# Phonon

A**phonon**is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because they contribute to many of the physical properties of materials, such as thermal and electrical conductivity. For example, the propagation of phonons is responsible for the conduction of heat in insulators, and the properties of long-wavelength phonons gives rise to sound in solids (hence the name

*phonon*).

According to a well-known result in classical mechanics, any vibration of a lattice can be decomposed into a superposition of normal modes of vibration. When these modes are analysed using quantum mechanics, they are found to possess some particle-like properties (see wave-particle duality.) When treated as particles, phonons are bosons possessing zero spin.

The following article provides an overview of the physics of phonons.

## Non-interacting phonons

### Modelling a lattice

We begin our investigation of phonons by examining the mechanical systems from which they emerge. 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 ions. *N* is some large number, say around 10^{23} (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:

*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 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 *V* as harmonic potentials, which 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.)

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

*R*is the position coordinate of the

_{i}*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 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 λ is marked.

It should be noted that 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*.

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 these normal modes in some detail.

### One-dimensional phonons

We begin by studying the simplest model of phonons, a one-dimensional quantum mechanical harmonic chain. The formalism for this one-dimensional model is readily generalizable to two and three dimensions. Consider a linear chain of *N* atoms. The Hamiltonian for this system is

*m*is the mass of each atom, and

*x*and

_{i}*p*are the position and momentum operators for the

_{i}*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 transforms of the

*x*'s, and

*N*"conjugate momenta" Π defined as the Fourier transforms of the

*p*'s:

*k*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

*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 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

*Q*'s and

*Π*'s 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.

(It is not *a priori* obvious that these excitations generated by the *a* operators are literally waves of lattice displacement, but one may convince oneself of this by calculating the *position-position correlation function*. Let |*k*> denote a state with a single quantum of mode *k* excited, i.e.

*j*and

*l*,

*ω*and wave number

_{k}*k*.)