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| {{For|[[KDE Software Compilation 4]]'s multimedia framework|Phonon (KDE)}}
| | Gail Edgell: I know you spoke about Their tea in [http://forskolinbellybuster.net/ Forskolin Belly Buster Review] a hot-tea formula, or a tea-bag situation. I know that there are hundreds around the market on industry. What kind of Green tea supplement should people be shopping?<br><br><br><br>The first one i want talk about is CLA; that is actually abbreviation for conjugated linoleic acid. Is actually because a connected with fatty [http://forskolinbellybuster.net/ Forskolin Belly Buster] acid that promotes fat loss and muscle growth. It takes place in some foods -- for example, it rrs extremely high in cottage mozerella. I prescribe 1,000 milligrams, twice a day, from a capsule as an ingredient of a weight-loss class. It increases the burning of body. That is how it also helps us drop more quickly. We are going to promote fat loss, but certainly are to be able to promote muscle growth. How come important? Muscle has an increased metabolic rate, so it keeps us lean. If you have more muscle mass, utilized eat more calories and yet stay very thin. You want to be using things does not only drop the weight but also build muscular. That is a real step to staying trim.<br><br>A associated with people are aware of a lot about Green tea and reduction. It is a useful herb to drop the weight because it, too, raises fat approach. Green Tea also reduces appetite. It's got a minute of caffeine and metabolic process and decreased medicinal component, ECGC. The decaffeinated form is and not as effective for weight reduction as the natural caffeinated form. It can be used in capsules, or it could be taken for a beverage tea. When you take Green tea in capsules, you want a fat-soluble capsule; it is much more proficient. Green Tea really, really increases thermogenics, or energy expenditure (calorie burn), that can boost your metabolism by 4 percent a celebration.<br><br>Dr. Chilkov: The point is having a lifelong normal weight. Maintaining a normal fat-to-lean-body-mass ratio is often a lifestyle. Then we deal specifically with metabolic issues with herbs and nutrients.<br><br>Here is really a brief report on some of this supplements usually are available: thermogenic fat burners, carb blockers, fat blockers, stimulant-free fat burner, thyroid hormone increasers, cortisol products and appetite suppressants. We probably should explain what almost all of these supplements are understanding that should a person a better idea with the items you should expect while taking each associated with these.<br><br>Use Supplements: Often times even info about the subject diets cannot supply us with every nutrient entire body need for running efficiently. Try supplementing with a or every one of the following: Multi-vitamin with spirulina, omega 3 and 6 fatty acids, [http://forskolinbellybuster.net/ Forskolin Belly Buster Reviews], guggul, advantra Z, L-tyrosine, chromium, and vanadyl sulfate.<br><br>Cellulite is caused via the fat cells that accumulate overtime and push resistant to the surrounding connective tissues. They normally start appearing located on the back of this thighs and bottom that could go for the hands and calves as well. Cellulite generally found in ladies but are are also frequent in a few men also. |
| {{Refimprove|date=February 2010}}
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| [[File:1D normal modes (280 kB).gif|thumb|275px|[[Normal mode]]s 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]].]]
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| In [[physics]], a '''phonon''' is a [[collective excitation]] in a periodic, [[Elasticity (physics)|elastic]] arrangement of [[atom]]s or [[molecule]]s in [[condensed matter physics|condensed matter]], such as [[solids]] and some [[liquid]]s. Often referred to as a [[quasiparticle]],<ref>F. Schwabel, ''Advanced Quantum Mechanics'', 4th Ed., Springer (2008), p. 253</ref> it represents an [[excited state]] in the [[quantum mechanical]] quantization of the [[mode of vibration|modes of vibrations]] of elastic structures of interacting particles.
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| Phonons play a major role in many of the physical properties of condensed matter, such as [[thermal conductivity]] and [[electrical conductivity]]. The study of phonons is an important part of condensed matter physics.
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| The concept of phonons was introduced in 1932 by Russian physicist [[Igor Tamm]]. The name ''phonon'' comes from the [[Greek language|Greek]] word ''φωνή'' (phonē), which translates as ''sound'' or ''voice'' because long-wavelength phonons give rise to [[sound]].
