Debye frequency calculation

An observation of motion of single atoms and single atomic clusters with STEM was reported by Isaacson et al,192 They observed atomic jumps of single uranium atoms on a very thin carbon film of —15 A thickness or less. Coupled motion of two to three atoms could also be seen. As the temperature of the thin film could not be controlled, no Arrhenius plot could be obtained. Instead, the Debye frequency , kTIh, was used to calculate the activation energy of surface diffusion, as is also sometimes done in field ion microscopy. That the atomic jumps were not induced by electron bombardment was checked by observing the atomic hopping frequencies as a function of the electron beam intensity. 

The values of djy which bring the experimental data into agreement with the Debye theory are usually about 100-400 K, corresponding to a frequency calculated from (13 66), of 2 x 10 to 8 x 10 . On the other hand, those substances which approach the Dulong and Petit value onjy at elevated temperatures have a much larger value of 0JD, Diamond, for example, has a value of 1860 K, corresponding to a classical frequency of 4 x 10 . 

V is the volume, and F is a factor of proportionality, which is calculable from the elastic properties of the solid. The connection with elasticity was in fact suspected by Sutherland in 1910 (Phil, May., 20, 657), who found that the infra-red frequency of a solid was of the same order as the frequency of an elastic transversal vibration with a wave length equal to the distance between two neighbouring atoms. To every degree of freedom Debye assigns an amount of energy ... 

Both the Einstein and Debye theories show a clear relationship between apparently unrelated properties heat capacity and elastic properties. The Einstein temperature for copper is 244 K and corresponds to a vibrational frequency of 32 THz. Assuming that the elastic properties are due to the sum of the forces acting between two atoms this frequency can be calculated from the Young s modulus of copper, E = 13 x 1010 N m-2. The force constant K is obtained by dividing E by the number of atoms in a plane per m2 and by the distance between two neighbouring planes of atoms. K thus obtained is 14.4 N m-1 and the Einstein frequency, obtained using the mass of a copper atom into account, 18 THz, is in reasonable agreement with that deduced from the calorimetric Einstein temperature. 

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. 

However, the obscure choice of frequencies in the visible and UV regions in the original calculations may have been guided by a desire to fit experimental heats. In fact, the Debye rotational and translational crystal frequencies relate to sublimation energies of the lattice, and, together with internal molecular vibrations, can be used to calculate thermodynamic functions (16). An indirect connection between maximum lattice frequencies (vm) and heats of formation may hold because the former is inversely related to interatomic dimensions (see Section IV,D,1) ... 

To appreciate the predictive properties of Kieffer s model, it is sufficient to compare calculated and experimental entropy values for several phases of geochemical interest in table 3.1, which also lists entropy values obtained through apphcation of Debye s and Einstein s models. One advantage of Kieffer s model with respect to the two preceding formulations is its wider T range of applicability (Debye s model is appropriate to low frequencies and hence to low T, whereas Einstein s model is appropriate to high frequencies and hence to high T). 

The preceding analysis provides a powerful method for determining the diffusivities of species that produce an anelastic relaxation, such as the split-dumbbell interstitial point defects. A torsional pendulum can be used to find the frequency, u>p, corresponding to the Debye peak. The relaxation time is then calculated using the relation r = 1/ojp, and the diffusivity is obtained from the known relationships among the relaxation time, the jump frequency, and the diffusivity. For the split-dumbbell interstitials, the relaxation time is related to the jump frequency by Eq. 8.63, and the expression for the diffusivity (i.e., D = ra2/12), is derived in Exercise 8.6. Therefore, D = a2/18r. This method has been used to determine the diffusivities of a wide variety of interstitial species, particularly at low temperatures, where the jump frequency is low but still measurable through use of a torsion pendulum. A particularly important example is the determination of the diffusivity of C in b.c.c. Fe, which is taken up in Exercise 8.22. 

The concentration dependence of ionic mobility at high ion concentrations and also in the melt is still an unsolved problem. A mode coupling theory of ionic mobility has recently been derived which is applicable only to low concentrations [18]. In this latter theory, the solvent was replaced by a dielectric continuum and only the ions were explicitly considered. It was shown that one can describe ion atmosphere relaxation in terms of charge density relaxation and the elctrophoretic effect in terms of charge current density relaxation. This theory could explain not only the concentration dependence of ionic conductivity but also the frequency dependence of conductivity, such as the well-known Debye-Falkenhagen effect [18]. However, because the theory does not treat the solvent molecules explicitly, the detailed coupling between the ion and solvent molecules have not been taken into account. The limitation of this approach is most evident in the calculation of the viscosity. The MCT theory is found to be valid only to very low values of the concentration. 

