The Debye model

The limiting angular vibrational frequency, D, that exists defines the Debye temperature, 0D, as 

Whereas the Debye heat capacity is equal to 3R at high temperatures, it reduces to 

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 

A dip in d versus temperature is typically observed for d/50 77K d/2. It follows that a constant Debye temperature is not able to reproduce the experimental observations over large temperature ranges. 

A somewhat more quantitative discussion of the behavior of the specific heat as a function of the temperature is afforded by the Debye model. In this model we assume all frequencies to be linear in the wave-vector magnitude, as acoustic modes near k 0 are. For simplicity we assume we are dealing with an isotropic solid, so we can take co = vk for all the different acoustic branches, with v, the sound velocity, being the same in all directions. The total number of phonon modes is equal to 3Nat, where Nat is the total number of atoms in the crystal, and for each value of k there are 3v normal modes in 3D (v being the number of atoms in 

With these definitions, we obtain the following expression for the specific heat at any temperature  

The physical meaning of the Debye temperature is somewhat analogous to that of the Fermi level for electrons. For T above the Debye temperature all phonon modes 

Within the Debye model, the density of phonon modes at frequency co per unit volume of the crystal, g co), can be easily obtained starting with the general expression 

Values of d (in K) and coo (in THz) were determined by fitting the observed value of the specific heat at a certain temperature to half the value of the Dulong-Petit law through Eq. (6.62). 

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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... 

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])... 

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. 

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. 

The second term in (5-4) is the second-order Doppler shift. This is the higher-order term of the Taylor expansion that we ignored in (5-3). Like , it can be calculated in the Debye model. Figure 5.6 shows plots of the second-order Doppler shift for the case of iron and for different values of the Debye temperature. Soft lattice vibrations are expected to decrease the isomer shift, although the effect becomes only significant at temperatures well above 80 K. 

Figure 5.6 Isomer shifts (relative to sodium nitroprusside) as a function of temperature in the Debye model, along with measurements on a-FeOOH (adapted from [11]).

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... 

The Debye temperature characterizes the rigidity of the lattice it is high for a rigid lattice but low for a lattice with soft vibrational modes. The mean squared displacement of the atom, , can be calculated in the Debye model and depends on the mass of the vibrating atom, the temperature and the Debye temperature. 

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

This approximation requires that cos. This behavior in fact follows from a Debye dielectric continuum model of the solvent when it is coupled to the solute nuclear motion [21,22] and then xs would be proportional to the longitudinal dielectric relaxation time of the solvent indeed, in the context of time dependent fluorescence (TDF), the Debye model leads to such an exponential dependence of the analogue... 

In the Debye model, the Brillouin zone (see section 3.3) is replaced by a sphere of the same volume in the reciprocal space (cf eq. 3.57) ... 

The heat capacity at constant volume in the Debye model is given by... 

The Debye model assumes that there is a single acoustic branch, the frequency of which increases with constant slope (proportional to the average velocity of sound in the crystal) as q increases, up to the boundary of the Brillouin zone. The boundary is assumed to be of spherical shape, with a radius qD determined by the total number of normal modes of the crystal. Thus,... 

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