In Physical of the condensed matter, a phonon (of the old Greek φονη/ phon , the voice) indicates a Quantum Vibration in a Solide Cristal flax, i.e. a “elementary package of vibration” or “elementary package of its”: when a mode of vibration of the crystal of definite Fréquence ν yields or gains energy, it can do it only per packages of energy hν , H being the Constante of Planck. This package is regarded as a quasi particle , namely a fictitious particle associated with the sound. The crystal is then considered to exchange phonons when it loses or gains energy. The concept allows a Analogie with the Lumière which has similar properties: it appears sometimes like a Onde, sometimes like a package of energy hν , which corresponds to a Elementary particle - nonfictitious this time - called Photon.

The phonon is a concept of quantum Mécanique calling upon the concept of Dualité wave-corpuscle: according to the experimental context it can appear either like a wave, or like an elementary package. If the study of the phonons takes a big part in the physics of the condensed matter, it is that they play a big role in a great number of physics solid state properties of which

• the Heat-storage capacity, or capacity to exchange the Chaleur;
• the thermal Conductivity, or capacity to conduct the heat;
• the electric Conductivity, or capacity to lead the Electric current;
• capacity to propagate the sound.

The traditional Mécanique, which takes into account only the vibratory access, is not able to explain these properties entirely.

Introduction

Crystalline solid in traditional mechanics

The phonons are the equivalent in quantum Mécanique of a particular category of movement vibratory known under the name of normal Modes of vibration in traditional Mécanique. A normal Mode of vibration is a mode in which each element of a network vibrates with same the Fréquence. These normal modes of vibration have a great importance, in particular because any movement of the vibration type in a solid can be represented like the superposition of a certain number of normal modes of vibration of different frequencies. The normal modes of vibration can thus to some extent be included/understood like the elementary vibrations of the network.

Quantification of the modes of vibration

Although the normal modes of vibration are of the entities of the undulatory type, they can acquire behavior of a particulate type partly when the network is studied through the laws of quantum mechanics (because of Dualité wave-corpuscle). They are then named phonons. The phonons are quasi-particles of Spin 0 (Boson S which thus obey the Statistique of Bump-Einstein).

Definition of the phonons

The phonons exist only within a crystal lattice comprising a great number of particles and the only known physical structures corresponding to this definition are the crystalline solids. In the continuation we will not thus treat phonons that in this framework and, for the clearness of the talk, we will call the particles constituting the network “Atome S”, although it can be a question of Ion S in a ionic Solide.

Mechanical aspects: movement of particles in a network

In a solid, there exist forces of interaction (Force of van der Waals, covalent forces, etc) which maintain each atom close to a position of balance. It is mainly of the forces of the electric type, the forces of the type Magnétique being generally negligible. The interaction between each pair of atoms can be characterized by a function of potential energy V which depends only on the distance between these atoms and which is the same one for all the pairs of atoms. The potential energy of the network as a whole is the sum of the potential energies of interaction of each pair:

$$

E = \ frac {1} {2} \ sum_ {I \ J} V (r_i - r_j) \,

where r_i is the position of the i^ème atom, and the factor 1/2 compensates for the fact that each pair is counted twice (like (I, J) and like (J, I) ).

This expression, characteristic of a Problème with NR body, does not lend itself to a resolution that it is in traditional Mécanique or in quantum Mécanique. It is thus necessary to carry out approximations to continue the analysis. The two approximations generally employed are:

• to restrict the summation with the close atoms. Indeed, although rigorously the electric forces have an infinite range in a real solid, this approximation is valid because the forces being exerted on distant atoms are écrantées and thus negligible.
• to consider that the potential V is a potential harmonic, which is valid when the atoms remain close to their positions of balance. (Formally, this assumption applies by carrying out a development of Taylor potential V around the value of balance and by keeping only the first not-constant term, which is of order 2.)

It is possible to slacken one or the other of the assumptions, for the first by considering the interaction with neighbors more moved away and for the second by adding terms of higher natures. In the majority of the cases, the inclusion of these terms does not modify the solution significantly.

The network can be visualized as a system of balls bound by Ressort S. the figure below illustrates two described types of network in this manner. The figure of left shows a cubic lattice (network corresponding to a big number of crystalline solids, of which in particular many metals). The figure of right-hand side shows a linear chain, a very simple network allowing an easy approach of the modeling of the phonons. For more information on the crystal lattices, to see the article Crystallography.

