# 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.

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 = EZ + EC + ES

### 2.1 Zeeman-energy

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

### 2.2 Crystal-anisotropy

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

Biaxial Anisotropy:
EC = 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:
ES = 1/2 µ0 MS2 (Na cos2(theta) + Nb sin2(theta))
Na: Demagnetization-coefficient parallel to the easy-axis
Nb: Demagnetization-coefficient perpendicular to the easy-axis

The coefficients for simple shapes are listed below:
Sphere:
Na = Nb (= Nc) = 1/3

Infinite layer:
Ni = 0 (in the layer)
Ns = 1 (perpendicular to the layer)

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