## 3.1 Lattice dynamics and phonons

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.