Forces and Phonon Dispersion Curves

Phonon dispersion curves w(q) can be measured by inelastic neutron scattering techniques this important method will be discussed in [1.35]. The curves determined by experiments are mainly of interest because they provide chance of testing various models of interatomic forces. The forces between the atoms are governed by the electronic structure of the atoms involved, and in simple cases it is possible to derive force constants from first principles which are based on the electronic wave functions [4.1-5]. With the exception of some qualitative remarks about metals, we shall not discuss these microscopic models here but consider only certain phenomenological models for the interatomic forces. A good model should contain only a few, but physically meaningful, parameters. 

A model which is often used for covalent or partly covalent crystals is the valence force model. This model has originally been developed for vibrations in molecules [4.19,20] but the formalism can be extended to crystals as well [4.21,22]. The potential energy is expressed in terms of changes in bond distances, bond angles and other so-called internal coordinates, we shall illustrate this model for the linear (SN) chain and discuss the results for diamond and 3-AgI. 

In molecular crystals or crystals containing complex ions, it is often possible to separate the internal vibrations within the molecules from their external vibrations [4.23]. External vibrations are translational or libra-tional motions of the whole molecules or complex ions which are regarded as being rigid units. If the separation between internal and external vibrations is possible, the problem can be greatly simplified. We shall illustrate the external vibrations for a simple one-dimensional system and for (ND )Cl. 

The theory of phonons in metals involves the electron-phonon interaction and, strictly speaking, lies outside the scope of this book. Nevertheless, we give a qualitative discussion of this topic which serves to illustrate the essential physics such as the screening of the ions by the conduction electrons. This screening is responsible for the Kohn anomaly observed in some dispersion curves, an effect which is particularly pronounced in onedimensional metals [1.35]. 

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