# 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} = -µ_{0} M_{S} H cos(theta-phi)

M_{S}: saturation magnetization

### 2.2 Crystal-anisotropy

**Uniaxial** Anisotropy:

**E**_{C} = K_{1} sin^{2}(theta) + K_{2} sin^{4}(theta) + ...

**Biaxial** Anisotropy:

**E**_{C} = K_{1} cos^{2}(theta) sin^{2}(theta) + ...

The order of magnitude of K_{1} is for cubic crystals about 10^{3} - 10^{4} [J/m^{3}] and for hexagonal crystals about 10^{5} - 10^{6} [J/m^{3}].

(see [Jäger] - note that the simulation now includes K_{1} and K_{2}.)

### 2.3 Shape-anisotropy

The energy due to shape-anisotropy is given by:

**E**_{S} = 1/2 µ_{0} M_{S}^{2} (N_{a} cos^{2}(theta) + N_{b} sin^{2}(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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