Debye frequency distribution

Subtraction of the resolution function reveals the vibrational contribution S (v) to the excitation probability. PHOENIX adjusts the subtraction weight to achieve the best match to the one-phonon contribution to the vibrational signal near Eq expected for a Debye frequency distribution (D (v) a v y The Fourier-log algorithm then yields the dominant first-order vibrational contribution... 

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... 

Tarassov (1955) and also Desorbo (1953) have considered these ideas in relation to a onedimensional crystal in which case the one-dimensional frequency distribution function predicts a T dependence of the specific heat at low temperatures. In the case of crystalline selenium, however, it has been found necessary to combine the one-dimensional theory with the three-dimensional Debye continuum model in order to obtain quantitative agreement with the data below about 40° K. Tem-perley (1956) has also concluded that the one-dimensional specific heat theory for high polymers would have to be combined with a three-dimensional Debye spectrum proportional to T3 at low temperatures. For a further discussion of one-dimensional models see Sochava and TRAPEZNrKOVA (1957). 

The last two terms in equation 1 are, in effect, linearly related to the frequency distribution, g(Ho). All of the remaining factors are known or are established by the experimental conditions, except the Debye-Waller factor, exp (—2 IF). It has been customary to set this factor equal to unity, because 2 IF is small under the conditions where the one-phonon approximation is valid (14). 

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 distribution of frequencies for one branch of the vibration spectrum, described in the Debye approximation by treating the crystal as an elastic continuum. The Debye frequency is cod-... 

Theoretical Estimates The use of the Debye model (Figure 3.2), which assumes that a solid behaves as a three-dimensional elastic continuum with a frequency distribution/(j ) = allows accurate prediction of the temperature dependence of the vibrational heat capacity C / of solids at low temperatures Cy oc r ), as well as at high temperatures (Cy = Wks). One may also use the same model with confidence to evaluate the temperature dependence of the surface heat capacity due to vibrations of atoms in the surface. 

Figure 3.2. Frequency distribution dn/dv of lattice vibrations u assumed by the Debye model.

From Eq. (3) the frequency distribution can be calculated following the Debye treatment by making use of the fact that an actual atomic system must have a limited number of frequencies, limited by the number of degrees of freedom N. The distribution p(v) is thus simply given by Eq. (4). This frequency distribution is drawn in the sketch on the right-hand side in Fig. 2.36. The heat capacity is calculated by using a properly scaled Einstein term for each frequency. The heat capacity function for one mole of vibrators depends only on Vj, the maximum frequency of the distribution, which can be converted again into a theta-temperature, j. Equation (5) shows that at temperature T is equal to R multiplied by the one-dimensional Debye... 

To conclude this discussion, Eqs. (8) and (9) of Fig. 2.38 represent the three-dimensional Debye function. The mathematical expression of the three-dimensional Debye function is also given in Fig. 2.37. Now the frequency distribution is quadratic in V, as shown in Fig. 2.38. The derivation of the three-dimensional Debye model is analogous to the one-dimensional and two-dimensional cases. The three-dimensional case is the one originally carried out by Debye [16]. The maximum frequency is Vj... 

The integration of equation 1 is evaluated in steps for the various regions of the frequency distributions found for macromolecules. The lowest vibrational frequencies (skeletal) usually follow a quadratic function up to a frequency limit called 0D or 03. This is the well-known Debye approximation (59) of the low temperature heat capacity at constant volume, Cy (B), which in this temperature range... 

For two- or one-dimensional structures, such as are found in crystals of layer and chain molecules, the frequency distribution changes to linear and constant functions, respectively. The corresponding integrals are called the two- and onedimensional Debye functions (60,61)... 

In the above analysis it is assumed that the atoms vibrate with a single frequency Q, as in the Einstein theory, whereas in a real crystal there is a distribution of vibrational frequencies with an appropriate cut-off, as considered in the Debye theory. In any case, it is clear that a measurement of the /-factor by Mossbauer spectroscopy can provide knowledge concerning phonon properties, such as their frequency distribution and density of states. Similar information can also be obtained from an analysis of the second-order Doppler shift. Unfortunately, the restriction imposed by the relative timescales, typically as discussed earlier, normally... 

Although the Debye theory predicts a heat-capacity curve which is often in good agreement with experiment, the assumptions in the theory with regard to the frequency spectrum are not necessarily correct. For certain simple types of lattice, Blackman (1937) was able to make a detailed calculation of the frequencies of the normal modes, allowing for the atomic structure of the system. It was found, as is assumed in the Debye theory, that there is an upper limit, to the possible frequencies, and it was also confirmed that the law should hold at very low temperatures. On the other hand, Blackman found that the frequency distribution does not have such a simple form (equation (13 54)) as was assumed by Debye and, in fact, it may have two or more peaks. In view of these results it seems that the agreement of the Debye theory with experiment is better than might reasonably have been expected. 

Next, it is useful to expand this analysis to two-dimensional vibrators. The frequency distribution is now linear, as shown in the next hgure, and is given in Eq. (6) of Fig. 5.14. The analogous two-dimensional Debye function has been tabulated also, so that heat capacities can be derived using Eq. (7). Note that in Eq. (7) it is assumed that there is a maximum of 2N vibrations for the two-dimensional vibrator, i.e., the atomic array is made up of N atoms, and vibration out of the plane are prohibited. In reality, this may not be so, and one would have to add additional terms to account for the omitted vibrations. The same reasoning applies for the one-dimensional case of Eq. 

Debye supposes that the frequencies are sufficiently similar so that the distribution can be supposed to be continuous, which enables us to replace the sum of equation [1.11] by an integral, which is written as follows for a frequency distribution g(v) ... 

Debye also supposes that this frequency distribution is of the same form as the elastic frequency distribution of the solid, supposed to be a continuum. Those frequencies themselves are linked to the propagation of sound in that solid. These frequencies range from the value 0 to a maximum frequency Vd defined by equation [1.13], where c is the celerity of sound in the solid. 

Figure 1.3. Frequency distribution according to Debye The frequency distribution g(v) is of the form (see Figure 1.3) ...

Debye s model, which allows for a frequency distribution given by relation [1.9], has only yielded correct values of the specific heat capacity at constant volume (see section 1.8) for fairly low temperatures. Other authors have improved the model by modifying that frequency distribution. For example, Bom and Karman took a new approach to the establishment of the frequency distribution, this time supposing that the solid was no longer a continuum, but instead was represented by a periodic lattice of particles, which led them to the distribution function as shown in Figure 1.4(a). The distribution function reaches its peak very near to the limit frequency. 

In fact in this types of models, with a frequency distribution modified compared to Debye s, we can keep the developments obtained using Debye s model, but as if the Debye temperature varied with temperature. 

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