Quantum mechanical description of lattice vibrations

3.6 Quantum mechanical description of lattice vibrations

Phonons are the quantum mechanical quasi particles which describe lattice vibrations. The Hamiltonian for an Eigenstate of the lattice vibration is

$H = ℏ ω (k, λ) (N + \frac{1}{2}) .$

(3.31)

Here $ω (k, λ)$ is the frequency of one Eigenvalue of the oscillation. $N$ is the number of phonons which occupy this state; since phonons are Bosons, each state can be occupied with an arbitrary number of particles. The factor $1 ∕ 2$ is the zero point energy of the vibration; this will be neglected in the further considerations. $k$ is the momentum and $λ$ the polarization. $λ$ indicates the different vibration modes (orientation in space , longitudinal, transverse, acoustic, optic). The vibrational Eigenstates we get by diagonalization of the Hamiltonian as described in the last section for the 1D example.
As usual we apply periodic boundary conditions; so each state occupies a $k$ space volume of ${(\frac{2 π}{L})}^{3}$ . With the often used approximation we find for the complete number of states with momentum values smaller than $| \vec{k} | = k$

$N (k) = {(\frac{L}{2 π})}^{3} \frac{4}{3} π k^{3} .$

(3.32)

Therefor the density of states is

$D (ω) = (\frac{V k^{2}}{2 π^{2}}) \frac{d k}{d ω} .$

(3.33)

In order to apply the Eq. (3.4) to (3.9) we must calculate the density of states or the dispersion relation

$ω = ω (k) .$

(3.34)

For this we can take the exact solutions or the approximation of section 3.5, i.e. the Einstein- and Debye-model.

[next] [prev] [prev-tail] [front] [up]