To understand the motion of atoms, we need first know the force:
(55) |
For stable solids at some temperature, it is much more useful and informative to cast the expressions in terms of an expansion of the energy in powers of displacements
(56) |
where Greek subscripts ,..., denote Cartesian components.
Within harmonic approximation, the vibrational modes at frequency are described by displacements
(57) |
so that equation 55 becomes:
(58) |
the solution of the equation is the set of independent oscillators, each with vibrational frequency ,
(59) |
In a crystal, the atomic displacement eigenvectors obey the Bloch theorem, i.e., the vibrations are classified by k with the displacement. Therefore, the vibrations can be reduced to the first BZ. For a unit cell with N atoms, the phonons at different k yield dispersion curves, which are solution of the determinant equation:
(60) |
There are three acoustic modes with for and the other modes are classified as opitc.