     

The potential energy of the network can be now written:

$$

E= \ frac {1} {2} \ sum_ {I \ J (statement)} {1 \ over2} m \ omega^2 (R_i - R_j) ^2

• ω is the own pulsation of the harmonic potentials

• m is the mass of the atoms.
• R_i is the coordinate of the i^ème atom, now considered compared to its position of balance .
• the symbol statement indicates that the summation is carried out only on the closest neighbors.

Waves in a network

Because of the force S being exerted between different the Atom S from the crystal lattice, the displacement of one or more atoms around their position of balance will involve a series of Onde S of vibrations being propagated in the network. The figure above shows a wave of vibration in a network. The Amplitude of the wave is given by the amplitude of the displacement of the atoms around their position of balance. The Wavelength corresponds to the smallest interval between two identical repetitions of the arrangement of the atoms. It is noted λ on the figure.

All the wavelengths of vibration are not possible. In particular, there exists a minimal wavelength given by the distance between the atoms has . We will further see than a wave wavelength lower than has is in fact identical to a wavelength larger than has .

All the possible vibrations of the network do not have necessarily a wavelength (or a Fréquence) well defined. It is however the case for the normal modes of vibration (elementary vibrations of the network), which we will examine more in details in the following paragraphs.

Phonons in a network 1D

Let us consider a unidimensional chain made up of NR atoms for which the potential are harmonic. This system is the simplest model for a crystal lattice. The mathematical formalism that we will develop in the continuation (within the framework of the quantum Mécanique) is easily generalizable with systems with two or three dimensions.

$\backslash \; mathbf\; \{H\}\; =\; \backslash \; sum\_\; \{i=1\}\; ^N\; \{P\_i^2\; \backslash \; over\; 2m\}\; +\; \{1\; \backslash \; over\; 2\}\; m\; \backslash \; omega^2\; \backslash \; sum\_\; \{\backslash \; \{ij\; \backslash \}\; (statement)\}\; (X\_i\; -\; X\_j)\; ^2$

* m is the mass of the atoms.

* X_i is the operator position .

* P_i is the operator Impulsion (who corresponds to the momentum)

A thorough description of this Hamiltonien is given in the article Oscillateur quantum harmonic

• Let us define NR " now; coordinates normales" Q_k defined as the transformed of Fourier of the operators X_i position.

• also Let us define NR " moments conjuguées" Π_k defined as the transformed of Fourier of the operators P_i impulse.

$\backslash begin\{matrix\}$

X_j &=& {1 \ over \ sqrt {NR}} \ sum_ {K} Q_k e^ {ikja} \ \ P_j &=& {1 \ over \ sqrt {NR}} \ sum_ {K} \ Pi_k e^ {- ikja} \ \ \end{matrix}

The quantity K is the Nombre of wave phonon, i.e. 2π divided by the Wavelength. This number takes quantified values because the number of atoms of the system is finished. The form of the quantification depends on the choice of the Boundary conditions. By preoccupations with a simplification, we impose in the continuation of the boundary conditions periodic (also called conditions of Born von Karman), i.e. we consider that the N+1 atom is equivalent to the first atom. Physically, that corresponds to form a loop with the chain while making meet the two ends. The result of the quantification is:

$K\; =\; \{N\; \backslash \; pi\; \backslash \; over\; Na\}$

\ quad \ hbox {for} \ N = 0, \ pm 1, \ pm 2,…, \ pm NR

The higher limit of N comes from the boundary condition selected (the atom in position x₁ is identical to the atom in position x_N+1.

By reversing the Transformed of Fourier to express Q_k in term of X_i and Π_k in term of P_i, and using the canonical relations of commutations between X_i and P_i, one can show that (see the article quantum Mécanique):

$$

\ Q_k left, \ Pi_ {k'} \ right = I \ hbar \ delta_ {K k'} \ quad ; \ quad \ left Q_k, Q_ {k'} \ right = \ left \ Pi_k, \ Pi_ {k'} \ right = 0

In other words, the normal coordinates Q_k and their moments combined Π_k obey the same relations of commutation as the operators position X_i and impulse P_i . According to these sizes, the Hamiltonien is written:

$\backslash \; mathbf\; \{H\}\; =\; \backslash \; sum\_k\; \backslash \; left\; ($

{\ Pi_k \ Pi_ {- K} \ over 2m} + {1 \ over2} m \ omega_k^2 Q_k Q_ {- K} \ right)

with

$\backslash \; omega\_k\; =\; \backslash \; sqrt\; \{2\; \backslash \; omega^2\; (1\; -\; \backslash \; cos\; (ka))\}$

One can note that the coupling between the variables positions was transformed. If the Q_k and the Π_k were Hermitien S (what is not the case), the Hamiltonien transformed would describe NR harmonic oscillators not coupled . In fact, this Hamiltonien describes a Quantum theory of the fields of Boson S not interacting.

