Why this method was called semiclassical? Even though it employs classical mechanics it has to use one additional postulate taken from quantum mechanics namely, that the energy of lattice vibrations is quantized.
The classical motions of any atom are determined by Newton's law of mechanics: force=mass x acceleration. Formally, if r(t) is the position of atom at time t, then
is the instantaneous potential energy of the atom.
This potential energy arises from the interaction of the atom with all the other atoms in the
crystal.
For the reason of mathematical convenience the discussion of the semiclassical treatment is
limited here by the harmonic approximation. In this case the representation of potential
energy in the Taylor
series is truncated at the quadratic term. In other words we assume that the force affecting the
atom is linearly proportional to the displacement of the atom from its equilibrium position (the
Hook's law).
We first consider monatomic linear chain to describe
acoustical phonons .
The optical phonons will appear in the discussion of diatomic
linear chain.