Lattice Dynamics and Heat Capacity of Crystals

The state of polarization, and hence the electrical properties, responds to changes in temperature in several ways. Within the Bom-Oppenheimer approximation, the motion of electrons and atoms can be decoupled, and the atomic motions in the crystalline solid treated as thermally activated vibrations. These atomic vibrations give rise to the thermal expansion of the lattice itself, which can be measured independendy. The electronic motions are assumed to be rapidly equilibrated in the state defined by the temperature and electric field. At lower temperatures, the quantization of vibrational states can be significant, as manifested in such properties as thermal expansion and heat capacity. In polymer crystals quantum mechanical effects can be important even at room temperature. For example, the magnitude of the negative axial thermal expansion coefficient in polyethylene is a direct result of the quantum mechanical nature of the heat capacity at room temperature." At still higher temperatures, near a phase transition, e.g., the assumption of stricdy vibrational dynamics of atoms is no... 

For the lattice dynamical evaluation of external contributions to crystal heat capacities, see Filippini, G. Gramaccioli, C. M. Simonetta, M. Suffritti, G. B. Thermodynamic functions for crystals of rigid hydrocarbon molecules a derivation via the Born-von Karman procedure, Chem. Phys. 1975, 8, 136-146. Harmonic dynamics works for crystals thanks to reduced molecular mobility. By contrast, liquids exhibit so-called instantaneous modes (Stratt, R. M. The instantaneous normal modes of liquids, Acc. Chem. Res. 1995, 28, 201-207) the eigenvalues of an instantaneous hessian for a liquid has a spectrum of imaginary frequencies, since any instantaneous frame of liquid stmcture is far from mechanical equilibrium because of collisions. Therefore, it is impossible to estimate heat capacities of liquids in this way, and dynamic simulation is necessary. 

The work reported here shows that the transformation temperature in the LaVxNbi x04 system is significantly affected by the simple, partial substitution of one 5+ ion for another at the center of the tetrahedron. This effect is connected closely to the dynamical characteristics of the crystal lattice. A study involving inelastic neutron scattering, heat capacity and Raman scattering is under way in this Laboratory to clarify these aspects of the problem. 

If a solid were classical, the heat capacity would be 3 Alfc. This is indeed the case at high temperatures and is called the law of Dulong and Petit. However, the experimental heat capacity goes to zero at low temperatures. This can be explained by regarding the solid as a collection of quantized oscillators. The only difficulty is to determine the spectrum of frequencies of the oscillators. For many purposes, the solid can be regarded as an elastic continuum. The result is the Debye theory. If something more sophisticated is needed one must solve for the normal modes of the crystal, i.e., the method of lattice dynamics. 

Elastic constants measured as a function of temperature are available for most of the lanthanides in polycrystalline form (Rosen, 1967, 1968) and for Tb, Dy, Ho and Er single crystals (Palmer, 1970 Palmer and Lee, 1973 and du Plessis, 1976). For a summary of the elastic properties of the lanthanides reference can be made to Taylor and Darby (1972, section 2.4) and to ch. 8, section 9. If a suitable lattice dynamical model were devised, we should be able to calculate Cl from first principles. This was done for Gd, Dy and Er metals (Sundstrom, 1968), but at the time of these calculations, elastic constants were available only for polycrystalline samples at a few fixed temperatures. Nevertheless the results obtained did indicate that Lounasmaa s (1964a) interpolation idea was reasonable. With the elastic constant data available today it should be possible to calculate Cl for the entire region of interest, although this appears not to have attracted much attention, presumably because the uncertainty involved in separating off the contributions in experimental heat capacity results makes comparison with theory unrewarding as far as Cl is concerned. 

The heat capacity, Cp, of an organic crystal can be subdivided into an internal part and an external part (recall Section 6.3 and the discussion around equation 6.19). The external part is the structure-sensitive part because it depends on the lattice vibrational frequencies and hence on the cry stal structure and the packing forces within the cry stal. External contributions can be estimated by lattice dynamics simulations [7], or can be derived from variable-temperature molecular dynamics simulations (recall Section 9.5 and especially equation 9.21). 

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