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| ==Definition==
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| A phonon is a [[quantum mechanics|quantum mechanical]] description of an elementary [[vibration]]al motion in which a [[lattice model (physics)|lattice]] of atoms or molecules uniformly oscillates at a single [[frequency]]. In [[classical mechanics]] this is known as a [[normal mode]]. Normal modes are important because any arbitrary lattice vibration can be considered as a [[superposition principle|superposition]] of these ''elementary'' vibrations (cf. [[Fourier analysis]]). While normal modes are [[wave|wave-like]] phenomena in classical mechanics, phonons have [[Elementary particle|particle-like]] properties as well in a way related to the [[wave–particle duality]] of quantum mechanics.
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| ==Lattice dynamics==
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| The equations in this section either do not use [[axiom]]s of quantum mechanics or use relations for which there exists a direct [[correspondence principle|correspondence]] in classical mechanics.
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| For example, a rigid regular, [[crystalline]], i.e. not [[amorphous solid|amorphous]], lattice is composed of ''N'' particles. These particles may be atoms, but they may be molecules as well. ''N'' is a large number, say ~10<sup>23</sup>, and on the order of [[Avogadro's number]], for a typical sample of solid. If the lattice is rigid, the atoms must be exerting [[force]]s on one another to keep each atom near its equilibrium position. These forces may be [[Van der Waals force]]s, [[covalent bond]]s, electrostatic attractions, and others, all of which are ultimately due to the [[electric field|electric]] force. [[magnetism|Magnetic]] and [[gravity|gravitational]] forces are generally negligible. The forces between each pair of atoms may be characterized by a [[potential energy]] function <math>\, V</math> that depends on the distance of separation of the atoms. The potential energy of the entire lattice is the sum of all pairwise potential energies:<ref name=latticemechanics3>{{cite book
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| | last = Krauth| first = Werner| title =Statistical mechanics: algorithms and computations | |
| | publisher =Oxford University Press| date =April 2006| location =International publishing locations| pages =231–232
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| | url =http://books.google.com/books?id=EnabPPmmS4sC&pg=RA1-PA231&dq=Mechanics+of+particles+on+a+lattice#v=onepage&q=Mechanics%20of%20particles%20on%20a%20lattice&f=false| isbn =978-0-19-851536-4}}</ref>
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| :<math>\,\sum_{i < j} V(r_i - r_j)</math>
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| where <math>\, r_i</math> is the [[space|position]] of the <math>\, i</math>th atom, and <math>\, V</math> is the [[potential energy]] between two atoms.
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| It is difficult to solve this [[many-body problem]] in full generality, in either classical or quantum mechanics. In order to simplify the task, two important approximations are usually imposed. First, the sum is only performed over neighboring atoms. Although the electric forces in real solids extend to infinity, this approximation is nevertheless valid because the fields produced by distant atoms are effectively [[electric field screening|screened]]. Secondly, the potentials <math>\, V</math> are treated as [[harmonic oscillator|harmonic potentials]]. This is permissible as long as the atoms remain close to their equilibrium positions. Formally, this is accomplished by [[Taylor series|Taylor expanding]] | |
| <math>\, V</math> about its equilibrium value to quadratic order, giving <math>\, V</math> proportional to the displacement <math>\, x^2</math> and the elastic force simply proportional to <math>\, x</math>. The error in ignoring higher order terms remains small if <math>\, x</math> remains close to the equilibrium position.
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| 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 modeling phonons. Other common lattices may be found under "[[crystal structure]]".
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| :[[File:Cubic.svg]]
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| <!-- Unsourced image removed: [[File:Linear crystal shape.png]] -->
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| The potential energy of the lattice may now be written as
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| :<math>\sum_{\{ij\} (nn)} {1\over2} m \omega^2 (R_i - R_j)^2.</math>
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| Here, <math>\,\omega</math> is the [[natural frequency]] of the harmonic potentials, which are assumed to be the same since the lattice is regular. <math>\, R_i</math> is the position coordinate of the <math>\, i</math>th atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted as ''(nn)''.
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| ===Lattice waves===
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| [[File:Lattice wave.svg|200px|thumb|right|Phonon propagating through a square lattice (atom displacements greatly exaggerated)]]
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| Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions give rise to a set of vibration [[wave]]s propagating through the lattice. One such wave is shown in the figure to the right. The [[amplitude]] of the wave is given by the displacements of the atoms from their equilibrium positions. The [[wavelength]] <math>\,\lambda</math> is marked.
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| There is a minimum possible wavelength, given by twice the equilibrium separation ''a'' between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than 2''a'', due to the periodicity of the lattice.
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| Not every possible lattice vibration has a well-defined wavelength and frequency. However, the [[normal mode]]s do possess well-defined wavelengths and [[frequency|frequencies]].
