Thermal and Elastic Properties of Crystals

Our starting point for the analysis of the thermal and elastic properties of crystals is an approximation. We begin with the assertion that the motions of the i atom, will be captured kinematically through a displacement field Uj which is a measure of the deviations from the atomic positions of the perfect lattice. It is presumed that this displacement is small in comparison with the lattice parameter, uj < C qq. Though within the context of the true total energy surface of our crystal (i.e. R2,. .., Rw)) this approximation is unnecessary, we will see... 

Althou, in principle, the general theory is superior to the band theory, the appropriate techniques for its application are not yet developed sufficiently well and a unified approach to a quantitative description of the structures and the physical properties of crystals is still lacking. The less generally valid band theory can at present give clearer and more convincing explanations of changes in the physical properties of crystals caused by variations in the temperature, pressure, magnetic and electric fields intensities, impurity concentrations, etc. However, many problems encoimtered in the study of chemical bonds in crystals cannot be considered within the framework of the standard band theory. They include, for example, determination of the elastic, thermal, and thermodynamic properties of solids, as well as the structure and properties of liquid and amorphous semiconductors. 

In this regime, the epitaxial film behaves as a perfect crystal that is elastically strained. Properties in this regime should thus, by definition, be predictable from the properties of bulk single crystals in the presence of uniform elastic strains. Unfortunately, data on the common thermal and mechanical properties of thin films are lacking for fliis range of thickness, owing mainly to the difficulty of such measurements. Transport properties are easier to measure but may be dominated by geometrical and interfacial effects. 

The glass-ceramic phase assemblage, ie, the types of crystals and the proportion of crystals to glass, is responsible for many of the physical and chemical properties, such as thermal and electrical characteristics, chemical durabiUty, elastic modulus, and hardness. In many cases these properties are additive for example, a phase assemblage comprising high and low expansion crystals has a bulk thermal expansion proportional to the amounts of each of these crystals. 

Physical properties of solid materials which are greatly influenced by the presence of defects of lattice order in real single crystals are called structural-sensitive properties, and are distinguished from intrinsic properties, which are determined by the elements constituting the crystal, for example the chemical bonds, the structure, etc. Color, plasticity, glide, and semiconductor properties are structural-sensitive properties, whereas density, hardness, elasticity, and optical, thermal, and magnetic properties are the intrinsic properties. Structural-sensitive... 

Figure 3. Ratio of the mean square root displacements derivatives along directions of weak and strong coupling, calculated in the model of a highly anisotropic layered crystal. Its anisotropy of interatomic interaction and elastic properties correspond to those of NbSe2. The pronounced maximum on this curve corresponds to a minimum on the temperature dependence of the thermal expansion along the layers.

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