Model Debye

Using the Planck distribution, the energy of an ensemble of oscillators can be found by 

Now the trick is to find the appropriate C( ). We start by finding the distribution in fc-space and use the relation C( a)d = W(fc)dfc to obtain C( a). To get W(fc), first take the number of states on the surface of a sphere in fc-space, which can be foimd by dividing the volume of a shell in fc-space, 4Tifc dfc, by the volume of an individual state, which we found from Equation 16.9 to be (2ti/L), and multiply by three polarization states to get 

To compute dfc/dw, we should use the dispersion relation developed in Chapter 16 for the chain of atoms. From Equation 16.7 

Here is where we run into a problem. As we saw before, it is necessary for 

Debye dodges this little problem by using the dispersion relation for a homogeneous solid which has no cutoff frequency and in which k = (ojvo and dk/do) = 1 /vq. 

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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... 

Low-temperature behaviour. In the Debye model, when T < 0q, the upper limit, can be approximately... 

This result is called the Debye law. Figure A2.2.4 compares the experimental and Debye model values for... 

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. 

Low-temperature behaviour. In the Debye model, when T upper limit, can be approximately replaced by co, die integral over v then has a value 7t /15 and the total phonon energy reduces to... 

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... 

Its value at 25°C is 0.71 J/(g-°C) (0.17 cal/(g-°C)) (95,147). Discontinuities in the temperature dependence of the heat capacity have been attributed to stmctural changes, eg, crystaUi2ation and annealing effects, in the glass. The heat capacity varies weakly with OH content. Increasing the OH level from 0.0003 to 0.12 wt % reduces the heat capacity by approximately 0.5% at 300 K and by 1.6% at 700 K (148). The low temperature (<10 K) heat capacities of vitreous siUca tend to be higher than the values predicted by the Debye model (149). 

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). 

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

Table A4.7 summarizes the thermodynamics properties of monatomic solids as calculated by the Debye model. The values are expressed in terms of d/T, where d is the Debye temperature. See Section 10.8 for details of the calculations. Tables A4.5 to A4.7 are adapted from K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 1995.

This procedure is applicable if the relaxation between the spin states is fast (t<1 X 10 s) and thus the quadrupole doublets of the two spin states collapse into one. It is found that, in the intermediate temperature range, the widths of the two lines are significantly enlarged. This shows that the assumption of fast relaxation strictly does not apply. In spite of this, the areas of the lines ean be well reproduced within the Debye model employing the same Debye temperature for both spin states, p 123 K. 

The temperature dependence of sod is related to that of the recoil-free fraction /(T) = Qxp[— x )Ey / Hc) ], where (x ) is the mean square displacement (2.14). Both quantities, (x ) and can be derived from the Debye model for the energy distribution of phonons in a solid (see Sect. 2.4). The second-order Doppler shift is thereby given as [20]... 

Fig. 4.2 Temperature dependence of the isomer shift due to the second-order Doppler shift, sod- The curves are calculated for different Mossbauer temperatures 0m by using the Debye model whereby the isomer shift was set to (5 = 0.4 mm s and the effective mass to Meff =100 Da, except for the dashed curve with Meff = 57 Da...

Several types of spin-lattice relaxation processes have been described in the literature [31]. Here a brief overview of some of the most important ones is given. The simplest spin-lattice process is the direct process in which a spin transition is accompanied by the creation or annihilation of a single phonon such that the electronic spin transition energy, A, is exchanged by the phonon energy, hcoq. Using the Debye model for the phonon spectrum, one finds for k T A that... 

With the observed temperature shift data for (dSldT)p and calculated (within the framework of the Debye model) numbers for the temperature shift of SOD and with the known thermal expansion coefficient as well as results from Ta Mossbauer experiments under pressure, the authors [191] were able to evaluate the true temperature dependence of the isomer shift, (dSisIdT) as —33 10 " and —26 10 " mm s degree for Ta and W host metal, respectively. 

Fig. 9.6 Measured NFS spectra of deoxymyoglobin at the indicated temperatures. The solid lines are the simulations obtained with SYNFOS [13, 14] using the Debye model for the effective thickness as described in the text. Taken from [15]...

Using the Debye model (9.5) for the temperature dependence of feff (with 0D = 215 K)... 

NFS spectra of the molecular glass former ferrocene/dibutylphthalate (FC/DBP) recorded at 170 and 202 K are shown in Fig. 9.12a [31]. It is clear that the pattern of the dynamical beats changes drastically within this relatively narrow temperature range. The analysis of these and other NFS spectra between 100 and 200 K provides/factors, the temperature dependence of which is shown in Fig. 9.12b [31]. Up to about 150 K,/(T) follows the high-temperature approximation of the Debye model (straight line within the log scale in Fig. 9.12b), yielding a Debye tempera-ture 6x) = 41 K. For higher temperatures, a square-root term / v/(r, - T)/T,... 

Fig. 9.12 (a) NFS spectra of FC/DBP with quantum beat and dynamical beat pattern, (b) Temperature-dependent /-factor. The solid line is a fit using the Debye model with 0D = 41 K below 150 K. Above, a square-root term / - V(Tc - T)/Tc was added to account for the drastic decrease of /. At Tc = 202 K the glass-to-liquid transition occurs. (Taken Ifom [31])... 

In real systems, there will always be a distribution of the relaxation time meaning that Equation 5.3 should more adequately be written as given in Ref. [8] (generalized Debye model) ... 

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. 

Although the Debye model reproduces the essential features of the low- and high-temperature behaviour of crystals, the model has its limitations. A temperature-dependent Debye temperature, d(F), can be calculated by reproducing the heat capacity at each single temperature using the equation... 

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. 

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... 

Here 0O is the characteristic temperature at volume V0. An average value for the volume dependence of the standard entropy at 298 K for around 60 oxides based on the Einstein model is 1.1 0.1 J K-1 cm-3 [15]. A corresponding analysis using the Debye model gives approximately the same numeric value. 

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. 

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