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| ===One dimensional lattice===
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| In order to simplify the analysis needed for a 3-dimensional lattice of atoms it is convenient to model a 1-dimensional lattice or linear chain. This model is complex enough to display the salient features of phonons.
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| ====Classical treatment====
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| The forces between the atoms are assumed to be linear and nearest-neighbour,
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| and they are represented by an elastic spring. Each atom is assumed to be a point particle and the nucleus and electrons move in step ([[adiabatic approximation]]). | |
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| ::::::::{{spaces|4}}n-1 {{pad|3em}} n {{pad|4em}} n+1 {{pad|8em}} ← {{pad|1em}} d {{pad|1em}} →
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| <math>\cdots</math>o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o<math>\cdots</math>
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| :::::::::→→{{pad|4em}}→{{pad|4em}}→→→
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| :::::::::<math>u_{n-1} \qquad\quad u_n \qquad\quad u_{n+1}</math>
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| Where '''<math>n</math>''' labels the <math>n</math>-th atom, '''<math>d</math>''' is the distance between atoms when the chain is in equilibrium and '''<math>u_n</math>''' the displacement of the <math>n</math>-th atom from its equilibrium position.<br />
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| If <math>C</math> is the elastic constant of the spring and <math>m</math> the mass of the atom then the equation of motion of the <math>n</math>-th atom is :
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| :<math>-2Cu_n + C(u_{n+1} + u_{n-1}) = m{\operatorname{d^2}u_n\over\operatorname{d}t^2}</math>
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| This is a set of coupled equations and since the solutions are expected to be oscillatory, new coordinates can be defined by a discrete Fourier transform, in order to de-couple them.<ref>Mattuck R. A guide to Feynman Diagrams in the many-body problem</ref>
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| Put
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| :<math>u_n = \sum_{k=1}^N U_k e^{iknd}</math>
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| Here '''<math>nd</math>''' replaces the usual continuous variable '''<math>x</math>'''. The '''<math>U_k</math>''' are known as the normal coordinates. Substitution into the equation of motion produces the following decoupled equations.(This requires a significant manipulation using the orthonormality and completeness relations of the discrete fourier transform <ref>Greiner & Reinhardt. Field Quantisation</ref>)
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| : <math> 2C(\cos\,kd-1)U_k = m{\operatorname{d^2}U_k\over\operatorname{d}t^2}</math>
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| These are the equations for [[harmonic oscillators]] which have the solution:
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| :<math>U_k=A_ke^{i\omega_kt};\qquad\quad \omega_k=\sqrt{ {2C \over m}(1-\cos{kd})}</math>
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| Each normal coordinate '''<math>U_k</math>''' represents an independent vibrational mode of the lattice with wavenumber '''<math>k</math>''' which is known as a [[normal mode]]. The second equation for '''<math>\omega_k</math>''' is known as the [[dispersion relation]] between the [[angular frequency]] and the [[wavenumber]].<ref>Donovan B. & Angress J.; Lattice Vibrations</ref>
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| ====Quantum treatment====
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| A one-dimensional quantum mechanical harmonic chain consists of ''N'' identical atoms. This is the simplest quantum mechanical model of a lattice that allows phonons to arise from it. The formalism for this model is readily generalizable to two and three dimensions.
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| As in the previous section, the positions of the masses are denoted by <math>x_1,x_2,...</math>, as measured from their equilibrium positions (i.e. <math>x_i=0</math> if particle <math> i</math> is at its equilibrium position.) In two or more dimensions, the <math>x_i</math> are vector quantities. The [[Hamiltonian (quantum mechanics)|Hamiltonian]] for this system is
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| :<math>\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\ </math>
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| where <math>\, m</math> is the mass of each atom (assuming is equal for all), and <math>\, x_i</math> and <math>\, p_i</math> are the position and [[momentum]] operators for the <math>\, i</math>th atom and the sum is made over the nearest neighbors (nn). However one expects that in a lattice there could also appear waves that behave like particles. It is customary to deal with [[waves]] in [[fourier space]] which uses [[normal modes]] of the [[wavevector]] as variables instead coordinates of particles. The number of normal modes is same as the number of particles. However, the [[fourier space]] is very useful given the [[Fourier series|periodicity]] of the system.
