Symmetry and Lattice Vibrations

Abstract

The other chapters in this book deal with the atoms in a crystalline solid being in their static equilibrium positions. This chapter focuses on the thermal vibrations of the atoms about their equilibrium positions. This motion is treated in the harmonic approximation. The symmetry of the lattice plays an important role in determining how the atoms move. The positions of neighboring atoms can inhibit motion in some directions while facilitating motion in other directions. This results in certain “normal modes” of vibration being allowed and other vibrational modes not allowed. Any state of vibration of the lattice can be expressed as a superposition of normal modes. The energy of the vibrational modes is quantized and can be described by eigenvectors and eigenvalues (frequencies). Each of these modes exhibit specific symmetry and can be associated with one of the irreducible representations of the crystallographic point group.