Phonon

A phonon is a quasiparticle, which is used in theoretical solid state physics to describe the properties of the lattice vibrations quantum mechanically described in a crystal with the help of a simplified model can. Phonons are delocalized, that is a phonon exists throughout the crystal lattice and can be assigned to any particular place.

The term phonon (after Greek " phon " = sound ), was chosen by analogy with the vibrational quantum of the electromagnetic field, the photon and for the first time of JI Frenkel ( 1894-1952 ) in 1932 in his book "Wave Mechanics, Elementary Theory" be used.

Classification

In a three-dimensional crystal with atoms in the primitive basis available at any given crystal symmetry compatible with the wave vector of possible vibration modes: acoustic (one longitudinal and two transverse), and optical:

  • Acoustic phonons (also known as phonons ) are sound waves that propagate through the crystal lattice, with adjacent atoms move in the same direction.
  • In contrast, the atoms move with optical phonons within the base against each other. The term " optical" is based on the fact that the oscillation frequencies of optical phonon often be in the range of infrared or visible light. Such designation is effected regardless of whether the phonons are actually optically active. (Optical activity as used herein means that a phonon can interact with a photon, ie that a phonon can be generated by a photon is absorbed, or, conversely, a photon can be emitted by a phonon is destroyed. Optical activity can only be if within the base is present electrical polarization, which is generally accurate will be the case when the base is composed of different atoms. crystals, which interact with the infrared photons are called infrared active. Examples of such gratings are ionic lattice, for example, in sodium chloride crystals. )

The model of lattice vibrations requires a crystalline order. Amorphous solids such as glasses show vibrations of the atoms with each other, but you do not refers to them as phononic oscillations. For long-wavelength acoustic vibrations of the influence of disorder is low.

Excitation energy and Statistics

The energy states of the phonons calculated equivalent to the levels of a harmonic oscillator by

This frequency depends on the vibration mode and the wave vector, see # dispersion.

Since phonons are bosons, the mean occupation number calculated in thermal equilibrium according to the Bose - Einstein distribution as

With

The chemical potential does not appear in the formula because the particle number of phonons is not a conserved quantity.

Usually ( as above) statistical mixtures of states with certain phonon number ( Fock states) are used. As Roy J. Glauber showed for photons in 1963, but there are also so-called phonon coherent states with an indefinite number of particles that resemble greatly classic lattice vibrations. While the expected value of the deflection is in Fock states 0, it is sufficient for coherent phonon states of the classical time dependence of lattice vibrations.

Proof

Experimentally determine the optical phonons can be by means of Raman or infrared spectroscopy. For the determination of the total spectrum of the phonons, both the information on the energy and on the momentum of the lattice vibrations is required. This requirement is met by the inelastic neutron scattering, X-ray scattering and by high-resolution electron energy loss spectroscopy ( HREELS ).

Dispersion

The dispersion relation is a function of the angular frequency of the wave number. For phonons, this relationship results from the Newton 's equation of motion. This one assumes that the atoms are in a periodic potential in which they perform oscillations.

Two adjacent atoms have a phase difference, wherein the distance between two adjacent atoms in the rest position. A phase difference corresponds to a non-zero; higher phase differences are therefore equivalent to a value between 0 and. By symmetry, we consider the interval between and. The corresponding values ​​of k from the first Brillouin zone, ie. This one has covered all physically relevant wave numbers.

Acoustic phonons

For the simple model of a linear chain of atoms, which are connected to each other by springs, is the dispersion relation

Where C ( in kg / s ^ 2 ) is the spring constant between the two adjacent layers, and m is the mass of the atom.

For low values ​​of the expression is approximately

Applies at the zone boundaries

The group velocity, ie the velocity of energy transport in the medium, results to

At the zone boundary, the group velocity is zero: the wave behaves like a standing wave.

Optical phonons

The optical branches exist (similar to the spring constant ), at a base consisting of different atoms or for a system with different force constants C. Both result in a lattice of atoms containing more than one atom in the base. The formula describes the dispersion relation for the model of a linear chain with two different atoms, which have the masses and. The force constant C remains constant.

The optical branch is of higher frequency than the acoustic and almost dispersionless.

648405
de