Anisotropy

The natural direction of magnetization in a microscopic ferromagnetic domain is usually constrained to lie along one or more easy axes. Since magnetism is associated with circulating electron currents, time reversal symmetry requires that a state with a certain magnetization distribution M(r) should have the same energy as the state with reversed magnetization along the same axis, −M(r). This tendency is represented by the anistropy energy E_a, of which the leading term is

........(1)

where θ is the angle between the direction of M and the easy axis. Here E_a and K_u, the anistropy constant, are measured in J m⁻³. Typical values range from less than 1 kJ m⁻³ to more than 10 MJ m₋₃. Anisotropy limits the coercivity available in hard magnets.

.........(2)

where the magnetic constant μ₀ is 4π × 10⁻⁷ J A⁻² m⁻¹. Anisotropy also leads to unwanted coercivity in soft magnets. It may be noted from the units that μ₀ always multiplies H₂ or MH in expressions for magnetic energy per unit volume.
Atomic densities in solids are around n = 10²⁹ m⁻³, so if anisotropy energy per atom is expressed in terms of an equivalent temperature using Ea/n= kBT , it is in the range 1 mK–10 K. The energy is usually small in relation to the Curie temperature, but it is nevertheless decisive in determining the hysteresis.