Über den einfluss der anharmonizität auf die eigenschaften der kristalle

Abstract

In the lattice theory of crystals we generally assume the validity of the harmonic approximation, in which only quadratic terms are taken into account in the expansion of the potential energy in the displacements. In the present paper the influence of higher terms (anharmonicity) is investigated, and a particular example (face-centred cubic lattice with central forces between nearest neighbours only) is discussed. The free energy, from which all the thermodynamic properties of a crystal may be evaluated, is computed with the aid of a method analogous to Dirac perturbation theory. It is possible to deduce general theorems for the temperature dependence of the isothermal and adiabatic elastic constants. In no case the Cauchy Relations are satisfied. It is also possible to derive relatively simple expressions for the contributions to the specific heat arising from anharmonicity. It is shown in particular, that the relation between elastic data and Debye temperature derived in the harmonic approximation is not valid even at absolute zero. An expression is also derived for the velocity of elastic waves taking into account anharmonicity. The relation between the wave velocity and the elastic constants is discussed. It appears from this that the relations deduced for cubic crystals are valid only for transverse waves. The simple example discussed would be valid for rare gases in the solid state, for which, however, only very scanty experimental data exist.