Phonon

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Normal modes of vibration progression through a crystal. The amplitude of the motion has been exaggerated for ease of viewing; in an actual crystal, it is typically much smaller than the lattice spacing.

In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny A solid' object is in the States of matter characterized by resistance to Deformation and changes of Volume. ^[1] The study of phonons is an important part of solid state physics, because phonons play a major role in many of the physical properties of solids, including a material's thermal and electrical conductivities. Solid-state physics, the largest branch of Condensed matter physics, is the study of rigid Matter, or Solids The bulk of solid-state physics theory and In Physics, thermal conductivity, k is the property of a material that indicates its ability to conduct Heat. Electrical conductivity or specific conductivity is a measure of a material's ability to conduct an Electric current. In particular, the properties of long-wavelength phonons give rise to sound in solids -- hence the name phonon from the Greek φωνή (phonē) = voice. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. Sound' is Vibration transmitted through a Solid, Liquid, or Gas; particularly sound means those vibrations composed of Frequencies ^[2] In insulating solids, phonons are also the primary mechanism by which heat conduction takes place. The term thermal insulation can refer to materials used to reduce the rate of Heat transfer, or the methods and processes used to reduce heat transfer In Physics, thermal conductivity, k is the property of a material that indicates its ability to conduct Heat.

Phonons are a quantum mechanical version of a special type of vibrational motion, known as normal modes in classical mechanics, in which each part of a lattice oscillates with the same frequency. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Vibration refers to mechanical Oscillations about an equilibrium point. A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Frequency is a measure of the number of occurrences of a repeating event per unit Time. These normal modes are important because, according to a well-known result in classical mechanics, any arbitrary vibrational motion of a lattice can be considered as a superposition of normal modes with various frequencies; in this sense, the normal modes are the elementary vibrations of the lattice. Although normal modes are wave-like phenomena in classical mechanics, they acquire certain particle-like properties when the lattice is analysed using quantum mechanics (see wave-particle duality. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and ) They are then known as phonons.

1 Repeating derivation of normal modes
2 Phonons
- 2.1 Crystal momentum
- 2.2 Thermodynamic properties
3 See also
4 References
5 External links

Repeating derivation of normal modes

The equations in this subsection either do not use axioms of quantum mechanics or use relations for which there exists a direct correspondence in classical mechanics. This article discusses quantum theory For other uses see Correspondence principle (disambiguation.

Mechanics of particles on a lattice

Consider a rigid regular (or "crystalline"; not amorphous) lattice composed of N particles. (We will refer to these particles as "atoms". In a real solid these atoms may be ions. An ion is an Atom or Molecule which has lost or gained one or more Valence electrons giving it a positive or negative electrical charge ) N is some large number, say around 10²³ (on the order of Avogadro's number) for a typical piece of solid. The Avogadro constant (symbols L, N A also called Avogadro's number, is the number of "elementary entities" (usually Atoms If the lattice is rigid, the atoms must be exerting forces on one another, so as to keep each atom near its equilibrium position. In Physics, a force is whatever can cause an object with Mass to Accelerate. In real solids, these forces include Van der Waals forces, covalent bonds, and so forth, all of which are ultimately due to the electric force; magnetic and gravitational forces are generally negligible. The Van der Waals equation is an Equation of state that can be derived from a special form of the potential between a pair of molecules (hard-sphere repulsion In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. Gravitation is a natural Phenomenon by which objects with Mass attract one another The forces between each pair of atoms may be characterized by some potential energy function V, depending on the separation of the atoms. Potential energy can be thought of as Energy stored within a physical system The potential energy of the entire lattice is the sum of all the pairwise potential energies:

$\,\sum_{i < j} V(r_i - r_j)$

where $\, r_i$ is the position of the $\, i$ th atom, and $\, V$ is the potential energy between two atoms. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another Potential energy can be thought of as Energy stored within a physical system

It is extremely difficult to solve this many-body problem in full generality, in either classical or quantum mechanics. The many-body problem may be defined as the study of the effects of interaction between bodies on the behaviour of a many-body system i In order to simplify the task, we introduce two important approximations. First, we perform the sum over neighboring atoms only. Although the electric forces in real solids extend to infinity, this approximation is nevertheless valid because the fields produced by distant atoms are screened. Screening is the damping of Electric fields caused by the presence of mobile charge carriers Secondly, we treat the potentials $\, V$ as harmonic potentials: this is permissible as long as the atoms remain close to their equilibrium positions. This article is about the harmonic oscillator in classical mechanics (Formally, this is done by Taylor expanding $\, V$ about its equilibrium value, which gives $\, V$ proportional to $\, x^2$ . In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives )

The resulting lattice may be visualized as a system of balls connected by springs. The following figure shows a cubic lattice, which is a good model for many types of crystalline solid. Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modelling phonons. Other common lattices may be found in the article on crystal structure. In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal.