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| A set of <math>\, N</math> "normal coordinates" <math>\, Q_k</math> may be introduced, defined as the [[discrete Fourier transform]]s of the <math>\, x</math>'s and <math>\, N</math> "conjugate momenta" <math>\,\Pi </math> defined as the Fourier transforms of the <math>\, p</math>'s:
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| :<math>
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| Q_k = {1\over\sqrt{N}} \sum_{l} e^{ikal} x_l</math>
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| :<math>
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| \Pi_{k} = {1\over\sqrt{N}} \sum_{l} e^{-ikal} p_l.
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| </math>
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| The quantity <math>\, k_n</math> turns out to be the [[Wavenumber|wave number]] of the phonon, i.e. <math>\, 2\,\pi</math> divided by the [[wavelength]]. | |
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| This choice retains the desired commutation relations in either real space or wave vector space
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| : <math> \begin{align}
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| \left[x_l , p_m \right]&=i\hbar\delta_{l,m} \\
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| \left[ Q_k , \Pi_{k'} \right] &={1\over N} \sum_{l,m} e^{ikal} e^{ik'am} [x_l , p_m ] \\
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| &= {i \hbar\over N} \sum_{m} e^{iam\left(k'-k\right)} = i\hbar\delta_{k,k'} \\
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| \left[ Q_k , Q_{k'} \right] &= \left[ \Pi_k , \Pi_{k'} \right] = 0
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| \end{align}</math>
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| From the general result
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| : <math> \begin{align}
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| \sum_{l}x_l x_{l+m}&={1\over N}\sum_{kk'}Q_k Q_k'\sum_{l} e^{ial\left(k+k'\right)}e^{iamk'}= \sum_{k}Q_k Q_{-k}e^{iamk} \\
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| \sum_{l}{p_l}^2 &= \sum_{k}\Pi_k \Pi_{-k}
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| \end{align}</math>
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| The potential energy term is
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| : <math>
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| {1\over 2} m \omega^2 \sum_{j} (x_j - x_{j+1})^2= {1\over 2}m\omega^2\sum_{k}Q_k Q_{-k}(2-e^{ika}-e^{-ika})= {1\over 2} \sum_{k}m{\omega_k}^2Q_k Q_{-k} </math>
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| where
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| :<math>\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))} = 2\omega\left|sin\left({{ka}\over 2 }\right)\right|\ </math>
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| The Hamiltonian may be written in wave vector space as
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| :<math>\mathbf{H} = {1\over {2m}}\sum_k \left(
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| { \Pi_k\Pi_{-k} } + m^2 \omega_k^2 Q_k Q_{-k}
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| \right)</math>
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| The couplings between the position variables have been transformed away; if the <math>\, Q</math>'s and <math>\,\Pi</math>'s were [[Hermitian operator|hermitian]](which they are not), the transformed Hamiltonian would describe <math>\, N</math> ''uncoupled'' harmonic oscillators.
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| The form of the quantization depends on the choice of boundary conditions; for simplicity, ''periodic'' boundary conditions are imposed, defining the <math>\, (N+1)</math>th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is
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| :<math>k=k_n = {2\pi n \over Na}
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| \quad \hbox{for}\ n = 0, \pm1, \pm2, ... , \pm {N \over 2}.\ </math>
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| The upper bound to <math>\, n</math> comes from the minimum wavelength, which is twice the lattice spacing <math>\, a</math>, as discussed above.
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| The harmonic oscillator eigenvalues or energy levels for the mode <math>\omega_k</math> are :
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| ::<math>E_n = \left({1\over2}+n\right)\hbar\omega_k \quad\quad\quad n=0,1,2,3 ......</math>
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| The levels are evenly spaced at:
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|
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| ::<math>{1\over2}\hbar\omega , \quad {3\over2}\hbar\omega ,\quad {5\over2}\hbar\omega \quad ......</math>
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| An '''exact''' amount of [[energy]] <math>\hbar\omega</math> must be supplied to the harmonic oscillator lattice to push it to the next energy level. In comparison to the [[photon]] case when the [[electromagnetic field]] is quantized, the quantum of vibrational energy is called a phonon.
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| All quantum systems show wave-like and particle-like properties simultaneously. The particle-like properties of the phonon are best understood using the methods of [[second quantization]] and operator techniques described later.<ref name="Mahan">{{cite book|last=Mahan|first=GD|authorlink=|title=many particle physics|publisher= springer|location=New York|isbn=0306463385|year=1981}}</ref>
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| ===Three-dimensional lattice===
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| This may be generalized to a three-dimensional lattice. The wave number ''k'' is replaced by a three-dimensional [[wave vector]] '''k'''. Furthermore, each '''k''' is now associated with three normal coordinates.
