In physics, a ferromagnetic material is said to have magnetocrystalline anisotropy if it takes more energy to magnetize it in certain directions than in others. These directions are usually related to the principal axes of its crystal lattice. It is a special case of magnetic anisotropy. In other words, the excess energy required to magnetize a specimen in a particular direction over that required to magnetize it along the easy direction is called crystalline anisotropy energy.
Causes The spin-orbit interaction is the primary source of magnetocrystalline anisotropy. It is basically the orbital motion of the electrons which couples with crystal electric field giving rise to the first order contribution to magnetocrystalline anisotropy. The second order arises due to the mutual interaction of the magnetic dipoles. This effect is weak compared to the exchange interaction and is difficult to compute from first principles, although some successful computations have been made.
Practical relevance Magnetocrystalline anisotropy has a great influence on industrial uses of ferromagnetic materials. Materials with high magnetic anisotropy usually have high coercivity, that is, they are hard to demagnetize. These are called "hard" ferromagnetic materials and are used to make permanent magnets. For example, the high anisotropy of rare-earth metals is mainly responsible for the strength of rare-earth magnets. During manufacture of magnets, a powerful magnetic field aligns the microcrystalline grains of the metal such that their "easy" axes of magnetization all point in the same direction, freezing a strong magnetic field into the material. On the other hand, materials with low magnetic anisotropy usually have low coercivity, their magnetization is easy to change. These are called "soft" ferromagnets and are used to make magnetic cores for transformers and inductors. The small energy required to turn the direction of magnetization minimizes core losses, energy dissipated in the transformer core when the alternating current changes direction.
Thermodynamic theory The magnetocrystalline anisotropy energy is generally represented as an expansion in powers of the direction cosines of the magnetization. The magnetization vector can be written M = Ms(α,β,γ), where Ms is the saturation magnetization. Because of time reversal symmetry, only even powers of the cosines are allowed. The nonzero terms in the expansion depend on the crystal system (e.g., cubic or hexagonal). The order of a term in the expansion is the sum of all the exponents of magnetization components, e.g., α β is second order.
Uniaxial anisotropy
More than one kind of crystal system has a single axis of high symmetry (threefold, fourfold or sixfold). The anisotropy of such crystals is called uniaxial anisotropy. If the z axis is taken to be the main symmetry axis of the crystal, the lowest order term in the energy is
E / V = K 1 ( α 2 + β 2 ) = K 1 ( 1 − γ 2 ) . {\displaystyle E/V=K_{1}\left(\alpha ^{2}+\beta ^{2}\right)=K_{1}\left(1-\gamma ^{2}\right).}
The ratio E/V is an energy density (energy per unit volume). This can also be represented in spherical polar coordinates with α = cos ϕ {\displaystyle \phi } sin θ, β = sin ϕ {\displaystyle \phi } sin θ, and γ = cos θ:
E / V = K 1 sin 2 θ . {\displaystyle \displaystyle E/V=K_{1}\sin ^{2}\theta .}
The parameter K1, often represented as Ku, has units of energy density and depends on composition and temperature. The minima in this energy with respect to θ satisfy
∂ E ∂ θ = 0 and ∂ 2 E ∂ θ 2 > 0. {\displaystyle {\frac {\partial E}{\partial \theta }}=0\qquad {\text{and}}\qquad {\frac {\partial ^{2}E}{\partial \theta ^{2}}}>0.}
If K1 > 0, the directions of lowest energy are the ± z directions. The z axis is called the easy axis. If K1 < 0, there is an easy plane perpendicular to the symmetry axis (the basal plane of the crystal). Many models of magnetization represent the anisotropy as uniaxial and ignore higher order terms. However, if K1 < 0, the lowest energy term does not determine the direction of the easy axes within the basal plane. For this, higher-order terms are needed, and these depend on the crystal system (hexagonal, tetragonal or rhombohedral).
Hexagonal system
In a hexagonal system the c axis is an axis of sixfold rotation symmetry. The energy density is, to fourth order,
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