Einstein and Debye models

In this chapter we examine the energetic contributions of the lattice vibrations. These are the most important, nonchemical, thermal mcdtations and involve motion of the nuclei. Lattice vibrations are quantized. In the same way as photons are, as the respective quasi-particles, equivalent to electromagnetic waves, there are quasiparticles allocated to these elastic waves termed phonons. 

If you bring a soUd from absolute zero to a finite t perature, the atomic constituents of the solid begin to vibrate around their equilibrium positions. In the simplest model, the Einstein model [77), all constituents vibrate with the same fi-e-quency (i ). Increasing the temperature effects an increase in the mean vibration amplitude, but not in i/q. As shown in Section 2.1.6 the expansion of the Mie function yields for small displacements a harmonic potential (see Eq. (2.21)) with spring constant nm / . Hence, the frequency of an oscillator vibrating in this potential is 

Edia is the dissociation energy, that corresponds — if the potential at infinily is taken as zero — to the (negative) minimum of the potential curve (— ) at f (see Fig. 2.3). The quantities cjig and f are related to the bonding parameters A, B, n, m via Eq. (2.19) Mred is the reduced mass. 

A crystal composed of N identical vibrators possesses 3N degrees of freedom if the degrees of freedom related to internal translation and rotation of the particles are neglected and the (six) external degrees of freedom of the total crystal are subtracted, 3N-6 degrees of vibrational freedom remain. In the case of a macroscopic solid 3N-6ci 3N, and the vibration energy of a monatomic soUd is 

Cvib is not the individual vibrator energy of a particular vibration under considerar tion, but rather a mean value because of the fact that not all constituents vibrate 

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First-order estimates of entropy are often based on the observation that heat capacities and thereby entropies of complex compounds often are well represented by summing in stoichiometric proportions the heat capacities or entropies of simpler chemical entities. Latimer [12] used entropies of elements and molecular groups to estimate the entropy of more complex compounds see Spencer for revised tabulated values [13]. Fyfe et al. [14] pointed out a correlation between entropy and molar volume and introduced a simple volume correction factor in their scheme for estimation of the entropy of complex oxides based on the entropy of binary oxides. The latter approach was further developed by Holland [15], who looked into the effect of volume on the vibrational entropy derived from the Einstein and Debye models. 

Even though the Einstein and Debye models are not exact, these simple one-parameter models illustrate the properties of crystals and should give reliable estimates of the volume dependence of the vibrational entropy [15]. The entropy is given by the characteristic vibrational frequency and is thus related to some kind of mean interatomic distance or simpler, the volume of a compound. If the unit cell volume is expanded, the average interatomic distance becomes larger and the... 

Entropies and heat capacities can thus now be calculated using more elaborate models for the vibrational densities of states than the Einstein and Debye models discussed in Chapter 8. We emphasize that the results are only valid in the quasiharmonic approximation and can only be as good as the accuracy of the underlying force-field calculation of such properties can thus be a very sensitive test of interatomic potentials. 

Table 6.3 Eliashberg function and phonon contribution to decay for Einstein and Debye models.

The experimental constant-pressure heat capacity of copper is given together with the Einstein and Debye constant volume heat capacities in Figure 8.12 (recall that the difference between the heat capacity at constant pressure and constant volume is small at low temperatures). The Einstein and Debye temperatures that give the best representation of the experimental heat capacity are e = 244 K and D = 315 K and schematic representations of the resulting density of vibrational modes in the Einstein and Debye approximations are given in the insert to Figure 8.12. The Debye model clearly represents the low-temperature behaviour better than the Einstein model. 

The important message from Einstein or Debye models is that vibrations of atoms in a crystal contribute to Entropy S and to Heat Capacity C therefore they affect the thermodynamic equilibrium of a crystal by modifying both the Eree energy F, which... 

The true specific heat capacity of a composite material was obtained by the mle of mixture and the mass fraction of each phase was determined by the decomposition and mass transfer model. The true specific heat capacity of resin or fiber was derived based on the Einstein or Debye model. The effective specific heat capacity was obtained by assembhng the trae specific heat capacity with the decomposition heat that was also described by the decomposition model. The modeling approach for effective specific heat capacity is useful in capturing the endothermic decomposition of resin and was further verified by a comparison to DSC curves. 

Figures 3.9-11 illustrate the importance of the background-lattice effects in a slightly different way. Shown are various results for the driven Einstein and Debye oscillator models of Sect.3.1. Figure 3.9 illustrates the ratio of the Einstein and Debye energy transfer (equivalently the ratio of the Einstein and Debye accommodation coefficients) as a function of the adiabaticity parameter For...

