The Static Lattice Model and Its Limitations

Many phenomena in solid-state physics can be understood on the basis of a static lattice model. In this model, the atoms of the solid are taken to constitute a fixed, rigid, immobile periodic array. Within this framework it is, for example, possible to account for a wealth of equilibrium properties of metals dominated by the behaviour of the conduction electrons. To some extent it is also possible to account for the equilibrium properties of ionic and molecular insulators. 

The static lattice model is, of course, an approximation to the actual ionic configuration, because the atoms or ions are not fixed to their equilibrium positions, but rather oscillate about them with an energy which is governed by the temperature of the solid. The reason for these oscillations is due to the fact that the ions are not infinitely massive, nor are they held in place by infinitely strong forces. In classical theory the static lattice model can therefore be valid only at zero temperature. At nonzero temperatures, each ion must have some thermal energy and therefore a certain amount of motion in the vicinity of its equilibrium position. In quantum theory, even at zero temperature, the static lattice model is incorrect, because according to the uncertainty principle AXAp h, localized ions possess some nonvanishing mean square momentum. 

The dynamics of atoms in solids is responsible for many phenomena which cannot be explained within the static lattice model. Examples are the specific heat of crystals, thermal expansion, thermal conductivity, displacive ferroelectric phase transitions, piezoelectricity, melting, transmission of sound, certain optical and dielectric properties and certain aspects of the interaction of radiation such as X-rays and neutrons with crystals. The theory of lattice vibrations, often called lattice dynamiosy and its implications for many of the above mentioned phenomena is the subject of this two-volume book. 

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