Magnetostriction, Forced Magnetostriction, and Anomalous Thermal Expansion in Ferromagnets

In previous papers we have formulated the theory of magnetostriction arising from single-ion crystal-field effects, for cubic crystals, and we have shown that the theory accounts extremely well for the temperature dependence of the magnetostriction in yttrium iron garnet. We here summarize that theory, extend it to arbitrary crystal symmetry, augment it by the inclusion of two-ion interactions, analyze the dependence on magnetic field strength (the "forced magnetostriction"), and discuss the totally symmetric component (which exhibits itself as an anomalous thermal expansion). The first part of this paper is concerned with the above matters, and the analysis is based entirely on symmetry considerations. It culminates in expressions relating the macroscopic magnetostriction coefficients to the product of microscopic magnetoelastic coupling constants and certain spin correlation functions. Only three such correlation functions appear, and all temperature and field dependence enters through these correlation functions. In the second part of the paper, we evaluate these correlation functions by various approximate theories: molecular field theory, a cluster theory, and the random-phase approximation. Applications to Dy and to EuS are cited, and a detailed application to Gd is given. In Gd the sign inversion of a particular magnetostriction coefficient, and its full temperature dependence, are accurately accounted for theoretically. The field dependence of the forced magnetostriction of Gd is also discussed, with special reference to the observed persistence of a pseudo-linearity slightly above the Curie temperature.