Normal modes of vibration progression through a crystal. In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid.^{[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. In particular, the properties of longwavelength phonons give rise to sound in solids  hence the name phonon from the Greek φονή (phonē) = voice.^{[2]} In insulating solids, phonons are also the primary mechanism by which heat conduction takes place. Image File history File links 1D_normal_modes_(280_kB). ...
Image File history File links 1D_normal_modes_(280_kB). ...
This is a discussion of a present category of science. ...
In physics, a quantum (plural: quanta) is an indivisible entity of energy. ...
Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...
For other uses, see Atom (disambiguation). ...
For other uses, see Solid (disambiguation). ...
Solidstate physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ...
In physics, thermal conductivity, k, is the intensive property of a material that indicates its ability to conduct heat. ...
Electrical conductivity or specific conductivity is a measure of a materials ability to conduct an electric current. ...
The wavelength is the distance between repeating units of a wave pattern. ...
Sound is a disturbance of mechanical energy that propagates through matter as a wave. ...
Thermal insulation on the Huygens probe 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 intensive 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. These normal modes are important because, according to a wellknown 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 wavelike phenomena in classical mechanics, they acquire certain particlelike properties when the lattice is analysed using quantum mechanics (see waveparticle duality.) They are then known as phonons. Phonons are bosons possessing integer spin. Fig. ...
Look up vibration in Wiktionary, the free dictionary. ...
For other types of mode, see mode. ...
Classical mechanics (also called Newtonian mechanics) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ...
FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...
The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational firstorder reasoning This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ...
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ...
In physics, waveparticle duality holds that light and matter exhibit properties of both waves and of particles. ...
In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ...
In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...
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. In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ...
Mechanics of particles on a lattice Consider a rigid regular (or "crystalline") lattice composed of N particles. (We will refer to these particles as "atoms", though in a real solid they may actually be ions.) N is some large number, say around 10^{23} (on the order of Avogadro's number) for a typical piece of solid. 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 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 forces between each pair of atoms may be characterized by some potential energy function V, depending on the separation of the atoms. The potential energy of the entire lattice is the sum of all the pairwise potential energies: This article is about the electrically charged particle. ...
Avogadros number, also called Avogadros constant (NA), named after Amedeo Avogadro, is formally defined to be the number of carbon12 atoms in 12 grams (0. ...
In physics, force is anything that can cause a massive body to accelerate. ...
In chemistry, the term van der Waals force originally referred to all forms of intermolecular forces; however, in modern usage it tends to refer to intermolecular forces that deal with forces due to the polarization of molecules. ...
Covalent bonding is a form of chemical bonding that is characterized by the sharing of pairs of electrons between atoms, or sometimes between atoms and other covalent bonds. ...
In physics, the space surrounding an electric charge or in the presence of a timevarying magnetic field has a property called an electric field. ...
For other senses of this word, see magnetism (disambiguation). ...
Gravity is a force of attraction that acts between bodies that have mass. ...
{{PortalEnergy}Potential energy is the energy available within a physical system due to an objects position in conjunction with a conservative force which acts upon it (such as the gravitational force or Coulomb force). ...
where is the position of the ^{th} atom, and is the potential energy between two atoms. Space has been an interest for philosophers and scientists for much of human history. ...
{{PortalEnergy}Potential energy is the energy available within a physical system due to an objects position in conjunction with a conservative force which acts upon it (such as the gravitational force or Coulomb force). ...
It is extremely difficult to solve this manybody problem in full generality, in either classical or quantum mechanics. In order to simplify the task, we introduce two important approximations. Firstly, we only perform the sum over neighbouring atoms. Although the electric forces in real solids extend to infinity, this approximation is nevertheless valid because the fields produced by distant atoms are screened. Secondly, we treat the potentials as harmonic potentials: this is permissible as long as the atoms remain close to their equilibrium positions. (Formally, this is done by Taylor expanding about its equilibrium value, which gives proportional to .) This article is about the manybody problem in quantum mechanics. ...
Screening is the damping of electric fields caused by the presence of mobile charge carriers. ...
In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ...
As the degree of the Taylor series rises, it approaches the correct function. ...
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. Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...
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The potential energy of the lattice may now be written as
Here, is the natural frequency of the harmonic potentials, which we assume to be the same since the lattice is regular. is the position coordinate of the ^{th} atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted as "(nn)". fdg220. ...
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. 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. The wavelength is marked. A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ...
It has been suggested that pulse amplitude be merged into this article or section. ...
The wavelength is the distance between repeating units of a wave pattern. ...
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. Image File history File links Lattice_wave. ...
Properly sampled image of brick wall. ...
Not every possible lattice vibration has a welldefined wavelength and frequency. However, the normal modes (which, as we mentioned in the introduction, are the elementary buildingblocks of lattice vibrations) do possess welldefined wavelengths and frequencies. We will now examine it in detail. For other types of mode, see mode. ...
FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...
Phonon dispersion of a onedimensional chain of atoms Consider a onedimensional quantum mechanical harmonic chain of N atoms. 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 Fig. ...
