Debye temperature general

Use of Debye Equation at Very Low Temperatures. Generally, it is assumed that the Debye equation expresses the behavior of the heat capacity adequately below about 20 K [9]. This relationship [Equation (4.68)],... 

The increased volume of a material with increasing temperature is a result of the same atomic vibration phenomenon which stores thermal energy. Consequently, changes in coefficient of thermal expansion generally parallel changes in heat capacity. Both increase rapidly at low temperature and approach nearly constant values above the Debye temperature (section 3.7). 

Effective Debye temperatures have been calculated for a number of alkyl and aryl tin halides [131], and although for instance the Debye temperature for polymeric MesSnF (102-8 K at 70 K) is larger than that for monomer Me Sn (71 K at 70 K), it remains true in general that lattice characteristics do not provide a means of distinguishing polymeric structures [131, 132]. 

Phonon surface bands of some insulators and semiconductors are given in Figs. 5.2-56-5.2-58. Surface phonon energies of alkali halide crytals are summarized in Table 5.2-23. Since insulators and semiconductors have in general more than one atom per unit cell, they display both acoustical and optical branches. Surface Debye temperatures of some semiconductors are given in Table 5.2-22. 

Variations in lattice vibrations in fine particles with respect to the bulk may arise from (i) the reduced volume leading to lattice softening with resultant decrease of the Debye temperature, (ii) surface effects since the surface atoms are probably more weakly bound than the atoms in the interior, or (iii) changes in the lower and upper cut-off frequencies of the phonon spectrum. The first two phenomena should decrease / while the latter could increase /. In general, one observes a recoil-free fraction in fine particle systems that is much smaller than that of bulk materials. However, most often this is not due to effects of the lattice vibrations but to the motion of the particle as a whole, which indeed drastically lowers the / factor. 

In fact, in general, the Debye temperature is much lower than the Einstein temperature, so we can divide the temperature scale into three... 

The variation in the Debye-temperature allows some general conclusions about the frequency... 

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