Draft:Normal Modes of Vibration in a Crystal

Like gases and liquids, crystals also exhibit vibrations about an axis, when they are exposed to electromagnetic radiations. A vibration is said to be a normal mode vibration, if all the atoms of the molecule vibrate in the same phase and frequency.

Assume that a crystal contains $N$ primitive unit cells. Each of them contains $n$ atoms. Hence, there will be $3Nn$ degree of freedom. The solution of the vibrational problem give rise to $3n$ frequencies. Three of these frequencies have zero value at the center of the Brillouin zone. These three frequencies are called the acoustic modes. The remaining $3n-3$ frequencies are called optical modes. Hence, at the zone center, we just need to consider the optical modes.

The optical modes are further divided into internal modes and external modes. Internal modes correspond to the stretching, bending etc. and external modes correspond to translation and rotation.

There exist two approaches to study the vibrational spectra of solids:

1. Unit cell approach
2. Site symmetry approach

Unit Cell Approach

This method is developed by the Indian physicists Bhagavantham and Venkatarayudu. In this method, we treat the unit cell as a large molecule. The modes are classified by applying space group operations. This method is more tedious as the complete arrangement of atoms are to be known. When there are more atoms, then the method is limited to be applied.

Site Symmetry Approach

In this approach, a site group is selected such that it is a subgroup of both free ion and factor group. The normal modes are obtained in the free ion symmetry and is correlated to the factor group using standard correlation tables.