Harmonic approximation of lattice vibrations ↩

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created: 2021-11-02 19:02:33
modified: 2022-01-10 04:13:03

In the harmonic approximation of lattice vibrations, we assume that the amplitude of the oscillation of atoms in a lattice is small enough that the system becomes linear.

The energy of atoms that make up the crystal:

Where the kinetic energy is just the sum of all the kinetic energies of the atoms:

Where is the index of the unit cell marked by the lattice vector and is the index of the atom inside the unit cell (and would give the position of the atom inside the unit cell). is the mass of the atom inside the cell of .

We describe the interaction of the atoms using the potential energy of the system:

Where is the pair potential and is the position of the corresponding atom at a given time.

In the harmonic approximation, we assume that the relative positions of atoms measured from their equilibrium position, is small. In this case the first order contribution vanishes of the relative positions because of the equilibrium. We expand the interaction potential energy until second order, which gives this:

Here we introduced the matrix that contains the second order derivatives of the pair potentials.

If the unit cell contains number of different atoms in a dimensional crystal, then this matrix is of size for every possible lattice vector.

For these considerations, Hamilton's canonical equations of motion go like this:

And the regular equations of motion:

Note that these number of equations are connected in complicated ways (where is the number of unit cells). When solving a practical problem, we start by writing down these equations.

Since this is a linear differential equation system, we can assume that the solution for the equations of motion can be written in a plane wave form:

With the wavenumbers of the Brillouin zone.

With this we get a linear system of equations instead of a system of linear differential equations. If the system is translation invariant, then so the equations of motion get decoupled by (the waves with different values become independent).

It is enough to describe the oscillations with the first Brillouin zone's wavenumbers, since with a reciprocal lattice vector, the wave with wavenumber gives the same values at the position of atoms as the wave with wavenumber .

Let and define the dynamical matrix: which is self-adjoint and positive definite.

The equations of motion in the wavenumber space can then be written as:

Which means that for every wavenumber we have a corresponding eigenvalue equation and by solving it we can determine the dispersion relation. The dynamical matrix is positive semi definite, so the oscillation frequencies are real. The eigenvectors give us the polarization of the lattice vibrations, i.e. how the atoms move relatively to each other inside a unit cell. Note that the matrix is dimensional, so we get different eigenvalues (i.e. dispersion branches), from which are acoustic and are optical. ^{^60345b}

The last equation has solutions, because a Brillouin zone contains number of wavenumbers.

We can further simplify this model by only considering the closest pair interactions of the atoms. In this case, we can model the forces by connecting the atoms with springs. This is the spring model of lattice vibrations.