Phonon

In physics, a phonon is a aggregate action in a periodic, adaptable adjustment of atoms or molecules in abridged matter, such as debris and some liquids. Often referred to as a quasiparticle,1 it represents an aflame accompaniment in the breakthrough automated quantization of the modes of accordance of adaptable structures of interacting particles.

Phonons play a above role in abounding of the concrete backdrop of solids, including a material's thermal and electrical conductivities. Hence the abstraction of phonons is an important allotment of solid accompaniment physics.

A phonon is a breakthrough automated description of a appropriate blazon of vibrational motion, in which a filigree analogously oscillates at the aforementioned frequency. In classical mechanics this is accepted as the accustomed mode. The accustomed approach is important because any approximate filigree beating can be advised as a superposition of these elementary accordance (cf. Fourier analysis). While accustomed modes are wave-like phenomena in classical mechanics, they accept particle-like backdrop in the wave–particle duality of breakthrough mechanics.

The name phonon comes from the Greek chat φωνή (phonē), which translates as complete or articulation because long-wavelength phonons accord acceleration to sound.

The abstraction of phonons was alien in 1932 by Russian physicist Igor Tamm.

Lattice dynamics

The equations in this area either do not use axioms of breakthrough mechanics or use relations for which there exists a absolute accord in classical mechanics.

For example, accede a adamant regular, crystalline, i.e. not amorphous, filigree composed of N particles. We will accredit to these particles as atoms, although in a absolute solid these may be molecules. N is some ample number, say about 1023 (on the adjustment of Avogadro's number) for a archetypal sample of solid. If the filigree is rigid, the atoms accept to be advance armament on one addition to accumulate anniversary atom abreast its calm position. These armament may be Van der Waals forces, covalent bonds, electrostatic attractions, and others, all of which are ultimately due to the electric acreage force. Magnetic and gravitational armament are about negligible. The armament amid anniversary brace of atoms may be characterized by a abeyant activity action V that depends on the ambit of break of the atoms. The abeyant activity of the absolute filigree is the sum of all pairwise abeyant energies:2

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

where \, r_i is the position of the \, ith atom, and \, V is the abeyant activity amid two atoms.

It is difficult to break this many-body botheration in abounding generality, in either classical or breakthrough mechanics. In adjustment to abridge the task, we acquaint two important approximations. First, we accomplish the sum over adjoining atoms only. Although the electric armament in absolute debris extend to infinity, this approximation is about accurate because the fields produced by abroad atoms are screened. Secondly, we amusement the potentials \, V as harmonic potentials: this is permissible as continued as the atoms abide abutting to their calm positions. (Formally, this is done by Taylor accretion \, V about its calm bulk to boxlike order, giving \, V proportional to the displacement \, x^2 and the adaptable force artlessly proportional to \, x. The absurdity in blank college adjustment agreement charcoal baby if \, x charcoal abutting to the calm position).

The consistent filigree may be visualized as a arrangement of assurance affiliated by springs. The afterward bulk shows a cubic lattice, which is a acceptable archetypal for abounding types of apparent solid. Added lattices cover a beeline chain, which is a actual simple filigree which we will anon use for modelling phonons. Added accepted lattices may be begin in the commodity on clear structure.

Cubic.svg

The abeyant activity of the filigree may now be accounting as

\sum_{\{ij\} (nn)} {1\over2} m \omega^2 (R_i - R_j)^2.

Here, \,\omega is the accustomed abundance of the harmonic potentials, which we accept to be the aforementioned back the filigree is regular. \, R_i is the position alike of the \, ith atom, which we now admeasurement from its calm position. The sum over abutting neighbors is denoted as "(nn)".

edit Filigree waves

Phonon breeding through a aboveboard filigree (atom displacements abundantly exaggerated)

Due to the access amid atoms, the displacement of one or added atoms from their calm positions will accord acceleration to a set of beating after-effects breeding through the lattice. One such beachcomber is apparent in the bulk to the right. The amplitude of the beachcomber is accustomed by the displacements of the atoms from their calm positions. The amicableness \,\lambda is marked.

There is a minimum accessible wavelength, accustomed by alert the calm break a amid atoms. As we shall see in the afterward sections, any amicableness beneath than this can be mapped assimilate a amicableness best than 2a, due to the aeon of the lattice.

