Magnetoelastic contribution

The free-energy contribution can be written as —lA(By2/Cv)EjhY2, showing explicitly the well-known magnetoelastic contribution to the magnetic anisotropy. 

Theory of single-ion magnetoelastic effects In the following we review a simple thermodynamic calculation for the magnetoelastic contribution to the temperature dependence of the elastic constants in the single-ion approximation. We start from eq. (22) and include the free energy F of CEF-split 4f states. For the symmetry constants we obtain... 

The fourth and fifth terms are the magnetoelastic contribution written down in the tensor form (Mason 1951). Here the terms, which are linear and squared on the mechanical stress tensor components Ty, were taken into account. The Cartesian axes x, y and z coincide with crystallographic axes a, b and c, respectively. The nonzero components of the tensors Myyi and Ryyi o for a crystal with a 6 mm symmetry class were derived by Fumi (1952). The sixth term is the elastic energy. 

The relative influence of the surface (or interface) effects, of course, must decrease with increasing thickness t of the layer(s). Since the surface effects contribute per unit surface area , one defines, for the layer, effective parameters such that Beff, or heff, equals gbuik 2bsurt/t. Here, the factor 2 is put in, because a layer has two surfaces. In practice, this simple l/t dependence works satisfactorily. For nanocrystallites, both the volume fraction and the volume to surface ratio of the crystallites (i.e. their radius) must be taken into account (see also section 8). In connection to these effects, (non-linear) contributions to the magnetoelastic coefficients due to surface strains and surface roughness are expected to be considerable. 

In the case, that the magnetoelastic interaction is dominated by the isotropic contribution... 

A contribution caused by spin-orbit coupling and closely related to magnetocrystalline anisotropy is magnetoelastic anisotropy. Mechanical stress creates a strain which amounts to a lattice distortion and yields a correction to the magnetocrystalline anisotropy. Surface anisotropy is a manifestation of magnetocrystalline anisotropy, too (sections below and Ch. 3). 

In case of a magnetoelast, the overall energy density is the sum of WM(ffeff) magnetic and Wei(A.z) elastic energy contributions ... 

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