The applet below calculates the Stoner-Wohlfarth astroid of a single domain particle for different magnetic anisotropies.

See also:

Magnetization reversal applet.

Magnetization dynamics applet.

Spin-torque applet.

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For the calculation of the Stoner-Wohlfarth astroid the energy of a single domain magnetic particle with uniform magnetization M_{S} is considered to consist of the **Zeeman energy** **E _{H}** due to interaction with the external magnetic field and an

The energy due to interaction with the external magnetic field is given by:

**E _{H} = - M_{S} H cos(θ-φ) **

M

θ: angle between the easy axis and the magnetization direction

φ: angle between the easy axis and the field direction

The basic features of the Stoner-Wohlfarth astroid are best explained in a system with a uniaxial anisotropy only [Stoner1948].
For a system with uniaxial anisotropy **K _{1}** the free energy of the particle can be rewritten in terms of dimensionless quantities as follows:

with

Many basic features of the magnetization process can be understood by analyzing the energy minima in the **h _{x}**-

One important property of the astroid is that tangents to the astroid represent magnetization directions with extremal energy, i.e. either local minima or local maxima. For the uniaxial case the tangent(s) that are closest to the easy axis (in the applet below the y-axis is the easy axis) lead to stable solutions, i.e. minimal energy. For the general case of an arbitrary anisotropy the determination which of the tangents minimize and which maximize the energy is not as easy but still possible, for details see the work of E.W. Pugh, W.D. Doyle and A. Thiaville.
The astroid can thus be used to geometrically determine the stable magnetization directions for any point in the **h _{x}**-

An astroid is a hypocycloid with four cusps and can be constructed by rolling a circle with radius 1/4 within a circle of radius 1 and tracing a point on its circumference. The parametric equation for the astroid is:

**
x= cos ^{3}φ
**

Note that the critical curves for anisotropies other than uniaxial can be constructed by tracing a point on the circumference of a circle with a different radius.

The Applet below was developed by Michael Vogel (mcsvogel@gmail.com) in spring 2006.

Once the calculation is finished, try clicking on the astroid! You can also drag the field vector around.

The field vector is shown in red while the stable magnetization directions are shown in black, tangents to the astroid are shown as green lines.
The color of a point in the astroid indicates how many energy minima exist for this point:

blue: one minimum

purple: two minima

orange: three minima

yellow: four minima

green: five minima

turquoise: six minima

If the option *Angle* of the *Plot Controls* is selected the brightness of the respective color represents the angle of the magnetization in the absolute minimum of the free energy - the brightest color corresponds to θ=0 (up) and the darkest color to θ=π (down).

If the option *Energy* of the *Plot Controls* is selected the brightness of the respective color represents the depth of the absolute minimum normalized to the range of energy values for this particular color - the darker the color the deeper is the absolute minimum at this position.

The applet uses the following dimensionless energy functionals:

Uniaxial anisotropy:

**
e = 1/2 sin ^{2}(θ) - h_{x}sin(θ) - h_{y}cos(θ)
**

Unidirectional and uniaxial anisotropy:

where

Biaxial anisotropy:

Uniaxial and biaxial anisotropy:

where

Triaxial anisotropy:

- Prof. J.W. Harrell for the idea for the applet.
- Prof. W. Doyle for invaluable advice on the Stoner-Wohlfarth astroid.

- Magnetization reversal Java-applet.
- E.C. Stoner, F.R.S. Wohlfarth, E.P. Wohlfarth, Phil. Trans. Roy. Soc. London, Ser. A 240, 599 (1948).
- J.C. Slonczewski, Research Memorandum RM 003.111.224, IBM Research Center Poughkeepsie (1956).
- L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon (1960).
- E.W. Pugh, Proc. Intermag Conf. 182, 15-16 (1963).
- W.D. Doyle, IEEE Transactions on Magnetics 2, 68-73 (1966).
- G. Bertotti, Hysteresis in Magnetism, Academic Press (1998).

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This page was last modified on 05/13/13

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