Atom vibrations

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves.

The study of phonons is an important part of condensed matter physics. 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 concept of phonons was introduced in 1932 by Soviet physicist Igor Tamm. The name ''phonon'' comes from the Greek word (), which translates to ''sound'' or ''voice'', because long-wavelength phonons give rise to sound. The name is analogous to the word ''photon''.

Definition

A phonon is the 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 wave–particle duality of quantum mechanics.

Lattice dynamics

The 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 10²³, or on the order of the Avogadro 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 multiplied by a factor of 1/2 to compensate for double counting:

:$\frac12\sum_ V\left(r_i - r_j\right)$

where ''r_i'' is the position of the ''i''th 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''² and the elastic force simply proportional to ''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 as

:$\sum_ \tfrac12 m \omega^2 \left(R_i - R_j\right)^2.$

Here, ''ω'' is the natural frequency of the harmonic potentials, which are assumed 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 neighbors is denoted (nn).

Lattice waves

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. 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 ''λ'' 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 2''a'', due to the periodicity of the lattice. This can be thought as one consequence of Nyquist–Shannon sampling theorem, the lattice points being 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.

One-dimensional lattice

In 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 treatment

The 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 theorem):

::::::::''n'' − 1 ''n'' ''n'' + 1 ← ''a'' →
···o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o···
::::::::→→→→→→
::::::::''u''_{''n'' − 1}''u_n'u''_{''n'' + 1}
where labels the th atom out of a total of , is the distance between atoms when the chain is in equilibrium, and the displacement of the th atom from its equilibrium position.

If ''C'' is the elastic constant of the spring and the mass of the atom, then the equation of motion of the th atom is
:$-2Cu_n + C\left(u_ + u_\right) = m\frac .$
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.

Put
:$u_n = \sum_^N Q_k e^.$
Here, corresponds and devolves to the continuous variable of scalar field theory. The are known as the ''normal coordinates'', continuum field modes .

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),
: $2C(\cos )Q_k = m\frac.$

These are the equations for decoupled harmonic oscillators which have the solution

:$Q_k=A_ke^;\qquad \omega_k=\sqrt.$

Each normal coordinate ''Q_k'' represents an independent vibrational mode of the lattice with wavenumber , which is known as a normal mode.

The second equation, for , is known as the dispersion relation between the angular frequency and the wavenumber.

In the continuum limit, →0, →∞, with held fixed, → , a scalar field, and $\omega(k) \propto k a$. This amounts to classical free scalar field theory, an assembly of independent oscillators.

Quantum treatment

A 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 ''u_i'', but, instead, by ''x''₁, ''x''₂…, as measured from their equilibrium positions (i.e. ''x_i'' = 0 if particle ''i'' is at its equilibrium position.) In two or more dimensions, the ''x_i'' are vector quantities. The Hamiltonian for this system is

:$\mathcal = \sum_^N \frac + \frac m\omega^2 \sum_ \left(x_i - x_j\right)^2$

where ''m'' is the mass of each atom (assuming it is equal for all), and ''x_i'' and ''p_i'' are the position and momentum operators, respectively, for the ''i''th atom and the sum is made over the nearest neighbors (nn). However one expects that in a lattice there could also appear waves that behave like particles. It is customary to deal with waves in Fourier space which uses normal modes of the wavevector as variables instead coordinates of particles. The number of normal modes is same as the number of particles. However, the Fourier space is very useful given the periodicity of the system.

A set of ''N'' "normal coordinates" ''Q_k'' may be introduced, defined as the discrete Fourier transforms of the ''x_k'' and ''N'' "conjugate momenta" ''Π_k'' defined as the Fourier transforms of the ''p_k'':

:$\begin Q_k &= \frac\sqrt \sum_ e^ x_l \\ \Pi_ &= \frac\sqrt \sum_ e^ p_l. \end$

The quantity ''k_n'' turns out to be the wavenumber of the phonon, i.e. 2 divided by the wavelength.

This choice retains the desired commutation relations in either real space or wavevector space

: \begin
\left _l , p_m \right=i\hbar\delta_ \\
\left Q_k , \Pi_ \right&=\fracN \sum_ e^ e^ \left _l , p_m \right\\
&= \fracN \sum_ e^ = i\hbar\delta_ \\
\left Q_k , Q_ \right&= \left \Pi_k , \Pi_ \right= 0
\end

From the general result

: $\begin \sum_x_l x_&=\fracN\sum_Q_k Q_\sum_ e^e^= \sum_Q_k Q_e^ \\ \sum_^2 &= \sum_\Pi_k \Pi_ \end$

The potential energy term is
: $\tfrac12 m \omega^2 \sum_ \left(x_j - x_\right)^2= \tfrac12 m\omega^2\sum_Q_k Q_(2-e^-e^)= \tfrac12 \sum_m^2Q_k Q_$
where

:$\omega_k = \sqrt = 2\omega\left, \sin\frac2\$

The Hamiltonian may be written in wavevector space as
:$\mathcal = \frac\sum_k \left( \Pi_k\Pi_ + m^2 \omega_k^2 Q_k Q_ \right)$

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.

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. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

:$k=k_n = \frac \quad \mbox n = 0, \pm1, \pm2, \ldots \pm \frac2 .\$

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

The harmonic oscillator eigenvalues or energy levels for the mode ''ω_k'' are:

:$E_n = \left(\tfrac12+n\right)\hbar\omega_k \qquad n=0,1,2,3 \ldots$

The levels are evenly spaced at:

:$\tfrac12\hbar\omega , \ \tfrac32\hbar\omega ,\ \tfrac52\hbar\omega \ \cdots$

where ''ħω'' is the zero-point energy of a quantum harmonic oscillator.

An exact amount of energy ''ħω'' must be supplied to the harmonic oscillator lattice to push it to the next energy level. In comparison to the photon case when the electromagnetic field