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| The new indices ''s = 1, 2, 3'' label the [[polarization (waves)|polarization]] of the phonons. In the one-dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to [[longitudinal wave]]s. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like [[transverse wave]]s. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.
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| ===Dispersion relation===
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| [[File:Diatomic phonons.png|thumb|Dispersion curves in linear diatomic chain]]
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| [[File:Optical & acoustic vibrations.png|thumb|250px|Optical and acoustic vibrations in linear diatomic chain.]]
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| [[File:Phonon dispersion relations in GaAs.png|thumb|250px|Dispersion relation ω=ω('''''k''''') for some waves corresponding to lattice vibrations in GaAs.<ref name=Cardona/>]]
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| For a one-dimensional alternating array of two types of ion or atom of mass ''m<sub>1</sub>, m<sub>2</sub>'' repeated periodically at a distance ''a'', connected by springs of spring constant ''K'', two modes of vibration result:<ref name=Misra/>
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| :<math>\omega_{\pm}^2 = K\left(\frac{1}{m_1} +\frac{1}{m_2}\right) \pm K \sqrt{\left(\frac{1}{m_1} +\frac{1}{m_2}\right)^2-\frac{4\sin^2(ka/2)}{m_1 m_2}} \ , </math>
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| where ''k'' is the wave-vector of the vibration related to its wavelength by ''k''=2π/λ.
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| The connection between frequency and wave-vector, ω=ω(''k''), is known as a [[dispersion relation]]. The plus sign results in the so-called ''optical'' mode, and the minus sign to the ''acoustic'' mode. In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together.
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| The speed of propagation of an acoustic phonon, which is also the [[speed of sound]] in the lattice, is given by the slope of the acoustic dispersion relation, <math>\,\tfrac{\partial\omega_k}{\partial k}</math> (see [[group velocity]].) At low values of <math>\, k</math> (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately <math>\,\omega a</math>, 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 <math>\, k</math>, i.e. short wavelengths, due to the microscopic details of the lattice.
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| For a crystal that has at least two atoms in its [[Wigner-Seitz cell#Primitive cell|primitive cell]] (which may or may not be different), the [[dispersion relation]]s exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper blue and lower red of curve in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the [[wave-vector]]. The boundaries at -π/a and π/a are those of the first [[Brillouin zone]].<ref name=Misra>
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| For a discussion see {{cite book |title=Physics of Condensed Matter |author=Prasanta Kumar Misra |url=http://books.google.com/books?id=J6rMISLVCmcC&pg=PA44 |pages=44 ''ff'' |chapter=§2.1.3 Normal modes of a one-dimensional chain with a basis |publisher=Academic Press |isbn=0-12-384954-3 |year=2010}}
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| </ref> It is also interesting that for a crystal with ''N'' ( > 2) different atoms in a [[primitive cell]], there are always three acoustic modes: one [[Longitudinal wave|longitudinal acoustic mode]] and two [[Transverse wave|transverse acoustic modes]]. The number of optical modes is 3''N'' – 3. The lower figure shows the dispersion relations for several phonon modes in GaAs as a function of wavevector '''''k''''' in the [[Brillouin zone#Critical points|principal directions]] of its Brillouin zone.<ref name=Cardona>
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| {{cite book |title=Fundamentals of Semiconductors: Physics and Materials Properties |author=Peter Y. Yu, Manuel Cardona |url=http://books.google.com/books?id=5aBuKYBT_hsC&pg=PA111 |page=111 |chapter=Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes |isbn=3-642-00709-0 |year=2010 |edition=4th |publisher=Springer}}
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| </ref> | |
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| Many phonon dispersion curves have been measured by [[neutron scattering]].
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| The physics of sound in [[fluid]]s 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 stress]]es (but see [[viscoelastic]] fluids, which only apply to high frequencies, though).
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| ===Interpretation of phonons using second quantization techniques===
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| In fact, the above-derived Hamiltonian looks like the classical Hamiltonian function, but if it is interpreted as an [[Operator (physics)|operator]], then it describes a [[quantum field theory]] of non-interacting [[boson]]s.
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| This leads to new physics.
