Born von Karman expansion

It is possible in principle to calculate all of these modes from the theory of the electronic structure, which is equivalent to the calculation of all the force constants. Indeed we will see that this is possible in practice for the simple metals by using pseudopotential theory. In covalent solids, even within the Bond Orbital Approximation, this proves extremely difficult because of the need to rotate and to optimize the hybrids, and it has not been attempted. The other alternative is to make a model of the interactions, which reduces the number of parameters. The most direct approach of this kind is to reduce the force constants to as few as possible by symmetry, and then to include only interactions with as many sets of neighbors as one has data to fit- for example, interactions with nearest and next-nearest neighbors. This is the Born-von Karman expansion, and it has somewhat surprisingly proved to be very poorly convergent. This simply means that in all systems there arc rather long-ranged forces. 

Up to this point, the analysis is rigorously correct and general it is simply a restatement of the Born-Von Karman expansion of the energy in terms of relative displacements—.see Eq. (8-17). However, we shall now make a major approximation in taking the force constants from the very simple valence force field that we described in Chapter 8. This will give us a clear and correct qualitative description of the vibration spectra and will even give semiquantitativc estimates of the frequencies. Afterward, we shall consider the influence of the many terms that are omitted in this simple model. 

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See also in sourсe #XX -- [ , ]

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