In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid.[1] The study of phonons is an important part of solid state physics, because phonons play a major role in many of the physical properties of solids, including a material's thermal and electrical conductivities. In particular, the properties of long-wavelength phonons give rise to sound in solids—hence the name phonon from the Greek φωνή (phonē) = voice.[2] In insulating solids, phonons are also the primary mechanism by which heat conduction takes place.
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 (compare Fourier transform); 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.)
The 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.
[edit] Mechanics of particles on a lattice
Consider a rigid regular (or "crystalline"; not amorphous) lattice composed of N particles. (We will refer to these particles as "atoms". In a real solid these atoms may be ions.) 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 forces on one another, so as to keep each atom near its equilibrium position. In real solids, these forces include Van der Waals forces, covalent bonds, 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 < j} V(r_i - r_j)
where \, r_i is the position of the \, ith atom, and \, V is the potential energy between two atoms.
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. First, we perform the sum over neighboring atoms only. 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 \, V as harmonic potentials: this is permissible as long as the atoms remain close to their equilibrium positions. (Formally, this is done by Taylor expanding \, 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. 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.
Image:Cubic.svg
The potential energy of the lattice may now be written as
\sum_{i \ne j (nn)} {1\over2} m \omega^2 (R_i - R_j)^2
Here, \,\omega 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 \, ith atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted as "(nn)".
[edit] 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 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 \,\lambda 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 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.
[edit] Phonon dispersion of a one-dimensional chain of identical atoms
Consider a one-dimensional quantum mechanical harmonic chain of N identical 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 \, ith 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 transforms of the \, x's and \, N "conjugate momenta" \,\Pi defined as the Fourier transforms of the \, p's:
\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. \, 2\,\pi 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 = {n\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 \, Q's in terms of the \, x's and the \,\Pi'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'} \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 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} \right)
where
\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))}
Notice that the couplings between the position variables have been transformed away; if the \, Q's and \,\Pi's were Hermitian (which they are not), the transformed Hamiltonian would describe \, N uncoupled harmonic oscillators.
[edit] Three-dimensional phonons
It is straightforward, though tedious, to generalize the above to a three-dimensional lattice. One finds that the wave number k is replaced by a three-dimensional wave vector k. Furthermore, each k is now associated with three normal coordinates.
The new indices s = 1, 2, 3 label the polarization of the phonons. In the one dimensional model, the atoms were restricted to moving along the line, so all the phonons corresponded to longitudinal waves. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular plane, like transverse waves. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.