The potential energy of the lattice may now be written as

$\sum_{i \ne j (nn)} {1\over2} m \omega^2 (R_i - R_j)^2$

Here, $\,\omega$ is the natural frequency of the harmonic potentials, which we assume to be the same since the lattice is regular. The fundamental tone, often referred to simply as the fundamental and abbreviated fo, is the lowest frequency in a harmonic series. $\, R_i$ is the position coordinate of the $\, i$ th atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted as "(nn)".

Lattice waves

Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions will give rise to a set of vibration waves propagating through the lattice. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. One such wave is shown in the figure below. The amplitude of the wave is given by the displacements of the atoms from their equilibrium positions. Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system The wavelength $\,\lambda$ is marked. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency.

There is a minimum possible wavelength, given by the equilibrium separation a between atoms. As we shall see in the following sections, any wavelength shorter than this can be mapped onto a wavelength longer than a, due to effects similar to that in aliasing. This article applies to signal processing including computer graphics

Not every possible lattice vibration has a well-defined wavelength and frequency. However, the normal modes (which, as we mentioned in the introduction, are the elementary building-blocks of lattice vibrations) do possess well-defined wavelengths and frequencies. A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency Frequency is a measure of the number of occurrences of a repeating event per unit Time. We will now examine it in detail.

Phonon dispersion of a one-dimensional chain of identical atoms

Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions. The Hamiltonian for this system is

$\mathbf{H} = \sum_{i=1}^N {p_i^2 \over 2m} + {1\over 2} m \omega^2 \sum_{\{ij\} (nn)} (x_i - x_j)^2$

where $\, m$ is the mass of each atom, and $\, x_i$ and $\, p_i$ are the position and momentum operators for the $\, i$ th atom. In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product A discussion of similar Hamiltonians may be found in the article on the quantum harmonic oscillator. The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator.

We introduce a set of $\, N$ "normal coordinates" $\, Q_k$ , defined as the discrete Fourier transforms of the $\, x$ 's and $\, N$ "conjugate momenta" $\,\Pi$ defined as the Fourier transforms of the $\, p$ 's:

$\begin{matrix} x_j &=& {1\over\sqrt{N}} \sum_{n=-N}^{N} Q_{k_n} e^{ik_nja} \\ p_j &=& {1\over\sqrt{N}} \sum_{n=-N}^{N} \Pi_{k_n} e^{-ik_nja} \\ \end{matrix}$

The quantity $\, k_n$ will turn out to be the wave number of the phonon, i. In Mathematics, the discrete Fourier transform (DFT is one of the specific forms of Fourier analysis. Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters e. $\, 2\,\pi$ divided by the wavelength. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. It takes on quantized values, because the number of atoms is finite. The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the $\, (N+1)$ th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

$k_n = {n\pi \over Na} \quad \hbox{for}\ n = 0, \pm1, \pm2, ... , \pm N$

The upper bound to $\, n$ comes from the minimum wavelength imposed by the lattice spacing $\, a$ , as discussed above.

By inverting the discrete Fourier transforms to express the $\, Q$ 's in terms of the $\, x$ 's and the $\,\Pi$ 's in terms of the $\, p$ 's, and using the canonical commutation relations between the $\, x$ 's and $\, p$ 's, we can show that

$\left[ Q_k , \Pi_{k'} \right] = i \hbar \delta_{k k'} \quad ;\quad \left[ Q_k , Q_{k'} \right] = \left[ \Pi_k , \Pi_{k'} \right] = 0$

In other words, the normal coordinates and their conjugate momenta obey the same commutation relations as position and momentum operators! Writing the Hamiltonian in terms of these quantities,

