Phonon

Phonons are a Quantum Mechanical version of a special type of vibrational motion, known as Normal Mode s 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 ''elementary'' vibrations of the lattice. Although normal modes are Wave-like phenomena in classical mechanics, they acquire certain Particle-like properties when the lattice is analysed using quantum mechanics (see Wave-particle Duality .) They are then known as ''phonons''. Phonons are Boson s possessing zero Spin . MODELLING PHONONS Mechanics of particles on a lattice Consider a rigid regular (or "crystalline") lattice composed of ''N'' particles. (We will refer to these particles as "atoms", though in a real solid they may actually be Ion s.) ''N'' is some large number, say around 10²³ (on the order of Avogadro's Number ) for a typical piece of solid. If the lattice is rigid, the atoms must be exerting Force s on one another, so as to keep each atom near its equilibrium position. In real solids, these forces include Van Der Waals Force s, Covalent Bond s, and so forth, 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 some Potential Energy function ''V'', depending on the separation of the atoms. The potential energy of the ''entire'' lattice is the sum of all the pairwise potential energies: : $\sum_{i e j} V(r_i - r_j) \,$ where ''r''_i is the Position of the ''i''th atom, and ''V'' is the Potential Energy between two atoms. It is extremely difficult to solve this ''V'' about its equilibrium value, which gives ''V'' proportional to ''x''².) The resulting lattice may be visualized as a system of balls connected by springs. Two such lattices are shown in the figures below. The figure on the left shows a cubic lattice, which is a good model for many types of crystalline solid. The figure on the right shows a linear chain, 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 : $\sum_{i e j (nn)} {1\over2} m \omega^2 (R_i - R_j)^2$ Here, ω is the Natural Frequency of the harmonic potentials, which we assume to be the same since the lattice is regular. ''R_i'' is the position coordinate of the ''i''th atom, which we now measure from its ''equilibrium'' position. The sum over nearest neighbours is denoted as "''(nn)''". Lattice waves Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions will give rise to a set of vibration Wave s 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 λ is marked. There is a ''minimum possible'' wavelength, given by the equilibrium separation ''a'' between atoms. As we shall see in the following sections, any wavelength shorter than this can be mapped onto a wavelength longer than ''a'', due to effects similar to that in Aliasing . Not every possible lattice vibration has a well-defined wavelength and frequency. However, the Normal Mode s (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 these normal modes in some detail. One-dimensional phonons Consider a one-dimensional Quantum Mechanical harmonic chain of ''N'' atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions. The Hamiltonian for this system 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 mass of each atom, and ''x_i'' and ''p_i'' are the position and Momentum operators for the ''i''th atom. A discussion of similar Hamiltonians may be found in the article on the Quantum Harmonic Oscillator . We introduce a set of ''N'' "normal coordinates" ''Q_k'', defined as the Discrete Fourier Transform s of the ''xs, and ''N'' "conjugate momenta" Π defined as the Fourier transforms of the ''ps: : $\begin{matrix} x_j &=& {1\over\sqrt{N}} \sum_{n=-N}^{N} Q_{k_n} e^{ik_nja} \ p_j &=& {1\over\sqrt{N}} \sum_{n=-N}^{N} \Pi_{k_n} e^{-ik_nja} \ \end{matrix}$ The quantity $k_n$ will turn out to be the Wave Number of the phonon, i.e. ''2π'' 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 ''periodic'' boundary conditions, defining the ''(N+1)''th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is : $k_n = {2n\pi \over Na} \quad \hbox{for}\ n = 0, \pm1, \pm2, ... , \pm N$ The upper bound to ''n'' comes from the minimum wavelength imposed by the lattice spacing ''a'', as discussed above. By inverting the discrete Fourier transforms to express the ''Qs in terms of the ''xs and the Π's in terms of the ''p'''s, and using the canonical commutation relations between the ''x'''s and ''p'''s, we can show that : $\left Q_k , \Pi_{k'} ight = i \hbar \delta_{k k'} \quad ;\quad \left Q_k , Q_{k'} ight = \left \Pi_k , \Pi_{k'} ight = 0$ 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, : $\mathbf{H} = \sum_k \left( { \Pi_k\Pi_{-k} \over 2m } + {1\over2} m \omega_k^2 Q_k Q_{-k} ight)$ where : $\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))}$ Notice that the couplings between the position variables have been transformed away; if the ''Qs and ''Πs were Hermitian (which they are not), the transformed Hamiltonian would describe ''N'' ''uncoupled'' harmonic oscillators. In fact, this Hamiltonian describes a Quantum Field Theory of non-interacting bosons. 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 : $\begin{matrix} a_k &=& \sqrt{m\omega_k \over 2\hbar} (Q_k + {i\over m\omega_k} \Pi_{-k}) \ a_k^\dagger &=& \sqrt{m\omega_k \over 2\hbar} (Q_{-k} - {i\over m\omega_k} \Pi_k) \end{matrix}$ The ladder operators satisfy the following identities: : $\mathbf{H} = \sum_k \hbar \omega_k \left(a_k^{\dagger}a_k + 1/2 ight)$ : $= \delta_{kk'}$ : $, a_{k'} =, a_{k'}^{\dagger} = 0.$ As with the quantum harmonic oscillator, we can then show that ''a''_k^† and ''a''_k respectively create and destroy one excitation of energy ℏω_k. These excitations are phonons. We can immediately deduce two important properties of phonons. Firstly, phonons are Boson s, since any number of identical excitations can be created by repeated application of the creation operator ''a''_k^†. 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.
	K Angle	a_k^\dagger 0 angle
	:<math>\langle K X J(t) X L(0) K Angle	rac{\hbar}{Nm\omega_k} \cos \left k(j-l)a - \omega_k t ight + \langle 0 x_j(t) x_l(0) 0 angle </math>
	It Is Usually Convenient To Consider Phonon Wave Vectors '''k''' Which Have The Smallest Magnitude ('''k''') In Their "family" The Set Of All Such Wave Vectors Defines The ''first	"http://wwwinformationdelightinfo/encyclopedia/entry/Brillouin_zone" class="copylinks">Brillouin Zone '' Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector

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Phonon