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| The [[energy]] [[Spectrum of an operator|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
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| :<math>b_k={1\over \sqrt{2}}\left({Q_k\over l_k}+i{\Pi_{-k}\over \hbar/l_k}\right)\quad,\quad Q_k=l_k{1\over \sqrt{2}}({b_k}^\dagger+b_{-k})</math>
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| :<math>{b_k}^\dagger={1\over \sqrt{2}}\left({Q_k\over l_k}-i{\Pi_{-k}\over \hbar/l_k}\right)\quad, \quad \Pi_k={\hbar\over l_k}{i\over \sqrt{2}}({b_k}^\dagger-b_{-k})</math>
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| :::<math> l_k=\sqrt{{\hbar \over m\omega_k}}</math>
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| By direct insertion on the Hamiltonian, it is readily verified that
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| ::<math>H_{ph} =\sum_k \hbar\omega_k \left({b_k}^\dagger b_k+{1\over2}\right)</math>
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| :<math>[b_k , b_{k'}^{\dagger} ] = \delta_{k,k'} , [b_k , b_{k'} ] = [b_k^{\dagger} , b_{k'}^{\dagger} ] = 0.</math>
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| As with the quantum harmonic oscillator, one can show that <math>\, b_k^\dagger</math> and <math>\, b_k</math> respectively create and destroy one excitation of energy <math>\,\hbar\omega_k</math>. These excitations are phonons.
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| Two important properties of phonons may be deduced. Firstly, phonons are [[boson]]s, since any number of identical excitations can be created by repeated application of the creation operator <math>\, b_k^\dagger</math>. 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.
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| It is not ''a priori'' obvious that these excitations generated by the <math>\, b</math> operators are literally waves of lattice displacement, but one may convince oneself of this by calculating the ''position-position [[correlation function]]''. Let <math>\, |k\rangle</math> denote a state with a single quantum of mode <math>\, k</math> excited, i.e.
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| :<math>\begin{matrix}
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| | k \rangle = b_k^\dagger | 0 \rangle.
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| \end{matrix}</math>
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| One can show that, for any two atoms <math>\, j</math> and <math>\,\ell</math>,
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| :<math>\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 </math>
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| which is exactly what we would expect for a lattice wave with frequency <math>\,\omega_k</math> and wave number <math>\, k</math>.
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| In three dimensions the Hamiltonian has the form
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| :<math>\mathbf{H} = \sum_k \sum_{s = 1}^3 \hbar \, \omega_{k,s}
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| \left( b_{k,s}^{\dagger}b_{k,s} + 1/2 \right).</math>
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| ==Acoustic and optical phonons==
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| Solids with more than one type of atom – either with different masses or bonding strengths – in the smallest [[unit cell]], exhibit two types of phonons: acoustic phonons and optical phonons.
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| Acoustic phonons are coherent movements of atoms of the lattice out of their equilibrium positions. The displacement as a function of position can be given by a cos(wx). If the displacement is in the direction of propagation, then in some areas the atoms will be closer, in others farther apart, as in a sound wave in air (hence the name acoustic). Displacement perpendicular to the propagation direction is comparable to waves in water. If the wavelength of acoustic phonons goes to infinity, this corresponds to a simple displacement of the whole crystal, and this costs zero energy. Acoustic phonons exhibit a linear relationship between frequency and phonon wavevector for long wavelengths. The frequencies of acoustic phonons tend to zero with longer wavelength. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively.
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| Optical phonons are out-of-phase movement of the atoms in the lattice, one atom moving to the left, and its neighbour to the right. This occurs if the lattice is made of atoms of different charge or mass. They are called ''optical'' because in ionic crystals, such as [[sodium chloride]], they are excited by [[infrared radiation]]. The electric field of the light will move every positive sodium ion in the direction of the field, and every negative chloride ion in the other direction, sending the crystal vibrating.
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| Optical phonons have a non-zero frequency at the [[Brillouin zone]] center and show no dispersion near that long wavelength limit. 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]]. Optical phonons that interact in this way with light are called ''infrared active''. Optical phonons that 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 modes respectively.
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| When measuring optical phonon energy by experiment, optical phonon frequencies, <math>\omega</math>, are often given in units of [[Centimetre|cm]]<sup>−1</sup>, which are the same units as the wavevector. This value corresponds to the inverse of the [[wavelength]] of a [[photon]] with the same energy as the measured phonon.<ref name="cmian">{{cite book | last = Mahan | first = Gerald | title = Condensed Matter in a Nutshell | publisher = Princeton University Press | location = Princeton | year = 2010 | isbn = 0-691-14016-2 }}</ref> The cm<sup>−1</sup> is a unit of energy used frequently in the dispersion relations of both acoustic and optical phonons, see [[Units of energy#Spectroscopy|units of energy]] for more details and uses.
