Origin of the magnetoelasticity in cubic rare-earth intermetallic compounds

The importance of the magnetoelastic effects in the cubic rare-earth intermetallic compounds motivates one to look for their microscopic origin. The magnetoelastic coefficients appear to be the strain derivatives of the cubic crystal-field parameters. As the latter ones, they originate from two contributions, the ligands and the conduction electrons. The aim of this paper is to study thoroughly these two contributions to the second-, fourth-, and sixth-order magnetoelastic coefficients associated with the tetragonal and trigonal symmetry-lowering modes. In particular the expressions of the direct and exchange Coulombic contribution of each type of conduction electron ($p,d$, or $f$ character) are derived. Numerical results are deduced for compounds DyZn and DyCu starting from self-consistent augmented-plane-wave functions. We verify that the magnetoelastic modifications of the cubic fourth- and sixth-order parameters are negligible in these intermetallic compounds and that the second-order magnetoelastic coefficients result from the competition of numerous opposite contributions. Comparisons are made with experimental data in the two isomorphous series of rare-earth compounds with zinc and copper.