Vibrations in solids
Abstract
This page is an introduction to vibrations in solids. Additional derivation and information will be required to obtain a full understanding of the material. Please submit corrections when you feel necessary.
1D dispersion relations
Dispersion relations is a relation between a waves frequency and wavelength, which describes how the wave propagates through a medium. In solids these waves are vibrations of the lattice called phonons. In a given unit cell there are two types of vibrations or phonons that exist, acoustic and optical phonons. There can be three modes of acoustic phonons, which include two transverse modes and one longitudinal mode. Optical phonons also have transverse and longitudinal modes. In a simple, 1D system (a chain of atoms) we can calculate a dispersion relation of the form:
where g is the assumed spring constant (inter-atomic potential), m is the atomic mass, k is the wave vector and a is the atomic spacing. This expression for the dispersion relation fits well with this simple 1D molecular dynamics tool.
Quantization and phonons
A phonon is a quantum of lattice vibrational energy being given by,
where n is the quantum number and the 1/2 is for the zero-point energy of the mode [1]. The full derivation of the quantum theory of phonons can be located here (new page needed).
Statistics of phonons
Similar to electrons, phonons also have a density of states and an occupation number for a given state. The density of states is dependent on the dimensionality of the system and it is the total number of allowed states for a given wave vector. The occupation number is the expected number of phonons at a given state at a specific temperature. Typically this is the Bose-Einstein distribution which is,
For more information, see notes on the quantum of thermal conductance.
Atomic Green’s functions
Phonon branches and modes
In a crystal with two or more atoms per primitive cell the phonon dispersion takes on new features. There exist two branches for each polarization mode, acoustical and optical. The possible polarization modes for each of these branches include longitudinal and transverse modes. The following figure shows the dispersion relation for each of these modes is shown here: 
. In optical branches, the atoms vibrate against each other (can be excited with the electric field of a light wave). In acoustical branches, the atoms move together like acoustic waves, propagating through the crystal.
Lattice specific heat
For now see notes on the quantum of thermal conductance.
Phonon scattering
Phonon-Phonon Scattering
There are two types of phonon-phonon scattering, elastic and inelastic. In an elastic scattering events both energy and momentum are conserved and are results of defects, dislocations and boundaries. In inelastic scattering events three or more phonons are involved. Within inelastic scattering there are two scattering processes known as N-process, normal, and U-process, Umklapp. N-processes always conserve energy and momentum. These N-processes do not impede phonon momentum and therefore do not have much impact on transport or thermal conductivity directly, though they do redistribute phonon frequencies. U-processes only conserve energy. In U-processes the phonon momentum is therefore impeded which directly impacts the thermal conductivity which is also why U-processes are also called intrinsic scattering. In inelastic scattering events phonon number is not always conserved. A diagram of inelastic scattering is shown here: 
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Phonon thermal conductivity
For now see notes on the quantum of thermal conductance.