# phonon

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## phonon

(fō`nŏn), quantum of vibrational energy. The atoms of any crystal are in a state of vibration, their average kinetic energy being measured by the absolute temperature**temperature,**

measure of the relative warmth or coolness of an object. Temperature is measured by means of a thermometer or other instrument having a scale calibrated in units called degrees. The size of a degree depends on the particular temperature scale being used.

**.....**Click the link for more information. of the crystal. In certain phenomena it becomes evident that this energy is divided into discrete bundles (see quantum theory

**quantum theory,**

modern physical theory concerned with the emission and absorption of energy by matter and with the motion of material particles; the quantum theory and the theory of relativity together form the theoretical basis of modern physics.

**.....**Click the link for more information. ); the energy bundles behave like particles in some respects and are termed phonons. These effects are most apparent at low temperatures where only a few phonons are present. For example, interactions between phonons and electrons are thought to be responsible for such phenomena as superconductivity

**superconductivity,**

abnormally high electrical conductivity of certain substances. The phenomenon was discovered in 1911 by Heike Kamerlingh Onnes, who found that the resistance of mercury dropped suddenly to zero at a temperature of about 4.

**.....**Click the link for more information. .

## Phonon

A quantum of vibrational energy in a solid or other elastic medium. This vibrational energy can be transported by elastic waves. The energy content of each wave is quantized. For a wave of frequency *f*, the energy is (*N* + ½)*h**f*, where *N* is an integer and *h* is Planck's constant. Apart from the zero-point energy, ½*h**f*, there are *N* quanta of energy *h**f*. In elastic or lattice waves, these quanta are called phonons. Quantization of energy is not related to the discreteness of the lattice, and also applies to waves in a continuum. *See* Quantum mechanics, Wave motion

The concept of phonons closely parallels that of photons, quanta of electromagnetic wave energy. The indirect consequences of quantization were established for phonons just as for photons in the early days of quantum mechanics—for example, the decrease of the specific heat of solids at low temperatures. Direct evidence that the energy of vibrational modes is changed one phonon at a time came much later than that for photons—for example, the photoelectric effect—because phonons exist only within a solid, are subject to strong attenuation and scattering, and have much lower quantum energy than optical or x-ray photons. *See* Photoemission

Like photons, phonons can be regarded as particles, each of energy *h**f* and momentum proportional to the wave vector of the elastic or lattice wave. Such a particle can be said to transport energy, thus moving with a velocity equal to the group velocity of the underlying wave. *See* Lattice vibrations, Photon

## Phonon

the quantum of vibrational motion of the atoms in a crystal. Because of interactions between the atoms in a crystal, atomic vibrations propagate through the crystal as waves. Every such wave may be characterized by a quasiwave vector k and a frequency ω that is a function of k: ω = ω_{v}(k); here, the subscript *v* = 1,2,. . ., 3*r*—where *r* is the number of atoms in a unit cell of the crystal—indicates the mode of vibration (*see*LATTICE VIBRATIONS). According to the laws of quantum mechanics, the vibrational energy of the atoms in a crystal may take on values equal to

where δ_{0} is the ground-state energy and ħ is Planck’s constant. Every wave can be identified with a quasiparticle, that is, with a phonon. The energy of a phonon is equal to ℰ = ħω_{v}(k), and the quasimomentum is *p*= ħ*k*. The number *n _{kv}* should be interpreted as the phonon density. A distinction is made between acoustic and optical phonons. For acoustic phonons, as

*p*→ 0, ℰ =

*sp*, where

*s*is the speed of sound; for optical phonons, as

*p*→ 0, ℰ

_{min}≠ 0. In simple crystals with

*r*= 1, optical phonons do not exist.

Phonons interact with each other; with other quasiparticles, such as conduction electrons and magnons; and with static crystal defects, for example, vacancies, dislocations, crystallite boundaries, the surface of a specimen, and foreign inclusions. The laws of conservation of energy and quasimomentum are satisfied in phonon collisions. The law of conservation of quasimomentum is more general than the law of conservation of momentum (*see*CONSERVATION LAW), since the total quasimomentum of colliding particles, particularly of phonons, may vary by the quantity 2iTftb, where b is a vector in the reciprocal lattice. Collisions in which b is not zero are called Umklapp processes and differ from normal collisions, in which b = 0. The possibility of an Umklapp process is a consequence of periodicity in the arrangement of the atoms in a crystal.

The mean phonon density <*n _{kv}*> is given by the Planck radiation formula

<*n _{kv}* = 1/(e

^{eħωv/kT}– 1)

where *T* is the temperature and *k* is the Boltzmann constant. The formula matches the distribution of gas particles that obey Bose-Einstein statistics when the chemical potential is equal to zero (*see*STATISTICAL MECHANICS). A chemical potential equal to zero means that the phonon number *N*_{p} in a crystal is not conserved but depends on temperature. For all solids, as *T* → 0, *N*_{p} ~ *T*^{3}; if *T* > θ_{D}, where θ_{D} is the Debye temperature, *N*_{p} ~ *T*. The concept of the phonon makes it possible to describe the thermal and other properties of crystals by using the methods of the kinetic theory of gases. In most cases, phonons are the primary heat reservoir of a solid. The heat capacity of a crystalline solid is practically the same as that of a phonon gas. The thermal conductivity of a crystal may be described as the thermal conductivity of a phonon gas whose thermal resistance owes its origin to Umklapp processes.

The scattering of conduction electrons in interactions with phonons is the primary mechanism that gives rise to the electrical resistance of metals and semiconductors. The ability of conduction electrons to emit and absorb phonons leads to the mutual attraction of the electrons; at low temperatures, this phenomenon causes various metals to become superconducting (*see*SUPERCONDUCTIVITY and COOPER EFFECT). The emission of phonons by excited atoms and molecules in solids makes it possible for nonradiative electronic transitions to occur. During relaxation processes in solids, phonons usually serve as a sink for the energy stored by other degrees of freedom of the crystal, for example, the electronic degrees of freedom.

As is the case with gases consisting of other quasiparticles, the mean energy of a phonon gas may be characterized by a quantity similar to the temperature of an ordinary gas. However, because of the relatively weak coupling between phonons and other quasiparticles, the phonon or lattice temperature may differ from the temperature of other quasiparticles, such as conduction electrons, magnons, and excitons. In the case of amorphous—that is, vitreous—solids, the concept of the phonon can be introduced only for long-wavelength acoustic vibrations, which are only slightly affected by the relative arrangement of the atoms.

Elementary excitations in superfluid helium, which describe the vibrational motion of a quantum fluid, are also called phonons (*see*SUPERFLUIDITY).

### REFERENCES

Ziman, J.*Elektrony I fonony*. Moscow, 1962. (Translated from English.)

Kosevich, A. M.

*Osnovy mekhaniki kristallicheskoi reshetki*. Moscow, 1972.

Reissland, J.

*Fizika fononov*. Moscow, 1975. (Translated from English.)

M. I. KAGANOV