Lattice Vibrations

So far we have been discussing equilibrium properties of crystal lattices. When the lattice is at equilibrium each atom is positioned exactly at its lattice site. Now suppose that an atom displaced from its equilibrium site by a small amount. Due to force acting on this atom, it will tend to return to its equilibrium position. This results in lattice vibrations. Due to interactions between atoms, various atoms move simultaneously, so we have to consider the motion of the entire lattice.

►The motion of atoms in a linear chain is coupled, giving rise topropagating waves

►The frequency of oscillation depends on the wavelength (i.e. the wave vector) of the propagating wave

►For an infinite chain, the possible frequency of oscillations is a continuous

► For a finite chain of quantum oscillator, only a discrete set of frequencies is possible

►Each propagating wave with a certain frequency and hence a certain group velocity is called a phonon phonon

►The frequency of atomic vibrations in a phonon depends on the phonon wave vector – k: This defines the dispersion relation.

►Phonon wave vectors for a 1D chain of length L are n·2π/L, where n is integer. Number of phonons is

►All phonon wave vectors lie between –π/a and π/a. Thereforethe number of phonons in a 1D chain is 2π/a / 2π/L = L/a

►Each phonon can be treated itself as a quantum oscillation. For low temperatures every atom can be approximated by an harmonic oscillator, the energy of the oscillation is

One-dimensional lattice

For simplicity we consider, first, a one-dimensional crystal lattice and assume that the forces between the atoms in this lattice are proportional to relative displacements from the equilibrium positions.

This is known as the harmonic approximation, which holds well provided that the displacements are small. One might think about the atoms in the lattice as interconnected by elastic springs.

Diatomic 1D lattice

Now we consider a one-dimensional lattice with two non-equivalent atoms in a unit cell. It appears that the diatomic lattice exhibit important features different from the monoatomic case. Shows a diatomic lattice with the unit cell composed of two atoms of masses M1 and M2 with the distance between two neighboring atoms a.

Three dimensional lattice

These considerations can be extended to the three-dimensional lattice. To avoid mathematical details we shall present only a qualitative discussion. Consider, first, the monatomic Bravais lattice, in which each unit cell has a single atom. The equation of motion of each atom can be written. The solution of this equation in three dimensions can be represented in terms of normal modes where the wave vector q specifies both the wavelength and direction of propagation. The vector A determines the amplitude as well as the direction of vibration of the atoms. Thus this vector specifies the polarization of the wave, i.e., whether the wave is longitudinal (A parallel to q) or transverse (A perpendicular to q). When we substitute into the equation of motion, we obtain three simultaneous equations involving Ax, Ay. and Az, the components of A. These equations are coupled together and are equivalent to a 3 x 3 matrix equation. The roots of this equation lead to three different dispersion relations, or three dispersion curves. All three branches pass through the origin, which means all the branches are acoustic. This is of course to be expected, since we are dealing with a monatomic Bravais lattice.

The three branches in differ in their polarization. When q lies along a direction of high symmetry - for example, the[100] or [110] directions − these waves may be classified as either pure longitudinal or pure transverse waves. In that case, two of the branches are transverse and one is longitudinal. One usually refers to these as the TA - transverse acoustic and LA − longitudinal acoustic branches, respectively. However, along non-symmetry directions the waves may not be pure longitudinal or pure transverse, but have a mixed character.

Shows the dispersion curves for Al in the [100] and [110] directions. Note that in certain high-symmetry directions, such as the [100] in Al, the two transverse branches coincide. The branches are then said to be degenerate. We turn our attention now to the non-Bravais three-dimensional lattice. Here the unit cell contains two or more atoms. If there are s atoms per cell, then on the basis of our previous experience we conclude that there are 3s dispersion curves. Of these, three branches are acoustic, and the remaining (3s −3) are optical. The mathematical justification for this assertion is as follows: We write the equation of motion for each atom in the cell, which results in s equations. Since these are vector equations, they are equivalent to 3s scalar equations, which have 3s roots. It can be shown that three of these roots always vanish at q = 0, which results in three acoustic branches. The remaining (3s −3) roots, therefore, belong to the optical branches, as stated above.

The acoustic branches may be classified, as before, by their polarizations as TA1, TA2, and LA. The optical branches can also be classified as longitudinal or transverse when q lies along a high-symmetry direction, and one speaks of LO and TO branches. As in the one-dimensional case, one can also show that, for an optical branch, the atoms in the unit cell vibrate out of phase relative to each other. As an example of a non-Bravais lattice, the dispersion curves. Since there are two atoms per unit cell in germanium, there are six branches: three acoustic and three optical. Note that the two transverse branches are degenerate along the [100] direction, as indicated earlier.

Phonons

So far we discussed a classical approach to the lattice vibrations. As we know from quantum mechanics the energy levels of the harmonic oscillator are quantized. Similarly the energy levels of lattice vibrations are quantized. The quantum of vibration is called a phonon in analogy with the photon, which is the quantum of the electromagnetic wave.

We see that there is a relationship between the amplitude of vibration and the frequency and the phonon occupation of the mode. In classical mechanics any amplitude of vibration is possible, whereas in quantum mechanics on discrete values are allowed. This is shown.

The lattice with s atoms in a unit cell is described by 3s independent oscillators. The frequencies of normal modes of these oscillators will be given by the solution of 3s linear equations as we discussed before.

Phonons can interact with other particles such as photons, neutrons and electrons. This interaction occurs such as if photon had a momentum qh. However, a phonon does not carry real physical momentum. The reason is that the center of mass of the crystal does not change it position under vibrations (except q=0). In crystals there exist selection rules for allowed transitions between quantum states. We saw that the elastic scattering of an x-ray photon by a crystal is governed by the wavevector selection rule, where G is a vector in the vector in the reciprocal lattice, k is the wavevector of the incident photon and ′ k is the wavevector of the scattered photon. This equation can be considered as condition for the conservation of the momentum of the whole system, in which the lattice acquires a momentum .

Referencias Bibliográficas:

http://physics.unl.edu/~tsymbal/tsymbal_files/Teaching/SSP-927/Section%2005_Lattice_Vibrations.pdf


http://users.physik.fu-berlin.de/~pascual/Vorlesung/SS09/slides/EPIV-09SS-SolSt_K3-Lattice%20vibrations.pdf