Shape anisotropy

The magnetostatic energy of a ferromagnetic ellipsoid with magnetization M_s is

Other simple shapes can be approximated to ellipsoids. The anisotropy energy is related to the difference in energy  Δε when the ellipsoid is magnetized along its hard and easy directions. N is the demagnetizing factor for the easy direction; N' = 1/2(1 − N) is the demagnetizing factor for the perpendicular, hard, directions. Hence Δε_m = 1/2 μ₀VM²_s [1/2 (1 − N) − N], which gives for a prolate ellipsoid

This is zero for a sphere (N = 1/3), as expected. Shape anisotropy is only fully effective in samples which are so small that they do not break up into domains. Non-ellipsoidal shapes are approximately described by an effective demagnetizing factor. In a multidomain state, each domain creates its own demagnetizing field, and is subject to the stray field of the other domains. The order of magnitude of shape anisotropy for a ferromagnet with μ₀M_s ≈ 1 T is 200 kJ m⁻³ when N = 0.

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