The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...
where is the mass of each atom, and and are the position and momentum operators for the ^{th} atom. A discussion of similar Hamiltonians may be found in the article on the quantum harmonic oscillator. In classical mechanics, momentum (pl. ...
The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...
We introduce a set of "normal coordinates" , defined as the discrete Fourier transforms of the 's and "conjugate momenta" defined as the Fourier transforms of the 's: In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finitedomain discretetime signals. ...
The quantity will turn out to be the wave number of the phonon, i.e. divided by the wavelength. 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 ^{th} atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (mâˆ’1). ...
The wavelength is the distance between repeating units of a wave pattern. ...
The upper bound to comes from the minimum wavelength imposed by the lattice spacing , as discussed above. By inverting the discrete Fourier transforms to express the 's in terms of the 's and the 's in terms of the 's, and using the canonical commutation relations between the 's and 's, we can show that
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,
where
Notice that the couplings between the position variables have been transformed away; if the 's and 's were Hermitian (which they are not), the transformed Hamiltonian would describe uncoupled harmonic oscillators. A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: selfadjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
Threedimensional phonons It is straightforward, though tedious, to generalize the above to a threedimensional lattice. One finds that the wave number k is replaced by a threedimensional wave vector k. Furthermore, each k is now associated with three normal coordinates. A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ...
The new indices s = 1, 2, 3 label the polarization of the phonons. In the one dimensional model, the atoms were restricted to moving along the line, so all the phonons corresponded to longitudinal waves. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular plane, like transverse waves. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons. In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ...
Longitudinal waves are waves that have vibrations along or parallel to their direction of travel. ...
A light wave is an example of a transverse wave. ...
Dispersion relation In the above discussion, we have obtained an equation that relates the frequency of a phonon, , to its wave number :
This is known as a dispersion relation. The relation between the energy of a system and its corresponding momentum is known as its dispersion relation. ...
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, (see group velocity.) At low values of (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately , 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 , i.e. short wavelengths, due to the microscopic details of the lattice. 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 waves amplitude (known as the modulation or envelope of the wave) propagate through space. ...
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 unper and lower sets of curves in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the wavevector. The boundaries at k_{m} and k_{m} are those of the first Brillouin zone. The blue, violet, and brown curves are those of longitudinal acoustic, transverse acoustic 1, and transverse acoutic 2 modes, respectively. In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...
The relation between the energy of a system and its corresponding momentum is known as its dispersion relation. ...
In mathematics and solid state physics, the first Brillouin zone is the primitive cell in the reciprocal lattice in momentum space. ...
The term, longitudinal means fronttoback or toptobottom as opposed to transverse which means sidetoside. In automotive engineering, the term, longitudinal refers to an engine in which the crankshaft is oriented along the long axis of the vehicle, front to back. ...
The term transverse means sidetoside, as opposed to longitudinal, which means fronttoback. In automotive engineering, the term transverse refers to an engine in which the crankshaft is oriented sidetoside relative to the wheels of the vehicle. ...
The term transverse means sidetoside, as opposed to longitudinal, which means fronttoback. In automotive engineering, the term transverse refers to an engine in which the crankshaft is oriented sidetoside relative to the wheels of the vehicle. ...
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. The number of optical modes is 3N  3. Many phonon dispersion curves have been measured by neutron scattering. In solid state physics and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum volume cell corresponding to a single lattice point. ...
The term Neutron Scattering encompasses all scientific techniques whereby neutrons are 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. This is because fluids can't support shear stresses. (but see viscoelastic fluids, which only apply to high frequencies, though). A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ...
Shear stress is a stress state where the stress is parallel or tangential to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ...
A viscoelastic material is one in which: hysteresis is seen in the stressstrain curve. ...
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. "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. In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...
"Optical phonons," which arise in crystals that have more than one atom in the smallest unit cell, always have some minimum frequency of vibration, even when their wavelength is large. They are called "optical" because in ionic crystals (like sodium chloride) they are excited very easily by light (in fact, infrared radiation). 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 timevarying electrical dipole moment. 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. Optical phonons are often abbreviated as LO and TO phonons, for the longitudinal and transverse varieties respectively. In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...
Jordanian and Israeli salt evaporation ponds at the south end of the Dead Sea Sodium chloride, also known as common salt, table salt, or halite, is a chemical compound with the formula NaCl. ...
Image of a small dog taken in midinfrared (thermal) light (false color) Infrared (IR) radiation is electromagnetic radiation of a wavelength longer than visible light, but shorter than microwave radiation. ...
In physics, the electric dipole moment is a measure of the polarity of a system of electric charges. ...
Raman scattering or the Raman effect is the inelastic scattering of a photon. ...
Phonons In fact, the above derived Hamiltonian looks like the classical Hamilton function, but if its interpreted as an operator it describes a quantum field theory of noninteracting bosons. This leads to new physics. Quantum field theory (QFT) is the quantum theory of fields. ...