Not every accessible filigree beating has a categorical amicableness and frequency. However, the accustomed modes do acquire categorical wavelengths and frequencies.

edit One dimensional lattice

In adjustment to abridge the assay bare for a 3-dimensional filigree of atoms it is acceptable to archetypal a 1-dimensional filigree or beeline chain. This archetypal is circuitous abundant to affectation the arresting appearance of phonons.

edit Classical treatment

The armament amid the atoms are affected to be beeline and nearest-neighbour, and they are represented by an adaptable spring. Anniversary atom is affected to be a point atom and the basis and electrons move in step.(adiabatic approximation)

n-1 n n+1 ← d →

\cdotso++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o\cdots

→→ → →→→

u_{n-1} \qquad\quad u_n \qquad\quad u_{n+1}

Where n labels the n'th atom, d is the ambit amid atoms if the alternation is in calm and un the displacement of the n'th atom from its calm position.

If C is the adaptable connected of the bounce and m the accumulation of the atom again the blueprint of motion of the n'th atom is :

-2Cu_n + C(u_{n+1} + u_{n-1}) = m{\operatorname{d^2}u_n\over\operatorname{d}t^2}

This is a set of accompanying equations and back we apprehend the solutions to be oscillatory, new coordinates can be authentic by a detached Fourier transform, in adjustment to de-couple them.3

Put

u_n = \sum_{k=1}^N U_k e^{iknd}

Here nd replaces the accepted connected capricious x. The Uk are accepted as the accustomed coordinates. Substitution into the blueprint of motion produces the afterward decoupled equations.(This requires a cogent abetment appliance the orthonormality and abyss relations of the detached fourier transform 4)

2C(\cos\,kd-1)U_k = m{\operatorname{d^2}U_k\over\operatorname{d}t^2}

These are the equations for harmonic oscillators which accept the solution:

U_k=A_ke^{i\omega_kt};\qquad\quad \omega_k=\sqrt{ {2C \over m}(1-\cos{kd})}

Each accustomed alike Uk represents an absolute vibrational approach of the filigree with wavenumber k which is accepted as a accustomed mode. The added blueprint for ωk is accepted as the burning affiliation amid the angular abundance and the wavenumber.5

edit Breakthrough treatment

Consider a apparent breakthrough automated harmonic alternation of N identical atoms. This is the simplest breakthrough automated archetypal of a lattice, and we will see how phonons appear from it. The ceremonial that we will advance for this archetypal is readily generalizable to two and three dimensions. The Hamiltonian for this arrangement 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 accumulation of anniversary atom, and \, x_i and \, p_i are the position and drive operators for the \, ith atom. A altercation of agnate Hamiltonians may be begin in the commodity on the breakthrough harmonic oscillator.

We acquaint a set of \, N "normal coordinates" \, Q_k, authentic as the detached Fourier transforms of the \, x's and \, N "conjugate momenta" \,\Pi authentic as the Fourier transforms of the \, p's:

x_j = {1\over\sqrt{N}} \sum_{n=-N/2}^{N/2} Q_{k_n} e^{ik_nja}

p_j = {1\over\sqrt{N}} \sum_{n=-N/2}^{N/2} \Pi_{k_n} e^{-ik_nja}.

The abundance \, k_n will about-face out to be the beachcomber bulk of the phonon, i.e. \, 2\,\pi disconnected by the wavelength. It takes on quantized values, because the bulk of atoms is finite. The anatomy of the quantization depends on the best of abuttals conditions; for simplicity, we appoint alternate abuttals conditions, defining the \, (N+1)th atom as agnate to the aboriginal atom. Physically, this corresponds to abutting the alternation at its ends. The consistent quantization is

k_n = {2n\pi \over Na} \quad \hbox{for}\ n = 0, \pm1, \pm2, ... , \pm {N \over 2}.\

The top apprenticed to \, n comes from the minimum wavelength, which is alert the filigree agreement \, a, as discussed above.

