Debye function three-dimensional

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). 

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... 

In Fig. 2.29 a number of examples of three-dimensional Debye functions for elements and salts are given [17]. A series of experimental heat capacities is plotted (calculated per mole of vibrators). Note that salts like KCl have two ions per formula... 

Examples of Three-dimensional Debye Functions for Crystals of Elements and 6aIts... 

Examples of Two- and Three-dimensional Debye Functions for Allotropes of Carbon... 

In Fig. 2.42 results from the ATHAS laboratory on group IV chalcogenides are listed [18]. The crystals of these compounds form a link between strict layer stractures whose heat capacities should be approximated with a two-dimensional Debye function, and crystals of NaCl stracture with equally strong bonds in all three directions of space and, thus, should be approximated by a three-dimensional Debye function. As expected, the heat capacities correspond to the structures. The dashes in the table indicate that no reasonable fit could be obtained for the experimental data to the given Debye function. For GeSe both approaches were possible, but the two-dimensional Debye function represents the heat capacity better. For SnS and SnSe, the temperature range for data fit was somewhat too narrow to yield a clear answer. 

The next step in the ATHAS analysis is to assess the skeletal heat capacity. The skeletal vibrations are coupled in such a way that their distributions stretch toward zero frequency where the acoustical vibrations of 20-20,000 Hz can be found, hi the lowest-frequency region one must, in addition, consider that the vibrations couple intermolecularly because the wavelengths of the vibrations become larger than the molecular anisotropy caused by the chain structure. As a result, the detailed molecular arrangement is of little consequence at these lowest frequencies. A three-dimensional Debye function, derived for an isotropic solid as shown in Figs. 2.37 and 38 should apply in this frequency region. To approximate the skeletal vibrations of linear macromolecules, one should thus start out at low frequency with a three-dimensional Debye function and then switch to a one-dimensional Debye function. 

T ables of the one, two, and three-dimensional Debye functions are available, respectively a. Wunderlich, B (1962) J Chem Phys 37 1207. 

Fig. 5.17 Three-dimensional crystal considered along one direction density function with equidistant maxima blurred by thermal fluctuations (a) and the angular spectrum of the structure factor (b). The height of the 5-type maxima is given by squared amplitudes of the density harmonics and is additionally modulated by both the molecular form-factor (MMF) and Debye-Waller factor (DWF)...

D (0jT) represents the three dimensional Debye function. D2(0jT) is the two dimensional Debye function tabulated by Tarasov (1959). Di 0IT) finally is the one dimensional Debye function, also tabulated [Wunderlich (1962)]. Eq. (8) is based on the assumption that a certain... 

The optical vibrations have also incteased in number by the introduction of the second CHg-group. A look at Tables III. 11 and III. 12 shows that there should only be little effect of the tical vibrations below 150° K. A try to fit a one (or three) dimensional Debye function (Eqs. 11.146 or 11.161) to the data of Table III.15 between 10 and 150° K failed. Neither leads to a constant 0j-temperature as is possible in case of polyethylene, polypropylene and also polystyrene. Of the 10 low frequency vibrations, an average of only 6 seem to be excited at 150° K, indicating that in comparison with polyethylene their average frequencies lie somewhat higher. They must also be higher than comparable frequencies in polypropylene. From the chemical structure of polyisobutylene one would like to conclude that this relative decrease in heat capacity is caused by steric hindrance of the two methyl groups bound to the same carbon atom. 

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