Advanced Solid State Physics

Figure 150: Vibration of atoms with the shortest wavelength

Figure 151: Phonon density of states of silicon

however is, that the vertical scale gets elongated. Take a look at the dispersion relations of diamond
(fig. 152), silicon (fig. 153), germanium (fig. 154) and tin (fig. 155). Although all these materials
do have the same crystal structure (see fig. 156 for the first Brillouin zone of these materials) and
the same amount of valence electrons, the dispersion relation of diamond reaches from− 40 to 10 eV,
whereas the dispersion relation of tin only goes from− 20 to 0 eV. The reason for this lies in the
different lattice constants of the crystal (table 2). The solutions of the equations of motion of the
electrons are Bloch-waves and they look the same for each of these materials. But because diamond
has a very short lattice constant compared to the others, the Bloch-waves get compressed, e.g. the
frequencies and therefore the energies are increasing.

material 10 −^10 m

diamond 3. 56

silicon 5. 43

germanium 5. 65

tin 6. 48

Table 2: lattice constants