Lattice vibrations

Lattice vibrations

The oscillations of atoms in a solid about their equilibrium positions. In a crystal, these positions form a regular lattice. Because the atoms are bound not to their average positions but to the neighboring atoms, vibrations of neighbors are not independent of each other. In a regular lattice with harmonic forces between atoms, the normal modes of vibrations are lattice waves. These are progressive waves, and at low frequencies they are the elastic waves in the corresponding anisotropic continuum. The spectrum of lattice waves ranges from these low frequencies to frequencies of the order of 10¹³ Hz, and sometimes even higher. The wavelengths at these highest frequencies are of the order of interatomic spacings. See Crystal structure, Vibration, Wave motion

At room temperature and above, most of the thermal energy resides in the waves of highest frequency. Because of the short wavelength, the motion of neighboring atoms is essentially uncorrelated, so that for many purposes the vibrations can be regarded as those of independently vibrating atoms, each moving about its average position in three dimensions with average vibrational energy of 3kT, where k is the Boltzmann constant and T the absolute temperature. The wave character of the vibrations is needed, however, to describe heat transport by lattice waves. Also, lattice vibrations interact with free electrons in a conducting solid and give rise to electrical resistance. The temperature variation at low temperatures provides evidence that this interaction is with waves. See Electrical resistivity

Scattering of lattice waves by defects increases with increasing frequency (f); its variation depends on the nature of the defect. Scattering by external and internal boundaries is almost independent of frequency, thus dominating at low frequencies and hence at low temperatures. A study of the thermal conductivity of nonmetallic crystals as function of temperature yields information about the defects present, and about the anharmonic nature of the interatomic forces in the crystal lattice. See Crystal defects, Thermal conduction in solids