Lattice vibrations Einstein approximation

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 g((o) can be used to provide a simple approximation to the lattice vibrations, the Einstein approximation. Let us begin by agreeing that single characteristic frequencies, could be chosen to individually represent each of the six types (three translational and three rotational) of external mode. The ( 6)y value is the density of states weighted mean value of all of the frequeneies over which that external mode, j, was dispersed. Normalising as discussed above ... 

Having adopted this convention we may ask. Does there exist a physical state of a substance for which the conventional entropy is actually zero Now perfect crystals are known to have a very orderly structure, and at very low temperatures the lattice vibrations will all be in their lowest states which correspond to the zero-point energy. Therefore it may be expected that a crystal will have a very low entropy at temperatures approaching the absolute zero, and in one of the original forms (Planck s version) of the third law it was asserted that the entropy of a pure substance is actually zero under such conditions. On the other hand, from (13 51), based on the Einstein approximation, it is seen that... 

In molecular crystals or in crystals composed of complex ions it is necessary to take into account intramolecular vibrations in addition to the vibrations of the molecules with respect to each other. If both modes are approximately independent, the former can be treated using the Einstein model. In the case of covalent molecules specifically, it is necessary to pay attention to internal rotations. The behaviour is especially complicated in the case of the compounds discussed in Section 2.2.6. The pure lattice vibrations are also more complex than has been described so far . In addition to (transverse and longitudinal) acoustical phonons, i.e. vibrations by which the constituents are moved coherently in the same direction without charge separation, there are so-called optical phonons. The name is based on the fact that the latter lattice vibrations are — in polar compounds — now associated with a change in the dipole moment and, hence, with optical effects. The inset to Fig. 3.1 illustrates a real phonon spectrum for a very simple ionic crystal. A detailed treatment of the lattice dynamics lies outside the scope of this book. The formal treatment of phonons (cf. e(k), D(e)) is very similar to that of crystal electrons. (Observe the similarity of the vibration equation to the Schrodinger equation.) However, they obey Bose rather than Fermi statistics (cf. page 119). 

The lattice component was calculated using the harmonic approximation, in which all the acoustic and low-frequency optical vibrations are included with the help of a single Debye fimction, while high-frequency crystal vibrations are taken into accoimt by Einstein s equation. According to Kelley s derivations (Gurvich et al., 1978-1984) based on the Born-von Karman d)mamic crystal lattice theory, we therefore have... 

The problem of behavior of the heat capacity of solids near 1°K has been treated by Einstein and by Debye to include the effects of vibration. For undergraduate treatment, it is sufficient to say that near 1°K heat capacities of lattices vary roughly as [2] so the heat capacity curve increases after TK. Other texts and monographs should be consulted for studies of materials at very low temperatures, but here the essential facts are that 5 = / In (HO gives an approximate value for low-temperature entropy due to isotope impurities and that the heat capacity varies as roughly in the 1°K range. There would be a constant A characteristic of the material and then Cp AT. As usual there are alternate verbal descriptions of the third law of thermodynamics but our summary would be... 

Debye s model gives only an approximate deseription of the vibrational properties of real solids, espeeially for solids eontaining different atoms or having certain lattice structures. However, it is very convenient in situations where an analytical expression for the distribution functions is necessary, because more rigorous models give analytical solutions for one- or two-dimensional systems only. Since the spectra of surface vibrations are much more complicated, this model is often used. There are also some empirical combinations of Debye and Einstein distributions (in the classical limit) ... 

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