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## Phonon*For KDE4's multimedia framework, see Phonon (KDE).*
In physics, a Phonons are a quantum mechanical version of a special type of vibrational motion, known as normal modes in classical mechanics, in which each part of a lattice oscillates with the same frequency. These normal modes are important because, according to a well-known result in classical mechanics, any arbitrary vibrational motion of a lattice can be considered as a superposition of normal modes with various frequencies; in this sense, the normal modes are the
## Repeating derivation of normal modesThe equations in this subsection either do not use axioms of quantum mechanics or use relations for which there exists a direct correspondence in classical mechanics. ## Mechanics of particles on a latticeConsider a rigid regular (or "crystalline") lattice composed of
where is the position of the It is extremely difficult to solve this many-body problem in full generality, in either classical or quantum mechanics. In order to simplify the task, we introduce two important approximations. Firstly, we only perform the sum over neighbouring atoms. Although the electric forces in real solids extend to infinity, this approximation is nevertheless valid because the fields produced by distant atoms are screened. Secondly, we treat the potentials as harmonic potentials: this is permissible as long as the atoms remain close to their equilibrium positions. (Formally, this is done by Taylor expanding about its equilibrium value, which gives proportional to .) The resulting lattice may be visualized as a system of balls connected by springs. The following figure shows a cubic lattice, which is a good model for many types of crystalline solid. Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modelling phonons. Other common lattices may be found in the article on crystal structure.
The potential energy of the lattice may now be written as
Here, is the natural frequency of the harmonic potentials, which we assume to be the same since the lattice is regular. is the position coordinate of the ## Lattice wavesDue to the connections between atoms, the displacement of one or more atoms from their equilibrium positions will give rise to a set of vibration waves propagating through the lattice. One such wave is shown in the figure below. The amplitude of the wave is given by the displacements of the atoms from their equilibrium positions. The wavelength is marked. There is a Not every possible lattice vibration has a well-defined wavelength and frequency. However, the normal modes (which, as we mentioned in the introduction, are the elementary building-blocks of lattice vibrations) do possess well-defined wavelengths and frequencies. We will now examine it in detail. ## Phonon dispersion of a one-dimensional chain of identical atomsConsider a one-dimensional quantum mechanical harmonic chain of
where is the mass of each atom, and and are the position and momentum operators for the We introduce a set of "normal coordinates" , defined as the discrete Fourier transforms of the 's and "conjugate momenta" defined as the Fourier transforms of the 's:
The quantity will turn out to be the wave number of the phonon, i.e. divided by the wavelength. It takes on quantized values, because the number of atoms is finite. The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose
The upper bound to comes from the minimum wavelength imposed by the lattice spacing , as discussed above. By inverting the discrete Fourier transforms to express the 's in terms of the 's and the 's in terms of the 's, and using the canonical commutation relations between the 's and 's, we can show that
In other words, the normal coordinates and their conjugate momenta obey the same commutation relations as position and momentum operators! Writing the Hamiltonian in terms of these quantities,
where
Notice that the couplings between the position variables have been transformed away; if the 's and 's were Hermitian (which they are not), the transformed Hamiltonian would describe ## Three-dimensional phononsIt is straightforward, though tedious, to generalize the above to a three-dimensional lattice. One finds that the wave number The new indices ## Dispersion relationIn the above discussion, we have obtained an equation that relates the frequency of a phonon, , to its wave number :
This is known as a dispersion relation. The speed of propagation of a phonon, which is also the speed of sound in the lattice, is given by the slope of the dispersion relation, (see group velocity.) At low values of (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately , independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of , i.e. short wavelengths, due to the microscopic details of the lattice. For a crystal that has at least two atoms in a unit cell (which may or may not be different), the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper and lower sets of curves in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the wave-vector. The boundaries at -k In some crystals the two transverse acoustic modes have exactly the same dispersion curve. It is also interesting that for a crystal with The physics of sound in fluids differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. This is because fluids can't support shear stresses. (but see viscoelastic fluids, which only apply to high frequencies, though). ## Acoustic and optical phononsIn solids with more than one atom in the smallest unit cell, there are two types of phonons: "acoustic" phonons and "optical" phonons. "Acoustic phonons", which are the phonons described above, have frequencies that become small at the long wavelengths, and correspond to sound waves in the lattice. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively. "Optical phonons," which arise in crystals that have more than one atom in the smallest unit cell, always have some minimum frequency of vibration, even when their wavelength is large. They are called "optical" because in ionic crystals (like sodium chloride) they are excited very easily by light (in fact, infrared radiation). This is because they correspond to a mode of vibration where positive and negative ions at adjacent lattice sites swing against each other, creating a time-varying electrical dipole moment. Optical phonons that interact in this way with light are called ## PhononsIn fact, the above derived Hamiltonian looks like the classical Hamilton function, but if its interpreted as an operator it describes a quantum field theory of non-interacting bosons. This leads to new physics. The energy spectrum of this Hamiltonian is easily obtained by the method of ladder operators, similar to the quantum harmonic oscillator problem. We introduce a set of ladder operators defined by
The ladder operators satisfy the following identities:
As with the quantum harmonic oscillator, we can then show that and respectively create and destroy one excitation of energy . These excitations are phonons. We can immediately deduce two important properties of phonons. Firstly, phonons are bosons, since any number of identical excitations can be created by repeated application of the creation operator . Secondly, each phonon is a "collective mode" caused by the motion of every atom in the lattice. This may be seen from the fact that the ladder operators contain sums over the position and momentum operators of every atom. It is not
One can show that, for any two atoms and ,
which is exactly what we would expect for a lattice wave with frequency and wave number . In three dimensions the Hamiltonian has the form
## Crystal momentum
It is tempting to treat a phonon with wave vector as though it has a momentum , by analogy to photons and matter waves. This is not entirely correct, for is not actually a physical momentum; it is called the
where
for any integer . A phonon with wave number is thus equivalent to an infinite "family" of phonons with wave numbers , , and so forth. Physically, the reciprocal lattice vectors act as additional "chunks" of momentum which the lattice can impart to the phonon. Bloch electrons obey a similar set of restrictions. It is usually convenient to consider phonon wave vectors which have the smallest magnitude in their "family". The set of all such wave vectors defines the It is interesting that similar consideration is needed in analog-to-digital conversion where aliasing may occur under certain conditions.
## Thermodynamic propertiesA crystal lattice at zero temperature lies in its ground state, and contains no phonons. According to thermodynamics, when the lattice is held at a non-zero temperature its energy is not constant, but fluctuates randomly about some mean value. These energy fluctuations are caused by random lattice vibrations, which can be viewed as a Unlike the atoms which make up an ordinary gas, thermal phonons can be created or destroyed by random energy fluctuations. In the language of statistical mechanics this means that the chemical potential for adding a phonon is zero. It is very important to note that this behaviour takes us away from the harmonic potential mentioned earlier, and into the anharmonic regime. The behaviour of thermal phonons is similar to the
where is the frequency of the phonons (or photons) in the state, is Boltzmann's constant, and is the temperature. ## See also
- Fracton
- Linear elasticity
- Rayleigh wave
- Surface acoustic wave
- Rigid Unit Modes a phonon where polyhedra move, by translation and/or rotation, without distorting
- Phononic crystal
- a vibron is for a molecule what a phonon is for a crystal
## References**^**International Union of Pure and Applied Chemistry. "phonon".*Compendium of Chemical Terminology*Internet edition.**^**Although φόνον (phonon) is literally the accusative case of φόνος (phonos) = murder!
Categories: Condensed matter physics | Bosons |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Phonon". A list of authors is available in Wikipedia. |