In several cases Mossbauer studies have revealed a recoil-free fraction with a very flat temperature dependence, which is significantly different from the predictions of either the Einstein or Debye models. Such results were mainly obtained with the Mossbauer probe atom bound within a volume large relative to the atomic size of the probe. They were analysed within a square well potential model, which yields a temperature-independent recoil-free fraction, since in a square well is bound for all states and is dictated by the width of the well, in contrast to the increasing width of the harmonic potential (Nussbaum, 1966). 

The Einstein and Debye crystal models describe vibrations in crystals. 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

The simplest model to describe lattice vibrations is the Einstein model, in which all atoms vibrate as harmonic oscillators with one frequency. A more realistic model is the Debye model. Also in this case the atoms vibrate as harmonic oscillators, but now with a distribution of frequencies which is proportional to o and extends to a maximum called the Debye frequency, (Oq. It is customary to express this frequency as a temperature, the Debye temperature, defined by... 

Table 3.1 Entropy values obtained by application of Einstein (E), Debye (D), and Kieifer (K) models, compared with experimental data at three diiferent temperatures. Data are expressed in J/(mole X K). Values in parentheses are Debye temperatures (d, ) and Einstein temperatures (9 ) adopted in the respective models (from Kieifer, 1985).

Equation (1) may be derived using a variety of microscopic models of the relaxation process. In the derivation of Eq. (1), Debye [1] used the theory of the Brownian motion developed by Einstein and Smoluchowski. Einstein s theory of Brownian motion [2] is based on the notion of a discrete time walk. The walk may be described in simple schematic terms as follows. Consider a two-dimensional lattice then, in discrete time steps of length At, the random walker is assumed to jump to one of its nearest-neighbor sites, displayed, for example [7], on a square lattice with lattice constant Ax, the direction being random. Such a process, which is local both in space and time, can be modeled [7] in the one-dimensional analogue by the master equation... 

The correlation time, in Eq. (4) is generally used in the rotational diffusion model of a liquid, which is concerned with the reorientational motion of a molecule as being impelled by a viscosity-related frictional force (Stokes-Einstein-Debye model). Gierer and Wirtz have introduced the idea of a micro viscosity, The reorientational... 

Cahill and Pohl [8,9] recently developed a hybrid model which has the essence of both the localized oscillators of the Einstein model and coherence of the Debye model. In the Cahill-Pohl model, it was assumed that a solid can be divided into localized regions of size A./2. These localized regions were assumed to vibrate at frequencies equal to to = 2kvsIX where v, is the speed of sound. Such an assumption is characteristic of the Debye model. The mean free time of each oscillator was assumed to be one-half the period of vibration or x = it/to. This implies that the mean free path is equal to the size of the region or XI2. Using these assumptions, they derived the thermal conductivity to be... 

For complex ions of similar size within a homologous series, it is expected that t//tc remains relatively constant for different solvents at a given temperature, in accordance with the Stokes-Einstein-Debye model. From the slope of the plot 5iso versus y/(Az/f/2)/A av shown in Fig. 16, the correlation time constants of a series of tra 5 -[Co(acac)2XY] complexes were estimated and the results were compared favourably with the Tc obtained from the relaxation measurements of the methene carbons (see Table 5). 

The temperature dependence of the resistivity of amorphous non-magnetic metals has been studied by many authors. The most popular theory is still Ziman s theory for the resistivity of liquid metals and several authors have extended this theory to include a dynamic structure factor. Cote and Meisel (1979) used an isotropic Debye spectrum for the phonons to calculate the dynamic structure factor. They obtained a quadratic dependence of the resistivity on temperature at low temperatures. Frobose and Jackie (1977) used both a Debye model and a model of uncorrelated Einstein oscillators in conjunction with a dynamic structure factor to analyse the resistivity. They found that the second model leads to a satisfactory fit for the temperature dependence of the resistivity of CuSn alloys for T 10 K. Ohkawa and Yosida (1977) predict a T ... 

As mentioned, many experimental results have shown that the specific heat for composites increases sHghtly with temperature before decomposition. In some previous models, the specific heat was described as a Hnear function. Theoretically, however, the specific heat capacity for materials wiU change as a function of temperature, as on the micro level, heat is the vibration of the atoms in the lattice. Einstein (1906) and Debye (1912) individually developed models for estimating the contribution of atom vibration to the specific heat capacity of a sohd. The dimensionless heat capacity is defined according to Eq. (4.32) and Eq. (4.33) and illustrated in Figure 4.12 [25] ... 

Figure 4.12 Debye model and Einstein model [12]. (With permission from Elsevier.)...

Diagrammatic representation of diffusion paths observed by numerical simulation (.with permission ) and their correspondence on the conductivity curve (ionic solid (a, Einstein nto el, b, Debye model), a superionic compound (c, dashed line), a viscous liquid (c, solid me) and a Brownian liquid (d). ... 

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