Phonons and reciprocal space

Vibrations

Recall that vibrations are dene as the oscillation (or movement) of atoms in a molecule (or crystal) in periodic motion. In the crystalline case, these periodic vibrations are known as phonons. The simplest approximation to describe this periodic motion (i.e. phonons) is that of the harmonic oscillator that is shown in Fig. 1 and described by Eq. 1.

	$x(t) = Acos(\omega t + \phi)$ (Eq. 1)
Fig. 1: Animation of a simple harmonic oscillator.	$A$ and $\phi$ is the amplitud and the phase respectively and $\omega$ is the angular frequency

If we consider the bonds between atoms as elastic strings, we can consider the Hooke's Law (Eq. 2) as a good approximation.

$F= -k \Delta x \qquad E = \frac{kx^2}{2} \qquad k = \omega^2 m$

Here $k$ denotes the spring constant and $m$ is the mass.

Vibrations in Solid State

In a simulation, we are first required to describe the system we are going to study. The "system" refers to the solid-state structure we want to investigate. Solid-state, by contrast to molecular science, makes use of the periodicity of a structure. Since the size of the crystal (or periodic tessellation of atoms) is significantly larger than the size of a single molecule in most cases, we assume that the number of atoms and structure of the crystal is infinite. Of course, not all solids have a perfect periodic arrangement and this makes things very complicated to simulate, however we will assume that our structure has a periodic arrangement.

As starting point, the simplest model where atoms are arranged periodically in space is the 1D chain of atoms with the same mass equally spaced by a (Notice that a is the equilibrium distance between atoms). Remember that in a vibration, the motion of the atoms can be described as a harmonic oscillator. Fig. 2 represent a 1D chain of atoms in a vibrational mode where the atoms are moving in anti-phase.

From Fig. 2 we note that when the atoms are far away from the equilibrium distance (Fig. 2 left and right), the energy will increase and when the distance between atoms is the same as the equilibrium distance (Fig. 2 centre), the energy is a minima. It is therefore possible to describe the change of the energy as a function of the position of the atoms in a vibrational mode with a function (Fig. 2 lower waves). To sum up, we have demonstrate that we only need a simple periodic function to describe the motion of the atoms in a crystal in a vibrational mode.

The next step is figuring out how many different vibrational modes are in a crystal. A simple rule to find the number of different vibrations is that the number of vibrational modes in a molecule is equal to 3N-6 (or 3N-5 if the molecule is linear) where N is the number of atoms. However, in a crystal we have an infinite number of atoms and therefore, we expected an infinite number of vibrational modes. In Fig. 2 we have define a vibrational wave that corresponds to the moment where all the atoms are moving in antiphase. The opposite case, where all the atoms are moving in phase is shown in Fig. 3. When all the atoms are moving in phase, there is no change in the distance between atoms and the energy is constant. Since there is no change in energy (the frequency is equal to zero), the function that describes this vibration is a straight line.

We have de�ned two boundary conditions for the vibrational frequency. As we shown above, these correspond to the anti-phase movement of atoms (where the resultant vibrational energy is at a max value) and to in phase movement of atoms (which corresponds to a resultant vibrational frequency of zero). Therefore, it follows that if we have a maximum and a minimum frequency boundary, all the other frequencies in the crystal must lie in between these two (Fig. 4 shows an example of a vibration).