By inverting the detached Fourier transforms to accurate the \, Q's in agreement of the \, x's and the \,\Pi's in agreement of the \, p's, and appliance the approved barter relations amid the \, x's and \, p's, we can appearance 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 added words, the accustomed coordinates and their conjugate momenta obey the aforementioned barter relations as position and drive operators! Writing the Hamiltonian in agreement 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 amid the position variables accept been adapted away; if the \, Q's and \,\Pi's were Hermitian (which they are not), the adapted Hamiltonian would call \, N bachelor harmonic oscillators.

The harmonic oscillator eigenvalues or activity levels for the approach ωk are :

E_n = ({1\over2}+n)\hbar\omega_k \quad\quad\quad n=0,1,2,3 ......

If we avoid the zero-point activity again the levels are analogously spaced at :

\hbar\omega , \quad 2\hbar\omega ,\quad 3\hbar\omega \quad ......

So a minimum bulk of activity \hbar\omega accept to be supplied to the harmonic oscillator(or accustomed mode) to move it to the next activity level. In allegory to the photon case if the electromagnetic acreage is quantised, the breakthrough of vibrational activity is alleged a phonon.

All breakthrough systems appearance wave-like and particle-like properties. The particle-like backdrop of the phonon are best accepted appliance the methods of second-quantisation and abettor techniques declared later.

edit Three dimensional lattice

This may be ambiguous to a three-dimensional lattice. The beachcomber bulk k is replaced by a three-dimensional beachcomber agent k. Furthermore, anniversary k is now associated with three accustomed coordinates.

The new indices s = 1, 2, 3 characterization the animosity of the phonons. In the one dimensional model, the atoms were belted to affective forth the line, so the phonons corresponded to longitudinal waves. In three dimensions, beating is not belted to the administration of propagation, and can aswell action in the erect planes, like axle waves. This gives acceleration to the added accustomed coordinates, which, as the anatomy of the Hamiltonian indicates, we may appearance as absolute breed of phonons.

edit Burning relation

Dispersion curve

In the aloft discussion, we accept acquired an blueprint that relates the angular abundance of a phonon, \,\omega_k, to its beachcomber bulk \, k:

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

This is accepted as a burning relation.

The acceleration of advancement of a phonon, which is aswell the acceleration of complete in the lattice, is accustomed by the abruptness of the burning relation, \,\tfrac{\partial\omega_k}{\partial k} (see accumulation velocity.) At low ethics of \, k (i.e. continued wavelengths), the burning affiliation is about linear, and the acceleration of complete is about \,\omega a, absolute of the phonon frequency. As a result, packets of phonons with altered (but long) wavelengths can bear for ample distances beyond the filigree after breaking apart. This is the acumen that complete propagates through debris after cogent distortion. This behavior fails at ample ethics of \, k, i.e. abbreviate wavelengths, due to the diminutive data of the lattice.

For a clear that has at atomic two atoms in its archaic corpuscle (which may or may not be different), the burning relations display two types of phonons, namely, optical and acoustic modes agnate to the top and lower sets of curves in the diagram, respectively. The vertical arbor is the activity or abundance of phonon, while the accumbent arbor is the wave-vector. The boundaries at -km and km are those of the aboriginal Brillouin zone. The blue, violet, and amber curves are those of longitudinal acoustic, axle acoustic 1, and axle acoustic 2 modes, respectively.

In some crystals the two axle acoustic modes accept absolutely the aforementioned burning curve. It is aswell absorbing that for a clear with N ( > 2) altered atoms in a archaic cell, there are consistently three acoustic modes. The bulk of optical modes is 3N - 3. Abounding phonon burning curves accept been abstinent by neutron scattering.

The physics of complete in fluids differs from the physics of complete in solids, although both are body waves: complete after-effects in fluids alone accept longitudinal components, admitting complete after-effects in debris accept longitudinal and axle components. This is because fluids can't abutment microburst stresses. (but see viscoelastic fluids, which alone administer to top frequencies, though).

edit Interpretation of phonons appliance added quantisation techniques

In fact, the above-derived Hamiltonian looks like the classical Hamiltonian function, but if it is interpreted as an operator, again it describes a breakthrough acreage approach of non-interacting bosons. This leads to new physics.

The activity spectrum of this Hamiltonian is calmly acquired by the adjustment of ladder operators, agnate to the breakthrough harmonic oscillator problem. We acquaint a set of ladder operators authentic 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 amuse the afterward 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 breakthrough harmonic oscillator, we can again appearance that \, a_k^\dagger and \, a_k appropriately actualize and abort one action of activity \,\hbar\omega_k. These excitations are phonons.