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| ==Crystal momentum==
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| {{Main|Crystal momentum}}
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| [[File:phonon k 3k.gif|right|thumb|250px|k-vectors exceeding the first Brillouin zone (red) do not carry any more information than their counterparts (black) in the first Brillouin zone.]]
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| It is tempting to treat a phonon with wave vector <math>\, k</math> as though it has a [[momentum]] <math>\,\hbar k</math>, by analogy to [[photon]]s and [[De Broglie wavelength|matter waves]]. This is not entirely correct, for <math>\,\hbar k</math> is not actually a physical momentum; it is called the ''crystal momentum'' or ''pseudomomentum''. This is because <math>\, k</math> is only determined up to addition of constant vectors (the [[reciprocal lattice|reciprocal lattice vector]]s and integer multiples thereof). For example, in our one-dimensional model, the normal coordinates <math>\, Q</math> and <math>\,\Pi</math> are defined so that
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| :<math>Q_k \ \stackrel{\mathrm{def}}{=}\ Q_{k+K} \quad;\quad \Pi_k \ \stackrel{\mathrm{def}}{=}\ \Pi_{k + K} \quad
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| </math>
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| where
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| :<math>\, K = 2n\pi/a</math>
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| for any integer <math>\, n</math>. A phonon with wave number <math>\, k</math> is thus equivalent to an infinite "family" of phonons with wave numbers <math>\, k\pm\tfrac{2\,\pi}{a}</math>, <math>\, k\pm\tfrac{4\,\pi}{a}</math>, and so forth. Physically, the reciprocal lattice vectors act as additional "chunks" of momentum which the lattice can impart to the phonon. [[Bloch wave|Bloch electron]]s obey a similar set of restrictions.
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| [[File:Brillouin zone.svg|thumb|Brillouin zones, a) in a square lattice, and b) in a hexagonal lattice]]
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| It is usually convenient to consider phonon wave vectors <math>\, k</math> which have the smallest magnitude <math>\, (|k|)</math> 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.
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| It is interesting that similar consideration is needed in [[Analog-to-digital converter|analog-to-digital conversion]] where [[aliasing]] may occur under certain conditions.
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| ==Thermodynamics==
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| The [[thermodynamics|thermodynamic]] properties of a solid are directly related to its phonon structure. The entire set of all possible phonons that are described by the above phonon dispersion relations combine in what is known as the phonon [[density of states]] which determines the heat capacity of a crystal.
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| At [[absolute zero]] temperature, a crystal lattice lies in its [[ground state]], and contains no phonons. A lattice at a non-zero [[temperature]] has an energy that is not constant, but fluctuates [[random]]ly about some [[Arithmetic mean|mean value]]. These energy fluctuations are caused by random lattice vibrations, which can be viewed as a gas of phonons. (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.
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| Unlike the atoms which make up an ordinary gas, thermal phonons can be created and destroyed by random energy fluctuations. In the language of statistical mechanics this means that the chemical potential for adding a phonon is zero. This behavior is an extension of the harmonic potential, mentioned earlier, into the anharmonic regime. The behavior 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 [[black body|Black-body radiation]]. 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:
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| :<math>n(\omega_{k,s}) = \frac{1}{\exp(\hbar\omega_{k,s}/k_BT) - 1}</math>
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| where <math>\,\omega_{k,s}</math> is the frequency of the phonons (or photons) in the state, <math>\, k_B</math> is [[Boltzmann's constant]], and <math>\, T</math> is the temperature.
