The Debye approximation

Proceeding in this way it can be shownf that the number of modes whose frequencies lie in the range v to v- dv is 

Within the scope of the present section we shall apply (13 54) to the internal energy and heat capacity only. The free energy and entropy on the Debye model may be worked out by similar methods. From (13 35) and (13 38) we have 

This expression takes a much simpler form at very low temperatures where Under such conditions the second term in (13 59) 

A very much simplified lattice-dynamical model is that of Debye. In the Debye approximation, discussed in the following section, a single phonon branch is assumed, with frequencies proportional to the magnitude of the wavevector q. 

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, 

The frequency vD at the edge of the Brillouin zone is thus equal to vsqD/2n. The Debye temperature 0D is defined as hvD/(kB). As shown below, 0D is an inverse measure for the vibrational mean-square amplitudes of the atoms in a crystal at a given temperature. 

As the normal modes are assumed to be uniformly distributed in reciprocal space, the frequency distribution g(a ) will be proportional to to2, that is, 

For a monatomic cubic crystal, the corresponding mean-square displacement is (Willis and Pryor 1975) 

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In the Ising-type model, the change of molecular volume AV due to the LS<->HS transformation leads to a change of phonon frequencies of the lattice. The effect may be treated within the Debye approximation which requires that the interaction parameters and J2 are replaced by J and J 2 where ... 

At higher temperatures, the two-phonon (Raman) processes may be predominant. In such a process, a phonon with energy hcOq is annihilated and a phonon with energy HcOr is created. The energy difference TicOq — ha>r is taken up in a transition of the electronic spin. In the Debye approximation for the phonon spectrum, this gives rise to a relaxation rate given by... 

The resonance width for low-frequency inodes rjq averaged over wave vectors is given in the Debye approximation as follows 143... 

Such an approximate description of acoustic vibrations is referred to as the Debye approximation and the limiting frequency coo is called the Debye frequency. The... 

The recoilless fraction, /, has been calculated (13) for monotomic lattices using the Debye approximation. When the specific heat Debye temperatures of the alkali iodides are inserted in the Debye-Waller factor, a large variation of f follows (from 0.79 in Lil to 0.15/xCsI). It is not... 

In any crystal, the low-frequency acoustic modes dominate at low temperatures, so that the approximation that w is proportional to q becomes increasingly valid as is evident from Fig. 2.2. In particular, the T dependence of the specific heat at very low temperatures is well predicted by the Debye approximation. 

Before turning to the applications of the Debye approximation, we should elaborate more fully on a point that was glossed over. This is the assumption —made at the outset, but explicated in going from Equation (58) to Equation (59) —that the scattering behavior of each scattering element is independent of what happens elsewhere in the particle. The approximation that the phase difference between scattered waves depends only on their location in the particle and is independent of any material property of the particle is valid as long as... 

Here /jn(f) is the intensity of the incident radiation and 0 is the phase of the interferometer in the dark. The functions N(< >) and M(< >) relate the intensities of the transmitted and intracavity fields to that of the incident light. The function 7ref (0 corresponds to the intensity of radiation from an additional source, which is very likely to be present in a real device to control the operating point. This description is valid in a plane-wave approximation, provided that we neglect transverse effects and the intracavity buildup time in comparison with the characteristic relaxation time of nonlinear response in the system. It has been shown that the Debye approximation holds for many OB systems with different mechanisms of nonlinearity. 

Because this result has been obtained by solving a generalized Poisson-Boltzmann equation with the linearization approximation, it is necessary to compare it with the DLVO theory in the limit where the Debye approximation holds. In this case, Verwey and Overbeek [2], working in cgs (centimeter-gram-second) units, derived the following approximate equation for the repulsive potential ... 

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 distribulion of fi cquencics for one branch of the vibration spectrum, described in the Debye approximation by treating the crystal as an elastic continuum. The Debye frequency is... 

Indeed, the Debye approximation is more appropriate in any problem where the modes of lowest frequency are important, as they are in thermal properties at low temperatures. In cases were all modes are important, such as in the evaluation of the total zero-point energy, the simpler Einstein model may be preferable. Notice that even within the Debye approximation the frequencies are concentrated near the highest frequency, called the Debye frequency. This is illustrated in Fig. 9-7. 

For longitudinal modes, we can therefore stale that the number of modes with frequency less than a> is 2 w/waY atom-pair, or a density of modes of 6w lwy, modes per atom-pair per frequency-range. Similarly, construct the spectrum for transverse modes and plot the total on the same abscissa as in Fig. 9-6 so that comparison can be made. (That histogram did not have a normalized scale on the ordinate, so you need not worry about the ordinate.) The principal discrepancies are understandable by comparison of the Debye approximation to the spectrum shown in Fig. 9-2. 

Thus, Eq. (16-23) may be used to estimate the high temperature resistivity of potassium. An analogous calculation, using the Debye approximation, is carried out in Problem 17-2. 

As noted above, the area of a peak in the VDOS provides a straightforward measure of the mode composition factor e j according to equation (6) (possibly summed over a number of unresolved modes). However, there are nontrivial approximations implicit in the calculation. In addition to the Debye approximation used to subtract the recoiUess contribution, the Fourier-log algorithm assumes a unique environment for the probe atom and neglects vibrational anisotropy. The resulting errors are often smaller than the experimental uncertainty, particularly for protein samples. However, there may be situations where these assumptions are questionable, for example, if the probe nucleus occupies... 

When reorganization of the medium is small, the rate constant in the Debye approximation is described (as shown in ref 161) by expression (62) at A = 0, but with doubled a , the contribution of zero-point vibrations is also doubled. 

The thermal shift 6j of the Mdssbauer spectrum is the sum of a contribution due to the second-order Doppler effect (6sod) and a possible contribution due to an intrinsic dependence of the isomer shift (8j) on temperature. The second-order Doppler shift is proportional to the mean square velocity of the Mdssbauer nucleus. For the purpose of a comparison with thermodynamic data, the SOD shift may be described in terms of the Debye approximation,6... 

The inadequacy of the Debye approximation in describing the details of the frequency distribution function in a real solid is well known. This results in noticeable disparities between Debye temperatures derived from the results of different experimental techniques used to elucidate this parameter on the same solid, or over different temperature ranges. Substantial discrepancies may be expected in solids containing two (or more) different atoms in the unit cell. This has been demonstrated by the Debye-Waller factors recorded for the two different Mdssbauer nuclei in the case of Snl4,7 or when the Debye-Waller factor has been compared with the thermal shift results for the same Mdssbauer nucleus in the iron cyanides.8 The possible contribution due to an intrinsic thermal change of the isomer shift may be obscured by an improper assignment of an effective Debye temperature. 

The Debye approximation is not valid for a diatomic cubic lattice, and both the recoil-free fraction and the elfective Debye temperature for each atom are strongly dependent on the mass-ratio of the two atoms. However, if the mass difference is small the difference in Debye temperature should also be small. This has been verified for SnTe using "Sn ( Sn) and (>2STe) sources and an SnTe absorber, as well as an Sn Te source (decaying... 

This is the correct contribution to sR p) e — 1) if the macroscopic dielectric constant is given by the Debye approximation ... 

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