Magnetoelastic second order

Usually the analysis is limited to the first order in the strain (harmonic approximation) and second order terms (anharmonic coupling) are neglected. This second order magnetoelasticity has not been analyzed for the compounds under consideration. 

E Ybung s modulus Rljjtlm second-order magnetoelastic interaction... 

The change of the interatomic distance caused by a shear soxmd wave is of the second order compared with a longitudinal wave. The main contribution to the attenuation arises in this case firom the single-ion magnetoelastic interaction. The Hamiltonian of this interaction can be expressed in the form proposed by Callen and Callen (1965) and is linear in strain and squared in spin variables ... 

In 3d atoms, the spin-orbit coupling is much smaller than the crystal-field energy, and the magnetic anisotropy is a perturbative effect [7, 8, 16]. Typical second- and fourth-order transition-metal anisotropies are of the orders of 1 MJ/m3 and 0.01 MJ/m3, respectively. A manifestation of magnetocrystalline anisotropy is magnetoelastic anisotropy, where the crystal field is changed by mechanical strain [5, 16]. 

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