2.5.2 Scattering on Phonons

The thermal motion of atoms in the crystal may be considered as normal oscillations of the crystal lattice. The mechanical properties of this system are described by the following Lagrangian [24]:

$$L=\frac{1}{2}\sum_{\vec{n}s}m_{s}\Dot{\vec{u}}^{2}_{s}(\vec{n})-\... ...Lambda_{ik}^{ss^{'}}(\vec{n}-\vec{n}^{'})u_{si}(\vec{n})u_{s^{'}k}(\vec{n}^{'})$$ (2.96)

which leads to equations of motion:

$$m_{s}\Ddot{u}_{si}=-\sum_{\vec{n}^{'}s^{'}}\Lambda_{ik}^{ss^{'}}(\vec{n}-\vec{n}^{'})u_{s^{'}k}(\vec{n}^{'}).$$ (2.97)

Here $\vec{n}=(n_{1},n_{2},n_{3})$, $m_{s}$ are the atomic masses, $\vec{u}_{s}$ are the atomic displacements. The solution of this equation represents a plane wave of the form:

$$\vec{u}_{s}(\vec{n})=\vec{e}_{s}(\vec{k})\exp\{i(\vec{k}\cdot\vec{r}_{n}-\omega t)\},$$ (2.98)

where $\vec{e}_{s}$ is the complex amplitude^2.33 which only depends on the position within the elementary cell.
S. Smirnov: