of the solid stationary phase. The phonon frequency can be evaluated from phonon spectra, from the standard entropy of solid metals, from the Debye temperatures or from the Lindemann equation [9]. 

The lattice component was calculated using the harmonic approximation, in which all the acoustic and low-frequency optical vibrations are included with the help of a single Debye fimction, while high-frequency crystal vibrations are taken into accoimt by Einstein s equation. According to Kelley s derivations (Gurvich et al., 1978-1984) based on the Born-von Karman d)mamic crystal lattice theory, we therefore have... 

The problem of behavior of the heat capacity of solids near 1°K has been treated by Einstein and by Debye to include the effects of vibration. For undergraduate treatment, it is sufficient to say that near 1°K heat capacities of lattices vary roughly as [2] so the heat capacity curve increases after TK. Other texts and monographs should be consulted for studies of materials at very low temperatures, but here the essential facts are that 5 = / In (HO gives an approximate value for low-temperature entropy due to isotope impurities and that the heat capacity varies as roughly in the 1°K range. There would be a constant A characteristic of the material and then Cp AT. As usual there are alternate verbal descriptions of the third law of thermodynamics but our summary would be... 

Debye s model gives only an approximate deseription of the vibrational properties of real solids, espeeially for solids eontaining different atoms or having certain lattice structures. However, it is very convenient in situations where an analytical expression for the distribution functions is necessary, because more rigorous models give analytical solutions for one- or two-dimensional systems only. Since the spectra of surface vibrations are much more complicated, this model is often used. There are also some empirical combinations of Debye and Einstein distributions (in the classical limit) ... 

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