The spectrum of clean energies of this Hamiltonien is obtained by using the assistant operators creation and annihilation a_k ^{definite †} and a_k like:

$\backslash begin\{matrix\}$

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}

For more precise details, to see the article Oscillating quantum harmonic. The assistant operators check the identity:

$\backslash \; mathbf\; \{H\}\; =\; \backslash \; sum\_k\; \backslash \; hbar\; \backslash \; omega\_k\; \backslash \; left\; (a\_k^\; \{\backslash \; dagger\}\; a\_k\; +\; 1/2\; \backslash \; right)$

$,\; a\_\; \{k\text{'}\}\; ^\; \{\backslash \; dagger\}\; =\; \backslash \; delta\_\; \{kk\text{'}\}$

$,\; a\_\; \{k\text{'}\}\; =,\; a\_\; \{k\text{'}\}\; ^\; \{\backslash \; dagger\}\; =\; 0.$

As in the case of the Oscillating quantum harmonic, one can show that the operators a_k ^† and a_k correspond respectively to the creation and the annihilation of an excitation of energy ℏ ω _k. This excitation is a phonon.

One can immediately deduce two important properties from them from the phonons. Initially, the phonons are Boson S: any number of identical excitations can be created by the repeated application of the operator creation a_k ^† . In the second place, each phonon is a " collectif" mode; corresponding to the movement of (quasi) the totality of the atoms of the network. This second conclusion is seen in the fact that the assistant operators contain summations on the positions and the impulses of all the atoms of the network.

It is not obvious a priori that the excitations generated by the assistant operators are literally waves of displacement of atoms of the network. One can be convinced some by calculating the Fonction of correlation position-position . That is to say | K > a state for which only one quantum of mode K is excited, i.e.:

$\backslash begin\{matrix\}$

| K \ rangle = a_k^ \ dagger | 0 \ rangle \end{matrix}

One can then show that for two atoms I and J unspecified:

$\backslash \; langle\; K\; |\; x\_i\; (T)\; x\_j\; (0)\; |\; K\; \backslash \; rangle\; =\; \backslash \; frac\; \{\backslash \; hbar\}\; \{Nm\; \backslash \; omega\_k\}\; \backslash \; cos\; \backslash \; left\; K\; (i-j)\; has\; -\; \backslash \; omega\_k\; T\; \backslash \; right\; +\; \backslash \; langle\; 0\; |\; x\_i\; (T)\; x\_j\; (0)\; |0\; \backslash \; rangle$

what is exactly the result awaited for a wave of the network of pulsation ω_k and of Nombre of wave K .

Phonons in a network 3D

Generalization with three dimensions of the preceding unidimensional model is easy (but rather heavy). The Nombre of wave K is replaced by a vector with three dimensions, the vector of wave $\backslash \; vec\; K$. Moreover, $\backslash \; vec\; K$, now is associated with three coordinated normal. The Hamiltonian with the form:

$$

\ mathbf {H} = \ sum_k \ sum_ {s=1} ^3 \ hbar \, \ omega_ {K, S} \ left (a_ {K, S} ^ {\ dagger} a_ {K, S} + \ frac {1} {2} \ right)

The new index s=1, 2,3 corresponds to the polarization of the phonons. Indeed, in a unidimensional model, the atoms can vibrate only on one line, and all the phonons correspond to a longitudinal wave. On the other hand in three dimensions, the vibration is not done solely any more in the direction of propagation, but can also be to him perpendicular. It corresponds then to a transverse wave. That gives rise to additional normal coordinates, which as the expression of the Hamiltonien indicates it, correspond to species independent of phonons.

Behavior and properties of the phonons

Curve of dispersion

In the discussion of the phonons in a unidimensional model, we obtained an equation binding the pulsation of a phonon ω_k to its number of wave K :

$\backslash \; omega\_k\; =\; \backslash \; sqrt\; \{2\; \backslash \; omega^2\; (1\; -\; \backslash \; cos\; (ka))\}$

This equation is known under the name of Relation of dispersion. The curve opposite described its behavior.

The propagation velocity of a phonon in the network, which corresponds in particular to the propagation velocity of the sound in a solid, is given by the slope of the relation of dispersion: ∂ω_k/∂k. With low values of K (i.e. with the big wavelengths), the relation of dispersion is almost linear, and the speed of sound is close to ωa, independently of the frequency of the phonon. Consequently, a package of phonons wavelengths different (but large) can be propagated on long distances in a network without the phonons separating. This is why the its is propagated in the solids without significant distortion (to some extent, the waves big wavelength are not influenced by the microscopic structure of material). This behavior is not true any more for great values of K (i.e. short wavelengths), for which the propagation velocity depends significantly on the wavelength.

