LATTICE VIBRATIONS

Atoms in crystal is thought of as residing at particular lattice sites, but in reality they undergo continuous fluctuations in the neighborhood of their regular positions in the lattice. These fluctuations arise from the heat or thermal energy in lattice, and become more pronounced at higher temperatures. Since the atoms are bound together by chemical bonds, the movement of one atom about tis site causes the neighboring atoms to respond to this motion. The chemical bonds act like springs that stretch and compress repeatedly during the oscillatory motion. The result is that many atoms vibrate in unison, and this collective motion spreads throughout the crystal. Every type of lattice has its own characteristic modes or frequencies of vibration called normal modes, and the overall collective vibrational motion of the lattice is a combination or superposition of many, many normal modes. For a diatomic lattice like GaAs, there are low frequency modes called acoustic modes, in which the heavy and light atoms tend to vibrate in phase or in unison with each other, and high frequency modes called optical modes, in which thye tend to vibrate out of phase.

A simple model for analyzing these vibratory modes is a linear chain of alternting atoms with a large mass M and a small mass m joined to each other by springs ~ as follows:

~ m ~ M ~ m ~ M ~ m ~ M ~

when one of the springs stretches or compresses by an amount Delta X, a force is exerted on the adjacent masses with the magnitude C delta x, where C is the spring constant. As the various springs stretch and compress in step with each other, longitudinal modes of vibration take place in which the motion of each atom is along the string direction. Each such normal mode has a particular frequency w and a wave vector k = 2 pi/ lambda, where lambda is the wavelength, and the energy E associated with the mode is given by E = hw. There are also transverse normal modes in which the atoms vibrate back and forth in directions perpendicular to the line of atoms.

......The atomic vibrations that we have been discussing correspond to standing waves. this vibrational motion can also produce traveling waves in which localized regions of vibratory atomic motion travel through the air, or seismic waves that start at the epicenter of an earthquake and travel thousands of miles to reach a seismograph detector that records the earthquake event many minutes later. Localized traveling waves of atomic vibrations in solids, called phonons, are quantized with the energy hw = hv, where v = w/2 pi is the frequency of vibration of the wave. Phonons play an important role in the physics of the solid state.

Owens F, The Physics and Chemistry of Nanosolids, WILEY, p 15