View source for 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.

{| class="wikitable"
![[File:Simple harmonic motion animation.gif | 300px]]
!<math> x(t) = Acos(\omega t + \phi) </math> (Eq. 1)
|-
!Fig. 1: Animation of a simple harmonic oscillator.
!<math>A</math> and <math>\phi</math> is the amplitud and the phase respectively and <math>\omega</math> 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.

<math> F= -k \Delta x \qquad E = \frac{kx^2}{2} \qquad k = \omega^2 m</math>

Here <math>k</math> denotes the spring constant and <math>m</math> 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 defined 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).

Once we have established a simple method to describe our system, we need to define a mathematical expression that describe the vibrations. Since the vibrational waves are also periodic functions, we can use a sine or cosine function (Eq. 3) to represent them.

This function is known as planewave, where A0 is the amplitude of the wave, k is the wavenumber, x is a given point in space, ! is the angular frequency, t is a given point in time and  is the phase shift (which is considered 0 here). Remember that the wavenumber  is defined as the number of wavelenghts per unit distance and k is just the number of radians per unit distance (Eq. 4).

We can also write this expression with a complex exponential form (Eq. 5).

Now we need a link between the function that describe the vibrations with the Hookes law (Eq. 2). By solving a series of equations it can be determined that the frequency of a particular vibration for a 1D chain of atoms is described by Eq. 6.

where J is the spring constant, M is the mass of the atom, a is the distance between atoms and k is the wavenumber (Eq. 4.) 

Now, when the chain of atoms is made of two atoms with different masses, after solving again a set of equations we found that now the vibrational frequency is described by Eq. 7.

Notice that for a particular value of k, there are two different vibrations for the same wavenumber. This will be discuss later when we introduce the concept of phonon band structures.


== Reciprocal space ==

Here we are going to introduce the concept of reciprocal space (also known as k space). The reciprocal space is the Fourier transform of real space. This has used in crystallography and in fact, we have already applied this concept in the previous section. We found that we can describe the motion of a 1D chain of atoms with a single function. This function is a periodic (wave) function which is described by a wavenumber k. We demonstrate that all the vibrations are in between two states, when the motion of all the atoms are in phase and when the motion of all the atoms is in anti-phase. Notice that when all the atoms are moving in anti-phase, the wave that describes the vibration of the the 1D chain of atoms has a wavelength � that is equal to the equilibrium distance of the atoms a (Fig. 2).

In the opposite case, when the motion of all the atoms is in phase, we demonstrate that the function that describe the vibration is a constant function. In this case, we can consider that � is infinity and therefore, we need an infinite number of atoms to represent it (Fig. 3).

Therefore � goes from a to infinity. One can intuitively deduce that there is a smarter way to represent a vibration that requires a infinity and this is done by representing the inverse of infinity. Instead of defining� we can define the inverse of the the wavelength which is the wavenumbers k.

We know that k = 2� and the minimum value of � is a, therefore

For the opposite case, � ! 1

And we know that any other vibration is in between these two values. Therefore, we have define a lattice (reciprocal lattice) that goes from 0 to 2� a where we can study the vibrations (and any other property that is periodic with the distance of the atoms). Notice that for a 1D chain of atoms, we only need to define one atom and the equilibrium distance between atoms. This is known as the primitive cell (minimum unit cell). When we move to more complicated system like an actual crystal, we are dealing with 3 dimensions. In a crystal we need to define three lattice parameters and therefore we will have three reciprocal lattice. The space define by this three lattices is known as reciprocal space.