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| ==Operator formalism==
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| The phonon Hamiltonian is given by
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| :<math>\mathbf{H} = \frac{1}{2}\sum_{\alpha}(p_{\alpha}^{2} + \omega^{2}_{\alpha}q_{\alpha}^{2} -\frac{1}{2}\hbar\omega_{\alpha})</math>
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| In terms of the operators, these are given by
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| :<math> \mathbf{H} = \sum_{\alpha}\hbar\omega_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}</math>
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| Here, in expressing the [[Hamiltonian (quantum mechanics)|Hamiltonian]] in operator formalism, we have not taken into account the <math>\frac{1}{2}\hbar \omega_{q}</math> term, since if we take an infinite lattice or, for that matter a continuum, the <math>\frac{1}{2}\hbar\omega_{q}</math> terms will add up giving an infinity. Hence, it is "renormalized" by putting the factor of <math>\frac{1}{2}\hbar\omega_{q}</math> to 0 arguing that the difference in energy is what we measure and not the absolute value of it. Hence, the <math>\frac{1}{2}\hbar\omega_{q}</math> factor is absent in the operator formalised expression for the [[Hamiltonian (quantum mechanics)|Hamiltonian]].<br>
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| The ground state also called the "vacuum state" is the state composed of no phonons. Hence, the energy of the ground state is 0. When, a system is in state <math>|n_{1}n_{2}n_{3}...\rangle</math>, we say there are <math>n_{\alpha}</math> phonons of type <math>\alpha</math>. The <math>n_{\alpha}</math> are called the occupation number of the phonons. Energy of a single phonon of type <math>\alpha</math> being <math>\hbar \omega_{q}</math>, the total energy of a general phonon system is given by <math>n_{1}\hbar\omega_{1} + n_{2}\hbar\omega_{2}+ ...</math>. In other words, the phonons are non-interacting. The action of creation and annihilation operators are given by
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| :<math>a^{\dagger}_{\alpha}|n_{1}...n_{\alpha -1}n_{\alpha}n_{\alpha +1}...\rangle = \sqrt{n_{\alpha} +1}|n_{1}...,n_{\alpha -1}, n_{\alpha}+1, n_{\alpha+1}...\rangle
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| </math>
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| and, | |
| :<math>a_{\alpha}|n_{1}...n_{\alpha -1}n_{\alpha}n_{\alpha +1}...\rangle = \sqrt{n_{\alpha}}|n_{1}...,n_{\alpha -1},(n_{\alpha}-1),n_{\alpha+1},...\rangle</math>
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| i.e. <math>a^{\dagger}_{\alpha}</math> creates a phonon of type <math>\alpha</math> while <math>a_{\alpha}</math> annihilates. Hence, they are respectively the [[creation and annihilation operator]] for phonons. Analogous to the [[Quantum harmonic oscillator]] case, we can define [[particle number operator]] as <math>N = \sum_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}</math>. The number operator commutes with a string of products of the creation and annihilation operators if, the number of <math>a</math>'s are equal to number of <math>a^{\dagger}</math>'s.<br>
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| Phonons are [[bosons]] since, <math>|\alpha,\beta\rangle = |\beta, \alpha\rangle</math> i.e. they are symmetric under exchange.<ref>{{cite book |title= Statistical Mechanics, A Set of Lectures |last= Feynman|first= Richard P. |authorlink= |coauthors= |year=1982 |publisher= The Benjamin/Cummings Publishing Company, Inc. |location=Reading, Massachusetts |isbn= Clothbound: 0-8053-2508-5, Paperbound: 0-8053-2509-3|page= 159|pages= |url= |accessdate=08.04.2010}}</ref>
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| ==Nonlinearity==
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| As well as [[photons]], phonons can interact via [[parametric down conversion]]<ref>Phonon-phonon interactions due to non-linear effects in a linear ion trap - Phonon-phonon interactions due to non-linear effects in a linear ion trap</ref> and form [[squeezed coherent state]]s.<ref>http://iopscience.iop.org/1742-6596/193/1/012121/pdf/1742-6596_193_1_012121.pdf - Generation of squeezed phonon states by optical excitation of a quantum dot</ref>
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| ==See also==
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| {{portal|Physics}}
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| {{colbegin|3}}
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| * [[Boson]]
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| * [[Brillouin scattering]]
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| * [[Fracton]]
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| * [[Linear elasticity]]
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| * [[Phonon scattering]]
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| * [[Acoustic metamaterials#Phononic crystal|Phononic crystal]]
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| * [[Rayleigh wave]]
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| * [[Relativistic heat conduction]]
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| * [[Rigid unit modes]]
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| * [[SASER]]
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| * [[Second sound]]
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| * [[Surface acoustic wave]]
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| * [[Surface phonon]]
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| * [[Thermal conductivity]]
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| {{colend}}
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| {{Refend}}
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| ==References==
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| {{Reflist|2}}
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| ==External links==
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| * [http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html Optical and acoustic modes]
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| * Phonons in a One Dimensional Microfluidic Crystal [http://www.nature.com/nphys/journal/v2/n11/abs/nphys432.html] and [http://arxiv.org/abs/1008.1155] with movies in [http://www.weizmann.ac.il/materials/barziv/project_1.htm].
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| {{particles}}
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| [[Category:Quasiparticles]]
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| [[Category:Bosons]]
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