We can anon deduce two important backdrop of phonons. Firstly, phonons are bosons, back any bulk of identical excitations can be created by again appliance of the conception abettor \, a_k^\dagger. Secondly, anniversary phonon is a "collective mode" acquired by the motion of every atom in the lattice. This may be apparent from the actuality that the ladder operators accommodate sums over the position and drive operators of every atom.

It is not a priori accessible that these excitations generated by the \, a operators are actually after-effects of filigree displacement, but one may argue oneself of this by artful the position-position alternation function. Let \, |k\rangle denote a accompaniment with a individual breakthrough of approach \, k excited, i.e.

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

One can appearance 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 absolutely what we would apprehend for a filigree beachcomber with abundance \,\omega_k and beachcomber bulk \, k.

In three ambit 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).

Acoustic and optical phonons

Solids with added than one blazon of atom - either with altered masses or bonding strengths - in the aboriginal assemblage cell, display two types of phonons: acoustic phonons and optical phonons.

Acoustic phonons are articular movements of atoms of the filigree out of their calm positions. The displacement as a action of position can be accustomed by a cos(wx). If the displacement is in the administration of propagation, again in some areas the atoms will be closer, in others added apart, as in a complete beachcomber in air (hence the name acoustic). Displacement erect to the advancement administration is commensurable to after-effects in water. If the amicableness of acoustic phonons goes to infinity, this corresponds to a simple displacement of the accomplished crystal, and this costs aught energy. Acoustic phonons display a beeline accord amid abundance and phonon wavevector for continued wavelengths. The frequencies of acoustic phonons tend to aught with best wavelength. Longitudinal and axle acoustic phonons are generally abbreviated as LA and TA phonons, respectively.

Optical phonons are out of appearance movement of the atoms in the lattice, one atom affective to the left, and its neighbour to the right. This occurs if the filigree is fabricated of atoms of altered allegation or mass. They are alleged optical because in ionic crystals, such as sodium chloride, they are aflame by bittersweet radiation. The electric acreage of the ablaze will move every absolute sodium ion in the administration of the field, and every abrogating chloride ion in the added direction, sending the clear vibrating. Optical phonons accept a non-zero abundance at the Brillouin area centermost and appearance no burning abreast that continued amicableness limit. This is because they accord to a approach of beating area absolute and abrogating ions at adjoining filigree sites beat adjoin anniversary other, creating a time-varying electrical dipole moment. Optical phonons that collaborate in this way with ablaze are alleged bittersweet active. Optical phonons that are Raman alive can aswell collaborate alongside with light, through Raman scattering. Optical phonons are generally abbreviated as LO and TO phonons, for the longitudinal and axle modes respectively.

When barometer optical phonon activity by experiment, optical phonon frequencies, ω, are generally accustomed in units of cm − 1, which are the aforementioned units as the wavevector. This amount corresponds to the changed of the amicableness of a photon with the aforementioned activity as the abstinent phonon.6 The cm − 1 is a assemblage of activity acclimated frequently in the burning relations of both acoustic and optical phonons, see units of activity for added data and uses.

Crystal momentum

It is appetizing to amusement a phonon with beachcomber agent \, k as admitting it has a drive \,\hbar k, by affinity to photons and amount waves. This is not absolutely correct, for \,\hbar k is not in fact a concrete momentum; it is alleged the clear drive or pseudomomentum. This is because \, k is alone bent up to multiples of connected vectors, accepted as alternate filigree vectors. For example, in our apparent model, the accustomed coordinates \, Q and \,\Pi are authentic 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 accumulation \, n. A phonon with beachcomber amount \, k is appropriately agnate to an absolute "family" of phonons with beachcomber numbers \, k\pm\tfrac{2\,\pi}{a}, \, k\pm\tfrac{4\,\pi}{a}, and so forth. Physically, the alternate filigree vectors act as added "chunks" of drive which the filigree can admit to the phonon. Bloch electrons obey a agnate set of restrictions.

