Magnetocrystalline anisotropy: beyond first-order perturbation theory

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Abstract
General expressions are derived for the magnetocrystalline anisotropy coefficients Knm(T), correct to second order in the single-ion crystal field Hamiltonian HCF. These expressions are subsequently examined using (i) the 'effective field/parametric equation' approach of Callen and Callen (1966) and (ii) the internal field approximation. Particular attention is paid to the case of the heavy rare earth metals Tb and Dy and it is suggested, on the basis of the Callen and Callen model, that the unexpected positive sign of K40(T) in Tb metal, is due to a second-order perturbation term involving (B20O20)2. A rather surprising result is found in the case of the internal field approximation. In this particular instance, second-order angular variations in the logarithmic term -kTln(exp(-H/kT)) are almost cancelled out by similar variations in the self energy term 1/2 lambda (J)2, as the magnetic field H is rotated.