Magnetization reversal applet

The applet below calculates the coherent magnetization reversal of an uniformly magnetized particle, also referred to as the Stoner-Wohlfarth model of magnetization reversal.

See also:
Stoner-Wohlfarth astroid applet.
Magnetization dynamics applet.
Spin-torque applet.
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1. Description

The energy-density of a ferromagnetic particle is calculated while a magnetic field H is applied to the particle. Searching the local minimum of the energy provides the angle that the magnetization forms with the easy-axis. The observed Hysteresis-loop is a result of the variation of strength the applied field.

1.1 Geometry

The geometry used for the simulation is as follows:



H: external field
M: magnetization of the particle
Theta: angle between easy-axis and magnetization
Phi: angle between easy-axis and external field
a,b: half-axes of the particle

2. Energy of the particle

The energy of the particle consists of the Zeeman-energy, the energy due to crystal-anisotropy and shape-anisotropy:

E = E_Z + E_C + E_S

2.1 Zeeman-energy

The energy due to interaction with the external magnetic field is given by:
E_Z = -µ₀ M_S H cos(theta-phi)
M_S: saturation magnetization

2.2 Crystal-anisotropy

Uniaxial Anisotropy:
E_C = K₁ sin²(theta) + K₂ sin⁴(theta) + ...

Biaxial Anisotropy:
E_C = K₁ cos²(theta) sin²(theta) + ...
The order of magnitude of K₁ is for cubic crystals about 10³ - 10⁴ [J/m³] and for hexagonal crystals about 10⁵ - 10⁶ [J/m³].
(see [Jäger] - note that the simulation now includes K₁ and K₂.)

2.3 Shape-anisotropy

The energy due to shape-anisotropy is given by:
E_S = 1/2 µ₀ M_S² (N_a cos²(theta) + N_b sin²(theta))
N_a: Demagnetization-coefficient parallel to the easy-axis
N_b: Demagnetization-coefficient perpendicular to the easy-axis

The coefficients for simple shapes are listed below:
Sphere:
N_a = N_b (= N_c) = 1/3

Infinite layer:
N_i = 0 (in the layer)
N_s = 1 (perpendicular to the layer)

Ellipsoid:
For an ellipsoid with the half-axes a > b >> c (see [Morrish]) the coefficients are:
N_a = pi c [1 - (a-b)/4a - 3 ((a-b)/4a)²]/4a
N_b = pi c [1 + 5(a-b)/4a + 21 ((a-b)/4a)²]/4a
N_a + N_b + N_c = 1

3 Simulation

If your browser is unable to run 1.5 Java applets, here is a picture of the applet

4 References

Stoner-Wohlfarth astroid Java-applet.
[Morish] : A.H. Morrish The physical principles of magnetism
[Stoner] : E.C. Stoner and E.P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A 240, 599 (1948).
[Wernsdorfer]: W. Wernsdorfer, Magnétométrie à micro-SQUID pour l'étude de particules ferromagnétiques isolées aux échelles sub-microniques, C.N.R.S.-Grenoble (1996)
[Jäger] : E. Jäger, R. Perthel, Magnetische Eigenschaften von Festkörpern, Akademie Verlag (1996)

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