In deriving the Debye heat capacity equation, one assumes that the atoms in an atomic solid are vibrating with a range or frequencies v varying from u = 0 to a maximum u — vm. The resulting equation for calculating Cv, m is... 

For a reasonable set of the parameters the calculated far-infrared absorption frequency dependence presents a two-humped curve. The absorption peaks due to the librators and the rotators are situated at higher and lower frequencies with respect to each other. The absorption dependences obtained rigorously and in the above-mentioned approximations agree reasonably. An important result concerns the low-frequency (Debye) relaxation spectrum. The hat-flat model gives, unlike the protomodel, a reasonable estimation of the Debye relaxation time td. The negative result for xD obtained in the protomodel is explained as follows. The subensemble of the rotators vanishes, if u —> oo. 

In our early work33 [50] the constant field model was applied to liquid water, where the harmonic law of particles motion, corresponding to a parabolic potential, was actually employed in the final calculations of the complex permittivity. In this work, qualitative description of only the libration band was obtained, while neither the R-band nor the low-frequency (Debye) relaxation band was described. Moreover, the fitted mean lifetime x of the dipoles, moving in the potential well, is unreasonably short ( ().02 ps)—that is, about an order of magnitude less than in more accurate calculations, which will be made here. 

The calculated spectra are illustrated by Fig. 25. In Fig. 25a we see a quasiresonance FIR absorption band, which, unlike water, exhibits only one maximum. Figure 25b demonstrates the calculated and experimental Debye-relaxation loss band situated at microwaves. Our theory satisfactorily agrees with the recorded a(v) and e"(v) frequency dependencies. Although the fitted form factor/is very close to 1 (/ 0.96), the hat-curved model gives better agreement with the experiment than does a model based on the rectangular potential well, where / = 1 (see Section IV.G.3). 

Using this calculation scheme, we have found the frequency dependencies a(v) and s"(v) for ordinary (H20) water at two temperatures. Figure 24 demonstrates for water a qualitative agreement in the calculated and experimental spectra in a very wide frequency band, comprising the microwave Debye relaxation region and the FIR range. This advance in application of the... 

The hat-curved model also gives a satisfactory description of the wideband dielectric/FIR spectra of a nonassociated polar fluid (CH3F) (Fig. 25). It is worthwhile mentioning that only a poor description of the low-frequency (Debye) spectrum could be accomplished, if the rectangular potential were used for such a calculation [32] see also Section IV.G.3. Unlike Fig. 25b, the estimated peak-loss frequency does not coincide38 in this case with the experimental frequency vD. 

Figs. 32a-c illustrate the absorption spectra, calculated, respectively, for water H20 at 27°C, water H20 at 22.2°C, and water D20 at 22.2°C dotted lines show the contribution to the absorption coefficient due to vibrations of nonrigid dipoles. The latter contribution is found from the expression which follows from Eqs. (242) and (255). The experimental data [42, 51] are shown by squares. The dash-and-dotted line in Fig. 32b represents the result of calculations from the empirical formula by Liebe et al. [17] (given also in Section IV.G.2) for the complex permittivity of H20 at 27°C comprising double Debye-double Lorentz frequency dependences. 

Figures 32d-f, placed on the right-hand side of Fig. 32, demonstrate a wideband dielectric-loss frequency dependence. This loss is calculated (solid lines) or measured [17, 42, 51, 54] (dashed lines) for water H20 and D20 at the same temperatures, as correspond to the absorption curves shown on the left-hand side of Fig. 32. Our theory gives a satisfactory agreement with the experimental data, obtained for the Debye region, R- and librational bands, to which three peaks (from left to right) correspond. However, in the submillimeter wavelength region (namely, from 10 to 100 cm ) the calculated loss is less than the recorded one. The fundamental reason for this difference will be discussed at the end of the next section.

In view of the calculations considered in Section V and in other publications (VIG), these interactions, giving rise to FIR absorption and to low-frequency Debye loss, resemble interactions pertinent to strongly polar nonassociated liquids. However, if we compare water with a nonassociated liquid (e.g., CH3F), then we shall find that in the latter (i) the R-band is absent (ii) the number mvjb of the reorientation cycles is much less, so that the reduced collision frequency y is substantially greater thus, molecular rotation is more damped and chaotic and (iii) the fitted form factor/is greater. 

Figure 36. The wideband loss frequency dependence (a) and the far-infrared part of Cole-Cole diagram (b) calculated (solid lines) and measured [51] (dashed lines) for liquid H20 at 22.2°C. The right and left vertical lines refer to the ends of the Debye and of the second-relaxation regions, respectively.

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