Lattice Symmetries And Thermal Expansion

Abstract

It is well known that many of the thermomechanical properties of crystals are due to anhannonic effects in the motion of the atoms. The anhannonic terms in the potential energy which lead to this behavior depend, in part, on the symmetry of the crystal lattice. Thus the thermomechanical properties of the crystal, including the coefficient of thermal expansion, depend on the point group of the lattice. In order to study this symmetry dependence the path integral representation of the density matrix in the adiabatic approximation is used here to evaluate the thermal expectation value of the atomic positions. From this result the approximate scaling behavior of the atomic displacement with temperature, mass, bond strength (harmonic coupling), and anharmonic coupling is determined. The path integral calculation is fully quantum mechanical and can be used, in principle, to derive all the thermomechanical properties of the crystal within the adiabatic approximation.