$\mathbf{H} = \sum_k \left( { \Pi_k\Pi_{-k} \over 2m } + {1\over2} m \omega_k^2 Q_k Q_{-k} \right)$

where

$\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))}$

Notice that the couplings between the position variables have been transformed away; if the $\, Q$ 's and $\,\Pi$ 's were Hermitian (which they are not), the transformed Hamiltonian would describe $\, N$ uncoupled harmonic oscillators. A number of Mathematical entities are named Hermitian, after the Mathematician Charles Hermite: Hermitian adjoint

Three-dimensional phonons

It is straightforward, though tedious, to generalize the above to a three-dimensional lattice. One finds that the wave number k is replaced by a three-dimensional wave vector k. A wave vector is a vector representation of a Wave. The wave vector has magnitude indicating Wavenumber (reciprocal of Wavelength) and the Furthermore, each k is now associated with three normal coordinates.

The new indices s = 1, 2, 3 label the polarization of the phonons. Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations In the one dimensional model, the atoms were restricted to moving along the line, so all the phonons corresponded to longitudinal waves. Longitudinal waves are waves that have vibrations along or parallel to their direction of travel that is waves in which the motion of the medium is in the same direction as the motion In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular plane, like transverse waves. A transverse wave is a moving Wave that consists of oscillations occurring perpendicular to the direction of energy transfer This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.

Dispersion relation

In the above discussion, we have obtained an equation that relates the frequency of a phonon, $\,\omega_k$ , to its wave number $\, k$ :

$\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))} = 2 \omega | \sin(ka/2) |$

This is known as a dispersion relation. Dispersion relations describe the ways that wave propagation varies with the Wavelength or Frequency of a wave.

The speed of propagation of a phonon, which is also the speed of sound in the lattice, is given by the slope of the dispersion relation, $\,\tfrac{\partial\omega_k}{\partial k}$ (see group velocity. Sound is a vibration that travels through an elastic medium as a Wave. The group velocity of a Wave is the Velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope ) At low values of $\, k$ (i. e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately $\,\omega a$ , independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of $\, k$ , i. e. short wavelengths, due to the microscopic details of the lattice.

For a crystal that has at least two atoms in a unit cell (which may or may not be different), the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper and lower sets of curves in the diagram, respectively. In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. Dispersion relations describe the ways that wave propagation varies with the Wavelength or Frequency of a wave. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the wave-vector. A wave vector is a vector representation of a Wave. The wave vector has magnitude indicating Wavenumber (reciprocal of Wavelength) and the The boundaries at -k_m and k_m are those of the first Brillouin zone. In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the The blue, violet, and brown curves are those of longitudinal acoustic, transverse acoustic 1, and transverse acoustic 2 modes, respectively.

In some crystals the two transverse acoustic modes have exactly the same dispersion curve. It is also interesting that for a crystal with N ( > 2) different atoms in a primitive cell, there are always three acoustic modes. In Geometry, Solid state physics and Mineralogy, particularly in describing Crystal structure, a primitive cell, is a minimum cell corresponding The number of optical modes is 3N - 3. Many phonon dispersion curves have been measured by neutron scattering. The term "Neutron Scattering" encompasses all scientific techniques whereby the deflection of Neutron radiation is used as a scientific probe

The physics of sound in fluids differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code This is because fluids can't support shear stresses. A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material (but see viscoelastic fluids, which only apply to high frequencies, though). Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation.

Acoustic and optical phonons

In solids with more than one atom in the smallest unit cell, there are two types of phonons: "acoustic" phonons and "optical" phonons. In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. "Acoustic phonons", which are the phonons described above, have frequencies that become small at the long wavelengths, and correspond to sound waves in the lattice. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively.

"Optical phonons," which also arise in crystals with more than one atom in the smallest unit cell, always have some minimum frequency of vibration, even when their wavelength is large. In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. They are called "optical" because in ionic crystals (like sodium chloride) they are excited very easily by light (in fact, infrared radiation). For sodium chloride in the diet see Salt. Sodium chloride, also known as common salt, table salt, or Halite, is a Infrared ( IR) radiation is Electromagnetic radiation whose Wavelength is longer than that of Visible light, but shorter than that of This is because they correspond to a mode of vibration where positive and negative ions at adjacent lattice sites swing against each other, creating a time-varying electrical dipole moment. In Physics, the electric dipole moment (or electric dipole for short is a measure of the polarity of a system of Electric charges. Optical phonons that interact in this way with light are called infrared active. Optical phonons which are Raman active can also interact indirectly with light, through Raman scattering. Raman scattering or the Raman effect (pronounced — is the inelastic scattering of a Photon. Optical phonons are often abbreviated as LO and TO phonons, for the longitudinal and transverse varieties respectively.

