FOMP

Question

The energy of a ferromagnetic crystal can be expressed as a power series of direction cosines of the magnetic moment with respect to the crystal axes. The coefficient of those terms are the anisotropy constant. In general the expansion is limited to few terms. Normally the magnetization curve is continuous respect to applied field up to saturation but, in certain intervals of the anisotropy constant values, irreversible field induced rotations of the magnetization are possible implying first order magnetization transition between equivalent magnetization minima, the so-called first order magnetization process (FOMP).

The total energy of an uniaxial magnetic crystal in an applied magnetic field can be written as a summation of the anisotropy term up to six order, neglecting the sixfold planar contribution,

and the field dependent Zeeman energy term

where:

so the total energy can be written

In order to determine the preferred directions the magnetization vector in the absence of the external magnetic field we analyze first the case of uniaxial crystal. The maxima and minima of energy with respect to θ must satisfy

while for the existence of the minima

For symmetry reasons the c-axis (θ=0) and the basal plane (θ=π/2) are always points of extrema and can be easy or hard directions depending on the anisotropy constant values. We can have two additional extrema along conical directions at angles given by:

The C ⁺ and C ^− are the cones associated to the + and - sign. It can be verified that the only C ⁺ is always a minimum and can be an easy direction, while C ^− is always a hard direction.

A useful representation of the diagram of the easy directions and other extrema is the representation in term of reduced anisotropy constant K₂ / K₁ and K₃ / K₁ . The following figure shows the phase diagram for the two cases K₁>0 and K₁<0 . All the information concerning the easy directions and the other extrema are contained in a special symbol that marks every different region. It simulate a polar type of energy representation indicating existing extrema by concave (minimum) and convex tips (maximum). Vertical and horizontal stems refer to the symmetry axis and the basal plane respectively. The left-hand and the right-hand oblique stems indicate the C ^− and C ⁺ cones respectively. The absolute minimum (easy direction) is indicated by filling of the tip.

EA=K1sin2⁡<!-- ⁡ -->θ<!-- θ -->+K2sin4⁡<!-- ⁡ -->θ<!-- θ -->+K3sin6⁡<!-- ⁡ -->θ<!-- θ -->{\displaystyle \displaystyle E_{A}=K_{1}\sin ^{2}\theta +K_{2}\sin ^{4}\theta +K_{3}\sin ^{6}\theta }

EH=−<!-- − -->HMscos⁡<!-- ⁡ -->(θ<!-- θ -->−<!-- − -->γ<!-- γ -->){\displaystyle \displaystyle E_{H}=-HM_{s}\cos(\theta -\gamma )}

K1,K2,K3{\displaystyle \,K_{1},K_{2},K_{3}}	 are the anisotropy constants up to six order,
H{\displaystyle \,H}	 is the applied magnetic field,
Ms{\displaystyle \,M_{s}}	 is the saturation magnetization,
θ<!-- θ -->{\displaystyle \,\theta }	 is the angle between the magnetization and the easy c-axis,
γ<!-- γ -->{\displaystyle \,\gamma }	 is the angle between the field and the easy c-axis,

ET=EA+EH{\displaystyle \displaystyle E_{T}=E_{A}+E_{H}}

∂<!-- ∂ -->E<m