## Thermal vibrations

The concepts of temperature and thermal equilibrium associated with crystal solids are based on individual atoms in the system possessing vibrational motion. The classical theory of thermal energy by atomic vibrations, though providing suitable explanations at elevated temperatures, has proved unsatisfactory at reduced or cryogenic temperatures. Quantum mechanics has subsequently provided theories based upon statistical probability that have provided possible mechanisms to explain some of the observed phenomena. A system of vibrating atoms in a crystal is highly complicated, and beyond the realm of any realisable theoretical method of analysis or calculations to verify spectral measurements from the total thermal energy of a crystalline substrate.

When a particle is bound to a crystal, the energy can only have discrete values as defined by the energy band structure. The quantum-mechanics of a one-dimensional simple harmonic oscillator gives permitted energies of (n+½)ħω where ω is the angular frequency and n is the permitted energy integer. At a position of minimum energy (0K) the energy can never be zero, but has energy of ½ħω(zero-point energy) and as such will still provide crystal vibration.

As an atom can vibrate independently in three dimensions it is equivalent to three separate oscillators. The total thermal energy for N atoms will then be 3NkT, ignoring the ½ħω term, the specific heat required to change the temperature by one degree will then be 3Nk where the specific heat of a solid for a given number of atoms is independent of temperature if N is the Avogadro number (6.02x1023). A detailed calculation of this form would require a knowledge of the number of atoms vibrating with frequencies ω1 ... ωn, which would depend on the density of states, and integration over the whole range of atomic vibrational frequencies would be required.

The thermal vibrations in a solid produce atomic displacements, which in a three dimensional lattice can be resolved into different states of polarization such that vibrations parallel to the wave vector are longitudinal waves and the two directions at right angles to the wave vector are transverse waves. As the rules of quantum mechanics apply to all the different atomic vibrations in the crystal, the lattice pulsates as a complete assembly in discrete energy steps of ħω(phonons). The phonon is related to both the frequency of vibration and the temperature. If the temperature is raised, the amplitude of atomic vibration increases, and in quantum terms this is considered as an increase in the number of phonons in the system.

The concept of the phonon is therefore considered as the quantum of lattice vibrational energy onto which is superimposed a complex pattern of standing and/or travelling waves that represent changes in temperature. If the crystal is at a uniform temperature the standing wave concept is adequate as the phonon vibrations are uniformly distributed.