The deviations of a lattice structure from its equilibrium configuration can be associated with vibrations that ultimately may be quantized to give origin to bosonic, wave-like, modes called phonons. The dispersion of these waves is determined and different branches are identified, depending on the unit cell structure, separating into sound waves and so-called optical modes, associated with finite energies and excitable coupling to radiation. The thermal properties of the various modes are considered, introducing simple models such as the Einstein and Debye models. These modes have a significant contribution to the specific heat of a material. In their presence the lattice is not exactly periodic, leading to an attenuation effect in the diffraction pattern.