S. Scandolo, ICTP

Lesson 19 (Fonons (vibrations))

The lesson starts with the discussion of the vibrations of a two-atom system. As the temperature goes to zero, the potential energy between the atoms realizes its minimum. When the temperature increases, the atoms move away from their equilibrium position as the statistics of the kinetic and potential energies follows Boltzmann distribution. If the temperature is high (e.g. is not true), simple vibrations around the minimum of the potential cannot fully describe the system.

The professor goes on to a one-dimensional chain of atoms. He models the interaction as springs, so the system becomes analogous to coupled oscillators. The equation of motion for the displacements can be solved easily by Fourier transforming along the time variable. The resulting matrix equation is a simple eigenvalue problem. Another approach involves Fourier transforming along the discrete spatial variable. This reduces the problem to an algebraic equation for the Fourier space variables. The solution is , where M is the mass, k is the spring constant, and q are the time- and space-like Fourier variables.