Phonons Normal Mode Solution

In order to describe lattice motion it is convenient to introduce phonons. The theoretical discussion of phonons dates back to the papers by Lord Rayleigh, who considered waves propagating along the surface of an elastic continuum [92]. These modes are also found among the low-frequency modes obtained when solving the normal mode problem of a solid. Within the harmonic approximation the hamiltonian for the solid is 

Here r is the number of particles in a unit cell and N the number of unit cells in the crystal. The phonons follow Bose-Einstein statistics, and hence the thermal population distribution at the temperature T is given by 

With the above distribution, the average energy of a given mode can be obtained as the Boltzmann weighted average over the harmonic oscillator energies 

We notice that for T 0 we have s ho /2 (the zero point energy) and for r - 00 we have e - kaT. This limit is a manifestation of the so-called equipartition theorem according to which the ensemble averaged energy in each degree of freedom in phase space (i.e., both momentum and coordinate space) is ( /2)kBT. 

We shall return to a dynamical treatment of phonon dynamics when dealing with atom/molecule surface scattering. 

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Every example of a vibration we have introduced so far has dealt with a localized set of atoms, either as a gas-phase molecule or a molecule adsorbed on a surface. Hopefully, you have come to appreciate from the earlier chapters that one of the strengths of plane-wave DFT calculations is that they apply in a natural way to spatially extended materials such as bulk solids. The vibrational states that characterize bulk materials are called phonons. Like the normal modes of localized systems, phonons can be thought of as special solutions to the classical description of a vibrating set of atoms that can be used in linear combinations with other phonons to describe the vibrations resulting from any possible initial state of the atoms. Unlike normal modes in molecules, phonons are spatially delocalized and involve simultaneous vibrations in an infinite collection of atoms with well-defined spatial periodicity. While a molecule s normal modes are defined by a discrete set of vibrations, the phonons of a material are defined by a continuous spectrum of phonons with a continuous range of frequencies. A central quantity of interest when describing phonons is the number of phonons with a specified vibrational frequency, that is, the vibrational density of states. Just as molecular vibrations play a central role in describing molecular structure and properties, the phonon density of states is central to many physical properties of solids. This topic is covered in essentially all textbooks on solid-state physics—some of which are listed at the end of the chapter. 

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