Phonons

In fact, the above derived Hamiltonian looks like the classical Hamilton function, but if it is interpreted as an operator, then it describes a quantum field theory of non-interacting bosons. In quantum field theory (QFT the forces between particles are mediated by other particles This leads to new physics.

The energy spectrum of this Hamiltonian is easily obtained by the method of ladder operators, similar to the quantum harmonic oscillator problem. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of Eigenvalues for matrices We introduce a set of ladder operators defined by

$\begin{matrix} a_k &=& \sqrt{m\omega_k \over 2\hbar} (Q_k + {i\over m\omega_k} \Pi_{-k}) \\ a_k^\dagger &=& \sqrt{m\omega_k \over 2\hbar} (Q_{-k} - {i\over m\omega_k} \Pi_k) \end{matrix}$

The ladder operators satisfy the following identities:

$\mathbf{H} = \sum_k \hbar \omega_k \left(a_k^{\dagger}a_k + 1/2\right)$

$[a_k , a_{k'}^{\dagger} ] = \delta_{kk'}$

$[a_k , a_{k'} ] = [a_k^{\dagger} , a_{k'}^{\dagger} ] = 0.$

As with the quantum harmonic oscillator, we can then show that $\, a_k^\dagger$ and $\, a_k$ respectively create and destroy one excitation of energy $\,\hbar\omega_k$ . These excitations are phonons.

We can immediately deduce two important properties of phonons. Firstly, phonons are bosons, since any number of identical excitations can be created by repeated application of the creation operator $\, a_k^\dagger$ . In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein Secondly, each phonon is a "collective mode" caused by the motion of every atom in the lattice. This may be seen from the fact that the ladder operators contain sums over the position and momentum operators of every atom.

It is not a priori obvious that these excitations generated by the $\, a$ operators are literally waves of lattice displacement, but one may convince oneself of this by calculating the position-position correlation function. Correlation functions contain information about the distribution of points or events or things across some space/time Let $\, |k\rangle$ denote a state with a single quantum of mode $\, k$ excited, i. e.

$\begin{matrix} | k \rangle = a_k^\dagger | 0 \rangle \end{matrix}$

One can show that, for any two atoms $\, j$ and $\,\ell$ ,

$\langle k | x_j(t) x_{\ell}(0) | k \rangle = \frac{\hbar}{Nm\omega_k} \cos \left[ k(j-\ell)a - \omega_k t \right] + \langle 0 | x_j(t) x_\ell(0) |0 \rangle$

which is exactly what we would expect for a lattice wave with frequency $\,\omega_k$ and wave number $\, k$ .

In three dimensions the Hamiltonian has the form

$\mathbf{H} = \sum_k \sum_{s = 1}^3 \hbar \, \omega_{k,s} \left( a_{k,s}^{\dagger}a_{k,s} + 1/2 \right)$

Crystal momentum

Main article: Crystal momentum

k-Vectors exceeding the first Brillouin zone (red) do not carry more information than their counterparts (black) in the first Brillouin zone. In solid state physics crystal momentum is a momentum -like vector associated with Electrons in a crystal lattice.

It is tempting to treat a phonon with wave vector $\, k$ as though it has a momentum $\,\hbar k$ , by analogy to photons and matter waves. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena In Physics, the de Broglie hypothesis (pronounced /brœj/ as French breuil close to "broy" is the statement that all Matter (any object has a Wave This is not entirely correct, for $\,\hbar k$ is not actually a physical momentum; it is called the crystal momentum or pseudomomentum. This is because $\, k$ is only determined up to multiples of constant vectors, known as reciprocal lattice vectors. In Crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that e^{i\mathbf{K}\cdot\mathbf{R}}=1 For example, in our one-dimensional model, the normal coordinates $\, Q$ and $\,\Pi$ are defined so that

$Q_k \ \stackrel{\mathrm{def}}{=}\ Q_{k+K} \quad;\quad \Pi_k \ \stackrel{\mathrm{def}}{=}\ \Pi_{k + K} \quad$

where

$\, K = 2n\pi/a$

for any integer $\, n$ . A phonon with wave number $\, k$ is thus equivalent to an infinite "family" of phonons with wave numbers $\, k\pm\tfrac{2\,\pi}{a}$ , $\, k\pm\tfrac{4\,\pi}{a}$ , and so forth. Physically, the reciprocal lattice vectors act as additional "chunks" of momentum which the lattice can impart to the phonon. Bloch electrons obey a similar set of restrictions. A Bloch wave or Bloch state, named after Felix Bloch, is the Wavefunction of a particle (usually an Electron) placed in a periodic potential

It is usually convenient to consider phonon wave vectors $\, k$ which have the smallest magnitude $\, (|k|)$ in their "family". The set of all such wave vectors defines the first Brillouin zone. In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector.

