## Lattice Vibrations

In this section we pretend to give a short review of vibrations in periodic systems such as crystals. The "adiabatic approximation" in solid state physics allows the separate study of those properties of materials, attributed to electrons, like the electrical conductivity, and those which depend on the vibrations of the atoms, such as the thermal properties. Suppose a mechanical wave, or a sound wave, travelling through a solid. If its wavelength X is much larger than the lattice constant of the crystal, then the medium behaves as an elastic continuum, although not necessarily isotropic. However, when X is comparable or smaller than the lattice unit cell, we have to consider the crystalline structure of the solid.

In order to treat the vibrations of the lattice atoms, we will usually follow the harmonic approximation. Forces between neighbouring atoms have their origin in a kind of potential which is mainly attractive, giving rise to the interatomic bonding (e.g. covalent, ionic, van der Waals). However, if the distance between atoms becomes very small, the electrons between two neighbouring atoms start to interact, and because of the Pauli exclusion principle, a repulsive interaction appears which increases very rapidly as the distance between them decreases. One of the better-known potentials which describes this interaction is the Lennard-Jones potential. This potential shows a minimum when the interatomic distance r is equal to the one at equilibrium, i.e. to the lattice constant a. For values of r close to a, the potential V(r) can be approximated by a parabolic or harmonic potential. At the beginning, we will assume that this harmonic approximation is valid around r = a.

### 2.7.1. One-dimensional lattice

The simplest model to study vibrations in a periodic solid is known as the one-dimensional monoatomic chain, which consists of a chain of atoms of mass m, equilibrium distance a, and harmonic interaction between atoms (Figure 2.9(a)). In this figure we call un the displacement of the atoms from the equilibrium position. The equation of motion of atom n, if we only consider interaction between closest neighbours, should be:

= C [(Un+l - Un) - (Un - Un-l)] = c(Un+1 + Un-l - 2Un) (2.67)

nth atom

Figure 2.9. (a) One-dimensional monoatomic chain of atoms in equilibrium (upper) and displaced from equilibrium (lower); (b) representation of the dispersion relation.

and similarly for any other atom in the chain. In order to solve Eq. (2.67), travelling plane waves of amplitude A, frequency m and wave number k are assumed, i.e.

where xn = na is the equilibrium position of the atoms. Substituting Eq. (2.68) for the atomic displacement un and the corresponding ones for un+1 and un-1 , after some algebra, one gets  from Eq. (2.67).

Eq. (2.69), known as the dispersion relation for vibrations in a one-dimensional lattice, is represented in Figure 2.9(b). One important consequence of this equation, difficult to imagine in continuous media, is the existence of a maximum frequency of value 2(c/m)l/2, over which waves cannot propagate. This frequency is obtained when k = n/a, i.e. X = 2a in Eq. (2.69). This condition is similar to that for a Bragg reflection for electrons in periodic structures (Section 2.6.2), and mathematically leads to standing waves, instead of travelling waves, which cannot propagate energy. This result should be expected since for k = n/a, the group velocity is zero. Note also in Eq. (2.69) that for ka ^ 0, m varies linearly with k and the group velocity coincides with the phase velocity of the wave, both having the value vs = a(c/m)1/2. This result is expected since for ka ^ 0, a/X ^ 0, i.e. the wavelength is much greater than the interatomic distance, and the medium can be considered as continuous. In this situation vs is equivalent to the speed of sound in the medium.

Following a similar procedure as in the case of electrons in periodic crystals (Section 2.3), we can establish periodic boundary conditions for the solutions given by the waves of Eq. (2.68). Physically these conditions could be obtained by establishing a fixed link or constraint forcing the first and last atoms of the chain to perform the same movement. This results (see Eq. (2.32)) in the following allowed values for k:

where L and N are the length of the chain and the total number of atoms, respectively, i.e. L = Na.

A final important consequence of the dispersion relation is that the value of m remains the same whenever the value of k changes in multiples of 2n/a. Therefore, it would be sufficient, as it happens for electrons, to consider only the values of k belonging to the first Brillouin zone, that is n n

The next level of complexity in the study of lattice vibrations comes when the crystal has more than one atom per primitive unit cell. Suppose then the diatomic linear chain of Figure 2.10(a) with two kinds of atoms of masses M and m, where M > m. The main difference arises now from the fact that the amplitudes of the atoms M and m are unequal.

nth atoms fWlfWWf i periodicity

 Optical branch^^" V Mm J 2c / m ■J2c/ M / Acoustic / branch k

Figure 2.10. (a) Diatomic linear chain of atoms in equilibrium (upper) and displaced from equilibrium (lower); (b) representation of the dispersion relation.

Figure 2.10. (a) Diatomic linear chain of atoms in equilibrium (upper) and displaced from equilibrium (lower); (b) representation of the dispersion relation.

If one amplitude is A, the other is aA, where a is in general a complex number that takes into account the relation between amplitudes as well as the phase difference. We can next proceed in an analogous way as we did with the linear chain. The solution of the problem becomes now more complicated but the calculation is straightforward and can be found in any elementary text on solid state physics.

For the diatomic linear chain, the dispersion relation is found to be :

The dispersion relation, represented in Figure 2.10(b), has now two branches, the upper and the lower corresponding to the ± signs of Eq. (2.72), respectively. As for the monoatomic chain, it is instructive to examine the solutions at certain values of k, close to the centre zone (k = 0) and at the boundaries (k = n/a). If ka ^ 1, then a = 1 for the lower or acoustic branch and a = -M/m for the upper or optical branch. If a = 1, the neighbouring atoms M and m vibrate essentially with the same phase, as it happens with the sound waves in solids for which X > a. If a = -M/m, both particles oscillate out of phase, and the upper branch presents a maximum frequency equal to:

The vibrational modes of this branch are called optical, because the value of the frequency is in the infrared range and in crystals such as NaCl, with a strong ionic character, the optical modes can be excited by an electromagnetic radiation. In these modes, the positive and negative ions move evidently out of phase when excited by the oscillating electric field of an electromagnetic radiation.

### 2.7.2. Three-dimensional lattice

The dispersion relations considered in Section 2.7.1 for the case of the 1D crystal can now be generalized to three dimensions . The number of acoustic branches for a 3D lattice is three, one longitudinal in which the atoms vibrate in one direction of the chain and two transverse ones, in which the atoms vibrate perpendicularly to the direction of the propagation of the wave. Therefore, there will be one longitudinal acoustic branch (LA) and two transverse acoustic branches (TA) which are often degenerated. As a note of caution, we would like to remark that one has to be careful with the above statement regarding the directions of the vibrations of the atoms, because if k is not along a direction with high symmetry, then the atomic displacements are not exactly along k or perpendicular to it.

As in 1D, if there is more than one atom, let us say p per primitive unit cell, the number of optical branches is, in general, 3p-3. If p = 2, as for an example in alkali halides, there are three acoustic branches and three optical ones. However in highly symmetric directions the two transverse modes might be degenerated. Figure 2.11 shows schematically the dispersion relations for a 3D crystal. Since there are no degeneracies we have assumed that the crystal is anisotropic. Note also that in general, as it happens with electrons in crystals, the dispersion curves cut the Brillouin zone boundaries perpendicularly, although there might be exceptions in the case of very complicated shapes of the Brillouin zones.

 3 * LO TO2 TOj LA TA2 Figure 2.11. Dispersion relations for a 3D crystal with two atoms per unit cell. Wavevector, k n/a Figure 2.11. Dispersion relations for a 3D crystal with two atoms per unit cell.
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