Phonons are the quantum mechanical quasi particles which describe lattice vibrations. The Hamiltonian for an Eigenstate of the lattice vibration is
|
| \begin{equation*} H = \hbar \omega(k,\lambda)\left(N+\frac{1}{2}\right) \qquad . \end{equation*} | (3.31) |
Here \(\omega(k,\lambda)\) 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 \(\lambda\) the polarization.
\(\lambda\) 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 \(\left(\frac{2\pi}{L}\right)^3\). With the often used approximation we find for the complete
number of states with momentum values smaller than \(|\vec{k}| = k\)
|
| \begin{equation*} N(k) = \left(\frac{L}{2\pi}\right)^3 \frac{4}{3} \pi k^3 \qquad . \label{n_k_phonon} \end{equation*} | (3.32) |
Therefor the density of states is
|
| \begin{equation*} D(\omega) = \left(\frac{V k^2}{2\pi^2}\right) \frac{dk}{d\omega} \qquad . \end{equation*} | (3.33) |
In order to apply the Eq. (3.4) to (3.9) we must calculate the density of states or the dispersion relation
|
| \begin{equation*} \omega = \omega(k) \qquad . \end{equation*} | (3.34) |
For this we can take the exact solutions or the approximation of section 3.5, i.e. the Einstein- and Debye-model.
|
|
|
|
© J. Carstensen (Stat. Meth.)