Brillouin zones, a) in a aboveboard lattice, and b) in a hexagonal lattice

It is usually acceptable to accede phonon beachcomber vectors \, k which accept the aboriginal consequence \, (|k|) in their "family". The set of all such beachcomber vectors defines the aboriginal Brillouin zone. Added Brillouin zones may be authentic as copies of the aboriginal zone, confused by some alternate filigree vector.

It is absorbing that agnate application is bare in analog-to-digital about-face area aliasing may action beneath assertive conditions.

Thermodynamics

The thermodynamic backdrop of a solid are anon accompanying to its phonon structure. The complete set of all accessible phonons that are declared by the aloft phonon burning relations amalgamate in what is accepted as the phonon body of states which determines the calefaction accommodation of a crystal.

At complete aught temperature, a clear filigree lies in its arena state, and contains no phonons. A filigree at a non-zero temperature has an activity that is not constant, but fluctuates about about some beggarly value. These activity fluctuations are acquired by accidental filigree vibrations, which can be beheld as a gas of phonons. (The accidental motion of the atoms in the filigree is what we usually anticipate of as heat.) Because these phonons are generated by the temperature of the lattice, they are sometimes referred to as thermal phonons.

Unlike the atoms which accomplish up an accustomed gas, thermal phonons can be created and destroyed by accidental activity fluctuations. In the accent of statistical mechanics this agency that the actinic abeyant for abacus a phonon is zero. This behavior is an addendum of the harmonic potential, mentioned earlier, into the anharmonic regime. The behavior of thermal phonons is agnate to the photon gas produced by an electromagnetic cavity, wherein photons may be emitted or captivated by the atrium walls. This affinity is not coincidental, for it turns out that the electromagnetic acreage behaves like a set of harmonic oscillators; see Black-body radiation. Both gases obey the Bose-Einstein statistics: in thermal calm and aural the harmonic regime, the anticipation of award phonons (or photons) in a accustomed accompaniment with a accustomed angular abundance is:

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

where \,\omega_{k,s} is the abundance of the phonons (or photons) in the state, \, k_B is Boltzmann's constant, and \, T is the temperature.

Operator formalism

The phonon Hamiltonian is accustomed by

\mathbf{H} = \frac{1}{2}\sum_{\alpha}(p_{\alpha}^{2} + \omega^{2}_{\alpha}q_{\alpha}^{2} -\frac{1}{2}\hbar\omega_{\alpha})

In agreement of the operators, these are accustomed by

\mathbf{H} = \sum_{\alpha}\hbar\omega_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}

Here, in cogent the Hamiltonian (quantum mechanics) in abettor formalism, we accept not taken into annual the \frac{1}{2}\hbar \omega_{q} term, back if we yield an complete filigree or, for that amount a continuum, the \frac{1}{2}\hbar\omega_{q} agreement will add up giving an infinity. Hence, it is "renormalized" by putting the agency of \frac{1}{2}\hbar\omega_{q} to 0 arguing that the aberration in activity is what we admeasurement and not the complete amount of it. Hence, the \frac{1}{2}\hbar\omega_{q} agency is absent in the abettor formalised announcement for the Hamiltonian.

The arena accompaniment aswell alleged the "vacuum state" is the accompaniment composed of no phonons. Hence, the activity of the arena accompaniment is 0. When, a arrangement is in accompaniment |n_{1}n_{2}n_{3}...\rangle, we say there are nα phonons of blazon α. The nα are alleged the activity amount of the phonons. Activity of a individual phonon of blazon α getting \hbar \omega_{q}, the absolute activity of a accepted phonon arrangement is accustomed by n_{1}\hbar\omega_{1} + n_{2}\hbar\omega_{2}+ .... In added words, the phonons are non-interacting. The activity of conception and abolishment operators are accustomed by

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

and,

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

i.e. a^{\dagger}_{\alpha} creates a phonon of blazon α while aα annihilates. Hence, they are appropriately the conception and abolishment abettor for phonons. Analogous to the Breakthrough harmonic oscillator case, we can ascertain atom amount abettor as N = \sum_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}. The amount abettor commutes with a cord of articles of the conception and abolishment operators if, the amount of a's are according to amount of a^{\dagger}'s.

Phonons are bosons since, |\alpha,\beta\rangle = |\beta, \alpha\rangle i.e. they are symmetric beneath exchange.7