User:Bolzoni/sandbox

FOMP
The magnetocrystalline anisotropy energy of a ferromagnetic crystal can be espressed 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, with respect to applied field, is continuous up to saturation but when the anisotropy constant fall within certain interval, irreversible field induced rotations of the magnetization are possible implying first order magnetization transition between nonequivalent magnetization states, the so called first order magnetization process (FOMP)

THEORY
For 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


 * $$\displaystyle E_A=K_1\sin^2\theta+K_2\sin^4\theta+K_3\sin^6\theta$$

neglecting the six fold planar contribution and the field dependent Zeeman energy term


 * $$\displaystyle E_H=-HM_s\cos(\theta-\gamma)$$

where:

 

so the total energy can be written
 * $$\displaystyle E_T=E_A+E_H $$

EQUILIBRIUM CONDITIONS FOR H=0
The minima of energy with respect to $&theta;$ satisfy


 * $$\frac{\partial E_A}{\partial \theta} = 0 \qquad \text{and} \qquad \frac{\partial^2 E_A}{\partial \theta^2} > 0.$$

For symmetry reasons the c-axis ($$\theta=0$$) and the basal plane ($$\theta=\pi/2$$) 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:


 * $$\displaystyle \sin\theta_c={[-K_2\pm (K_2^2-3K1K_3)^{1/2}]/3K_3}^{1/2} $$

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