Atoms Software Download' title='Atoms Software Download' />Michigan Tech Information Technology develops and supports technology solutions that enable Michigan Techs faculty, students, and staff to excel in teaching.Atoms Software Download' title='Atoms Software Download' />Phonon Wikipedia.In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, like solids and some liquids.Often designated a quasiparticle,1 it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles.Phonons play a major role in many of the physical properties of condensed matter, like thermal conductivity and electrical conductivity.The study of phonons is an important part of condensed matter physics.The concept of phonons was introduced in 1.Soviet physicist Igor Tamm.The name phonon comes from the Greek word phon, which translates to sound or voice because long wavelength phonons give rise to sound.Shorter wavelength higher frequency phonons are responsible for the majority of the thermal capacity of solids.DefinitioneditA phonon is a quantum mechanical description of an elementary vibrational motion in which a lattice of atoms or molecules uniformly oscillates at a single frequency.In classical mechanics this designates a normal mode of vibration.Normal modes are important because any arbitrary lattice vibration can be considered to be a superposition of these elementary vibration modes cf.Fourier analysis.While normal modes are wave like phenomena in classical mechanics, phonons have particle like properties too, in a way related to the waveparticle duality of quantum mechanics.Lattice dynamicseditThe equations in this section do not use axioms of quantum mechanics but instead use relations for which there exists a direct correspondence in classical mechanics.For example a rigid regular, crystalline not amorphous, lattice is composed of N particles.These particles may be atoms or molecules.N is a large number, say of the order of 1.Avogadros number for a typical sample of a solid.Since the lattice is rigid, the atoms must be exerting forces on one another to keep each atom near its equilibrium position.These forces may be Van der Waals forces, covalent bonds, electrostatic attractions, and others, all of which are ultimately due to the electric force.Magnetic and gravitational forces are generally negligible.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.The potential energy of the entire lattice is the sum of all pairwise potential energies 3ij.Vrirjdisplaystyle sum ineq jVleftri rjrightwhere ri is the position of the ith atom, and V is the potential energy between two atoms.It is difficult to solve this many body problem explicitly 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 still valid because the fields produced by distant atoms are effectively screened.Secondly, the potentials V are treated as harmonic potentials.This is permissible as long as the atoms remain close to their equilibrium positions.Formally, this is accomplished by Taylor expanding.V about its equilibrium value to quadratic order, giving V proportional to the displacement x.The error in ignoring higher order terms remains small if x remains close to the equilibrium position.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.For other common lattices, see crystal structure.The potential energy of the lattice may now be written asijnn1.RiRj2. displaystyle sum ijmathrm nn tfrac 12momega 2leftRi Rjright2.Here, is the natural frequency of the harmonic potentials, which are assumed to be the same since the lattice is regular.Ri is the position coordinate of the ith atom, which we now measure from its equilibrium position.The sum over nearest neighbors is denoted nn.Lattice wavesedit.Phonon propagating through a square lattice atom displacements greatly exaggeratedDue to the connections between atoms, the displacement of one or more atoms from their equilibrium positions give rise to a set of vibration waves 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 is marked.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 2a, due to the periodicity of the lattice.This can be thought as one consequence of NyquistShannon sampling theorem, the lattice points are viewed as the sampling points of a continuous wave.Not every possible lattice vibration has a well defined wavelength and frequency.However, the normal modes do possess well defined wavelengths and frequencies. Cover Collaborations Vol 1 Boyce Avenue Download Free . One dimensional latticeeditIn 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.Classical treatmenteditThe forces between the atoms are assumed to be linear and nearest neighbour, 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 n 1 n n 1 a oooooooooo un 1 un un 1where n labels the nth atom out of a total of N, a is the distance between atoms when the chain is in equilibrium, and un the displacement of the nth atom from its equilibrium position.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 is2.CunCun1un1md. CunCleftun1un 1rightmfrac d2undt2.This is a set of coupled equations.Since the solutions are expected to be oscillatory, new coordinates are defined by a discrete Fourier transform, in order to decouple them.PutunNak21. NQkeiknadisplaystyle unsum Nak2pi 1NQkeiknaHere, na corresponds and devolves to the continuous variable x of scalar field theory.The Qk are known as the normal coordinates, continuum field modes k.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 transform5,2.Ccoska1Qkmd. Qkdt.Ccos ka 1Qkmfrac d2Qkdt2.These are the equations for harmonic oscillators which have the solution.QkAkeikt k2. Cm1coskadisplaystyle QkAkeiomega kt qquad omega ksqrt frac 2.Cm1 cos kaEach normal coordinate Qk represents an independent vibrational mode of the lattice with wavenumber k which is known as a normal mode.The second equation, for k, is known as the dispersion relation between the angular frequency and the wavenumber.In the continuum limit, a0, N, with Na held fixed, un x, a scalar field, and kkadisplaystyle omega kpropto ka.This amounts to free scalar classical field theory.Quantum treatmenteditA 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.In some contrast to the previous section, the positions of the masses are not denoted by ui, but, instead, by x.In two or more dimensions, the xi are vector quantities.The Hamiltonian for this system is.Hi1. Npi. 22m1. Hsum i1Nfrac pi22mfrac 12momega 2sum ijmathrm nn leftxi xjright2where m is the mass of each atom assuming it is equal for all, and xi and pi are the position and momentum operators, respectively, for the ith atom and the sum is made over the nearest neighbors nn.
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