The energy spectrum of this Hamiltonian is easily obtained by the method of ladder operators, similar to the quantum harmonic oscillator problem. We introduce a set of ladder operators defined by In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinitedimensional spaces. ...
The ladder operators satisfy the following identities: As with the quantum harmonic oscillator, we can then show that and respectively create and destroy one excitation of energy . 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 . 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. In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ...
It is not a priori obvious that these excitations generated by the operators are literally waves of lattice displacement, but one may convince oneself of this by calculating the positionposition correlation function. Let denote a state with a single quantum of mode excited, i.e. For stochastic processes, including those that arise in statistical mechanics and Euclidean quantum field theory, a correlation function is the correlation between random variables at two different points in space or time. ...
One can show that, for any two atoms and ,
which is exactly what we would expect for a lattice wave with frequency and wave number . In three dimensions the Hamiltonian has the form
Crystal momentum 
kVectors exceeding the first Brillouin zone (red) do not carry more information than their counterparts (black) in the first Brillouin zone. It is tempting to treat a phonon with wave vector as though it has a momentum , by analogy to photons and matter waves. This is not entirely correct, for is not actually a physical momentum; it is called the crystal momentum or pseudomomentum. This is because is only determined up to multiples of constant vectors, known as reciprocal lattice vectors. For example, in our onedimensional model, the normal coordinates and are defined so that Crystal momentum is a term used to describe the momentumlike quantum number associated with electrons in a crystal. ...
Image File history File links kVectors exceeding the first Brillouin zone (red) do not carry more information than their counterparts (black) in the first Brillouin zone. ...
Image File history File links kVectors exceeding the first Brillouin zone (red) do not carry more information than their counterparts (black) in the first Brillouin zone. ...
In classical mechanics, momentum (pl. ...
In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...
The wavelength is the distance between repeating units of a wave pattern. ...
In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ...
where
for any integer . A phonon with wave number is thus equivalent to an infinite "family" of phonons with wave numbers , , 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 is the wavefunction of a particle (usually, an electron) placed in a periodic potential. ...
It is usually convenient to consider phonon wave vectors which have the smallest magnitude in their "family". The set of all such wave vectors defines the first Brillouin zone. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector. In mathematics and solid state physics, the first Brillouin zone is the primitive cell in the reciprocal lattice in momentum space. ...
It is interesting that similar consideration is needed in analogtodigital conversion where aliasing may occur under certain conditions. This article or section should include material from AD converters In electronics, an analogtodigital converter (abbreviated ADC, A/D, or A to D) is a device that converts continuous signals to discrete digital numbers. ...
Properly sampled image of brick wall. ...
Image File history File links This is a lossless scalable vector image. ...
Image File history File links This is a lossless scalable vector image. ...
Thermodynamic properties A crystal lattice at zero temperature lies in its ground state, and contains no phonons. According to thermodynamics, when the lattice is held at a nonzero temperature its energy is not constant, but fluctuates randomly about some mean value. 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.) Because these phonons are generated by the temperature of the lattice, they are sometimes referred to as thermal phonons. Absolute zero is the lowest possible temperature where nothing could be colder, and no heat energy remains in a substance. ...
In physics, the ground state of a quantum mechanical system is its lowestenergy state. ...
Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dunamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...
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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 by the number of items in the list. ...
For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another as a result of a difference in temperature. ...
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. This similarity is not coincidental, for it turns out that the electromagnetic field behaves like a set of harmonic oscillators; see Blackbody radiation. Both gases obey the BoseEinstein statistics: in thermal equilibrium and within the harmonic regime, the probability of finding phonons (or photons) in a given state is with a given angular frequency is: An electromagnetic cavity is a cavity which acts as a container for electromagnetic fields such as photons, in effect containing their wave function inside it. ...
As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ...
In statistical mechanics, BoseEinstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ...
where is the frequency of the phonons (or photons) in the state, is Boltzmann's constant, and is the temperature. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...
See also Image File history File links Portal. ...
There are very few or no other articles that link to this one. ...
// Linear elasticity The linear theory of elasticity models the macroscopic mechanical properties of solids assuming small deformations. ...
Rayleigh waves, also known as the RayleighLamb Wave or ground roll, are a type of surface wave. ...
A surface acoustic wave (SAW) is a kind of wave used in piezoelectric devices called SAW devices in electronics circuits. ...
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. ...
3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ...
References  ^ International Union of Pure and Applied Chemistry. "phonon". Compendium of Chemical Terminology Internet edition.
 ^ Although φόνον (phonon) is literally the accusative case of φόνος (phonos) = murder!
IUPAC logo The International Union of Pure and Applied Chemistry (IUPAC) (Pronounced as eyeyoupack) is an international nongovernmental organization established in 1919 devoted to the advancement of chemistry. ...
Compendium of Chemical Terminology (ISBN 0865426848) is a book published by IUPAC containing internationally accepted definitions for terms in chemistry. ...
External links  PHONONS 2007: 12th International Conference on Phonon Scattering in Condensed Matter [1].