It is interesting that similar consideration is needed in analog-to-digital conversion where aliasing may occur under certain conditions. An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic integrated circuit which converts continuous signals to This article applies to signal processing including computer graphics

Brillouin zone

Thermodynamic properties

A crystal lattice at zero temperature lies in its ground state, and contains no phonons. Absolute zero is the point at which molecules do not move (relative to the rest of the body more than they are required to by a quantum mechanical effect called Zero-point In Quantum mechanics, a stationary state is an Eigenstate of a Hamiltonian, or in other words a state of definite energy According to thermodynamics, when the lattice is held at a non-zero temperature its energy is not constant, but fluctuates randomly about some mean value. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature Randomness is a lack of order Purpose, cause, or predictability In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided These energy fluctuations are caused by random lattice vibrations, which can be viewed as a gas of phonons. (Note: the random motion of the atoms in the lattice is what we usually think of as heat. In Physics, heat, symbolized by Q, is Energy transferred from one body or system to another due to a difference in Temperature ) Because these phonons are generated by the temperature of the lattice, they are sometimes referred to as thermal phonons.

Unlike the atoms which make up an ordinary gas, thermal phonons can be created or destroyed by random energy fluctuations. In the language of statistical mechanics this means that the chemical potential for adding a phonon is zero. It is very important to note that this behaviour takes us away from the harmonic potential mentioned earlier, and into the anharmonic regime. The behaviour of thermal phonons is similar to the photon gas produced by an electromagnetic cavity, wherein photons may be emitted or absorbed by the cavity walls. An Electromagnetic cavity is a cavity that acts as a container for Electromagnetic fields such as Photons in effect containing their Wave function inside This similarity is not coincidental, for it turns out that the electromagnetic field behaves like a set of harmonic oscillators; see Black-body radiation. In Physics, a black body is an object that absorbs all light that falls on it Both gases obey the Bose-Einstein statistics: in thermal equilibrium and within the harmonic regime, the probability of finding phonons (or photons) in a given state with a given angular frequency is:

$n(\omega_{k,s}) = \frac{1}{\exp(\hbar\omega_{k,s}/k_BT) - 1}$

where $\,\omega_{k,s}$ is the frequency of the phonons (or photons) in the state, $\, k_B$ is Boltzmann's constant, and $\, T$ is the temperature. In Statistical mechanics, Bose - Einstein statistics (or more colloquially B-E statistics determines the statistical distribution of Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics

References

^ International Union of Pure and Applied Chemistry. A fracton is a collective quantized vibration on a substrate with a Fractal structure Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions Rayleigh waves, also known as the Rayleigh-Lamb Wave or "ground roll" are a type of Surface wave. A surface acoustic wave ( SAW) is an Acoustic wave traveling along the surface of a material having some elasticity, with an Amplitude that Rigid Unit Modes (RUMs represent a class of lattice vibrations or Phonons that exist in network materials such as Quartz, Cristobalite or Zirconium A phononic crystal is a material which exhibits Stop bands for Phonons preventing phonons of selected ranges of frequencies from being transmitted through the material In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by The International Union of Pure and Applied Chemistry ( IUPAC) (aɪjuːpæk or ay-yoo-pec) is an international Non-governmental organization "phonon". Compendium of Chemical Terminology Internet edition. Compendium of Chemical Terminology (ISBN 0-86542-684-8 is a book published by IUPAC containing internationally accepted definitions for terms in Chemistry.
^ Although φόνον (phonon) is literally the accusative case of φόνος (phonos) = murder!

External links

PHONONS 2007: 12th International Conference on Phonon Scattering in Condensed Matter [1].

Dictionary

-noun

(physics) The quantum of acoustic or vibrational energy (sound), considered a discrete particle.

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