• [10] For a one-dimensional alternating array of two types of ion or atom of mass m1, m2 repeated periodically at a distance a, connected by springs of spring constant K, two
    modes of vibration result:[11] where k is the wavevector of the vibration related to its wavelength by .

  • 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.

  • The potential energy of the lattice may now be written as Here, ω is the natural frequency of the harmonic potentials, which are assumed to be the same since the lattice is

  • [12] Finally, using the position–position correlation function, it can be shown that phonons act as waves of lattice displacement.

  • Actually, in general, the wave velocity in a crystal is different for different directions of k. In other words, most crystals are anisotropic for phonon proagation.

  • For a crystal that has at least two atoms in its primitive cell, the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to
    the upper blue and lower red curve in the diagram, respectively.

  • [citation needed] This technique is readily generalized to three dimensions, where the Hamiltonian takes the form:[12][2] Which can be interpreted as the sum of 3N independent
    oscillator Hamiltonians, one for each wave vector and polarization.

  • The forces between each pair of atoms may be characterized by a potential energy function V that depends on the distance of separation of the atoms.

  • It is customary to deal with waves in Fourier space which uses normal modes of the wavevector as variables instead coordinates of particles.

  • Not every possible lattice vibration has a well-defined wavelength and frequency.

  • In the one-dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to longitudinal waves.

  • [13] Crystal momentum By analogy to photons and matter waves, phonons have been treated with wavevector k as though it has a momentum ħk;[14] however, this is not strictly
    correct, because ħk is not actually a physical momentum; it is called the crystal momentum or pseudomomentum.

  • The entire set of all possible phonons that are described by the phonon dispersion relations combine in what is known as the phonon density of states which determines the
    heat capacity of a crystal.

  • This choice retains the desired commutation relations in either real space or wavevector space From the general result The potential energy term is where The Hamiltonian may
    be written in wavevector space as The couplings between the position variables have been transformed away; if the Q and Π were Hermitian (which they are not), the transformed Hamiltonian would describe N uncoupled harmonic oscillators.

  • Lattice waves[edit] Phonon propagating through a square lattice (atom displacements greatly exaggerated) Due to the connections between atoms, the displacement of one or more
    atoms from their equilibrium positions gives rise to a set of vibration waves propagating through the lattice.

  • Although the electric forces in real solids extend to infinity, this approximation is still valid because the fields produced by distant atoms are effectively screened.

  • long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately ωa, independent of the phonon frequency.

  • This similarity is not coincidental, for it turns out that the electromagnetic field behaves like a set of harmonic oscillators, giving rise to Black-body radiation.

  • 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, (see group velocity.)

  • The lower figure shows the dispersion relations for several phonon modes in GaAs as a function of wavevector k in the principal directions of its Brillouin zone.

  • Interpretation of phonons using second quantization techniques[edit] The above-derived Hamiltonian may look like a classical Hamiltonian function, but if it is interpreted
    as an operator, then it describes a quantum field theory of non-interacting bosons.

  • 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),[8] These are the equations for decoupled harmonic oscillators which have the solution Each normal coordinate Qk represents an independent vibrational mode of the lattice with wavenumber k, which is known
    as a normal mode.

  • The energy of a single phonon of type α is given by ħωq and the total energy of a general phonon system is given by As there are no cross terms, the phonons are said to be

  • [17] This type of heat transfer works between distances too large for conduction to occur but too small for radiation to occur and therefore cannot be explained by classical
    heat transfer models.

  • In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together.

  • This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.

  • They play a major role in many of the physical properties of condensed matter systems, such as thermal conductivity and electrical conductivity, as well as in models of neutron
    scattering and related effects.

  • The particle-like properties of the phonon are best understood using the methods of second quantization and operator techniques described later.

  • Given the Hamiltonian, , as well as the conjugate position, , and conjugate momentum defined in the quantum treatment section above, we can define creation and annihilation
    operators:[12] and The following commutators can be easily obtained by substituting in the canonical commutation relation: Using this, the operators bk† and bk can be inverted to redefine the conjugate position and momentum as: and Directly
    substituting these definitions for and into the wavevector space Hamiltonian, as it is defined above, and simplifying then results in the Hamiltonian taking the form:[2] This is known as the second quantization technique, also known as the
    occupation number formulation, where nk = bk†bk is the occupation number.

