Equations of Motion and Atomic Force Constants

We consider a crystal composed of an infinite number of unit cells, each of which is a parallelepiped defined by three noncoplanar vectors a, I12 and a (Fig.3.L). The equilibrium position vector of the unit cell relative to an origin located at some atom is denoted by [3.1] 

As a result of thermal fluctuations, the atoms vibrate about their equilibrium positions and the actual position of the atom ( ) is given by 

For small displacements, we can expand about its equilibrium value using the three-dimensional form of Taylor s theorem  

Tn the harmonic approximation all cubic and higher-order terms are neglected. Since q is just the static potential energy of the crystal (i.e., independent of the displacement coordinates), it can be ignored for the time 

---