One can note that the physique of the sound in the solids is very different from the physique of the sound in the Air, although it acts in both cases of waves of vibration. This is due to the fact that in the air, the sound is propagated in a random Gaz formed of Molécule S actuated by movement, and not in an organized network.

Acoustic phonons and optical phonons

In a real solid, there are two types of phonons: phonons " acoustic " and " optical ". The acoustic phonons, which are those that we described in the preceding parts, correspond typically to the sound waves in the network. The acoustic phonons of type longitudinal and transverse are often written manners shortened IT and MT respectively.

The optical phonons are present in the solids which comprise several Atome S by Maille. They are called " optiques" because in the ionic crystals (as for example sodium chloride) they are very easily excited by light waves (in the field of the infra-red). This is due to the fact that they correspond to modes of vibration for which the Ion S positive and negative located on adjacent sites of the network approach and move away from/to each other by creating a Dipole moment electric oscillating with time. The optical phonons which interact in this manner with the light are known as credits in the Infrarouge. The optical phonons which are active in spectrometry Raman can also interact with the light through the Raman Diffusion. The optical phonons of type longitudinal and transverse are often written manners shortened LO and TO respectively.

It is possible to find more information on the modes of vibration in articles treating of the Théorie of groups.

Pseudo-moment

It is trying to consider a phonon of vector of wave $\backslash \; vec\; K$ as if it had a moment $\backslash \; hbar\; \backslash \; vec\; K$, by analogy with the Photon S, or all waves corresponding to a corpuscle (Dualité wave-corpuscle). It is not completely correct, because $\backslash \; hbar\; \backslash \; vec\; K$ is not really a physical moment. It is named pseudo-moment or moment of vibration . This is due to the fact that $\backslash \; vec\; K$ is given only with one multiple of constant vector near, vector of the reciprocal Réseau. For example, in a unidimensional model, the normal coordinates Q and Π are defined in such a way that:

$$

Q_k \equiv Q_{k+K} \quad;\ quad \ Pi_k \ equiv \ Pi_ {K + K} \ quad \ mbox {with} \; K = 2n \ pi/a

whatever the integer N. A phonon of number of wave K is thus equivalent to an infinite number of other phonons of the same family of numbers of wave k±2π/a, k±4π/a (etc). The electrons of Bloch obey the same type of restrictions.

Generally, one considers only the phonons of vectors of wave $\backslash \; vec\; K$ of each family having more " petit" vector $\backslash \; vec\; K$. The whole of these vectors defines the first Zone of Brillouin. Other zones of Brillouin can be defined like copies of the first zone, shifted of a multiple of vectors of the reciprocal network.

Thermodynamic properties

A crystal lattice with the Absolute zero is in its basic state, and no phonon method is excited. According to the laws of the Thermodynamic , when a crystal lattice is with an higher temperature with the absolute zero, its energy is not constant but it fluctuates in a random way around a median value. These fluctuations of energy are due to vibrations random of the network, which can be seen as a gas of phonons (let us note that the random movement of the Atome S of the network corresponds to the Chaleur). As these phonons are related to the temperature of the network, they are sometimes named thermal phonons .

Contrary to the molecules which form a ordinary Gaz, the thermal phonons can be created or destroyed by random fluctuations of energies. Their behavior is similar to gas of Photon S produced by a electromagnetic Cavité, for which the photons can be absorptive or emitted by the walls of the cavity. This similarity is not a coincidence: the electromagnetic Champ behaves indeed like a group of harmonic oscillators (see radiation of the black Corps). The two gases obey the Statistique of Bump-Einstein, i.e. with thermal balance, the median number of phonons or photons in a given state is:

$$

\ langle n_ {K, S} \ rangle = \ frac {1} {\ exp (\ hbar \ omega_ {K, S} /k_BT) - 1}

* ω_{k, s} is the pulsation of the phonon or of the photon in the state

* k_B is the Boltzmann constant

* T is the temperature (in Kelvin)

One can notice that the chemical potential of a gas of Photons or Phonons is null.

This type of considerations led to the Modèle of Debye describing the behavior of the Heat capacity of the crystalline solids thanks to the phonons which they contain. This model presents a better agreement with the experimental results than the preceding models: the Law of Dulong and Petit and the Model of Einstein.

See too

For a better comprehension of this article, it is interesting to consult:

• Mechanical quantum
• Physical of the particles
• Physical statistics
• Crystallography
• List of particles

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