  • The value is obtained by dividing the frequency by the speed of light in vacuum.

  • Classical treatment[edit] The forces between the atoms are assumed to be linear and nearest-neighbour, and they are represented by an elastic spring.

  • If the wavelength of acoustic phonons goes to infinity, this corresponds to a simple displacement of the whole crystal, and this costs zero deformation energy.

  • [2] The second quantization technique, similar to the ladder operator method used for quantum harmonic oscillators, is a means of extracting energy eigenvalues without directly
    solving the differential equations.

  • While normal modes are wave-like phenomena in classical mechanics, phonons have particle-like properties too, in a way related to the wave–particle duality of quantum mechanics.

  • Quantum treatment[edit] A one-dimensional quantum mechanical harmonic chain consists of N identical atoms.

  • Optical phonons are out-of-phase movements of the atoms in the lattice, one atom moving to the left, and its neighbor to the right.

  • An exact amount of energy must be supplied to the harmonic oscillator lattice to push it to the next energy level.

  • This may be seen from the fact that the creation and annihilation operators, defined here in momentum space, contains sums over the position and momentum operators of every
    atom when written in position space (See position and momentum space).

  • In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like transverse waves.

  • 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:[16] where ωk,s is the frequency of the phonons (or photons) in the state, kB is the Boltzmann constant, and T is the temperature.

  • The velocity of the wave also is given in terms of ω and k .

  • For example, in the one-dimensional model, the normal coordinates Q and Π are defined so that where for any integer n. A phonon with wavenumber k is thus equivalent to an
    infinite family of phonons with wavenumbers and so forth.

  • [9] See also: Canonical quantization § Real scalar field Three-dimensional lattice[edit] This may be generalized to a three-dimensional lattice.

  • The form of the quantization depends on the choice of boundary conditions; for simplicity, periodic boundary conditions are imposed, defining the (N + 1)th atom as equivalent
    to the first atom.

  • When measuring optical phonon energy experimentally, optical phonon frequencies are sometimes given in spectroscopic wavenumber notation, where the symbol ω represents ordinary
    frequency (not angular frequency), and is expressed in units of cm.

  • The number of normal modes is same as the number of particles.

  • As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart.

  • One-dimensional lattice[edit] Animation showing 6 normal modes of a one-dimensional lattice: a linear chain of particles.

  • However, the Fourier space is very useful given the periodicity of the system.

  • [12] As with the quantum harmonic oscillator, one can show that bk† and bk respectively create and destroy a single field excitation, a phonon, with an energy of ħωk.

  • [20] Predicted properties Recent research has shown that phonons and rotons may have a non-negligible mass and be affected by gravity just as standard particles are.

  • 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).

  • Second, each phonon is a “collective mode” caused by the motion of every atom in the lattice.

  • The resulting quantization is The upper bound to n comes from the minimum wavelength, which is twice the lattice spacing a, as discussed above.

  • [2] Hence, they can be 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, causing the crystal to vibrate.

  • [25] They have been also shown to form “phonon winds” where an electric current in a graphene surface is generated by a liquid flow above it due to the viscous forces at the
    liquid–solid interface.

  • [citation needed] At absolute zero temperature, a crystal lattice lies in its ground state, and contains no phonons.

  • However, the normal modes do possess well-defined wavelengths and frequencies.

  • By the nature of this distribution, the heat capacity is dominated by the high-frequency part of the distribution, while thermal conductivity is primarily the result of the
    low-frequency region.

  • The following figure shows a cubic lattice, which is a good model for many types of crystalline solid.

  • If C is the elastic constant of the spring and m the mass of the atom, then the equation of motion of the nth atom is This is a set of coupled equations.

  • Analogous to the quantum harmonic oscillator case, we can define particle number operator as The number operator commutes with a string of products of the creation and annihilation
    operators if and only if the number of creation operators is equal to number of annihilation operators.


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