In that case it is found that anharmonic effects are negligible and therefore it is assumed here that they can also be neglected. Phonon dispersion relations are calculated by looking for wavelike solutions to the classical equations of motion of atoms under a small displacement from their equilibrium sites. Firstly, a crystal is composed of an infinite number of primitive unit cells in three dimensions labelled where are integers. Each primitive unit cell is defined by three linearly independent vectors and forming a parallelipiped. This unit cell defines the periodic structure of the crystal by repeating its primitive unit cell throughout space.
Vibrations occur when the atoms are displaced from their equilibrium positions.
The subscript 0 means that the coefficients are evaluated at the equilibrium positions, . The first term in the series is constant, that is, it is independent of the displacements of the atoms from their equilibrium sites and so can be taken as the zero of the potential energy. The force on the atom is given by which is the coefficient of the second term in the series. Provided the derivative is evaluated at the equilibrium position, this term will be zero since at equilibrium the total force on any atom is zero. In the harmonic approximation all terms in cubic or higher order are assumed negligible, therefore the potential can be expressed simply as the third term in the expansion.