# Stoner-Wohlfarth astroid

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

Magnetization reversal applet.
Magnetization dynamics applet.
Spin-torque applet.
Tim Mewes - Mainpage

## Energy density

For the calculation of the Stoner-Wohlfarth astroid the energy of a single domain magnetic particle with uniform magnetization MS is considered to consist of the Zeeman energy EH due to interaction with the external magnetic field and an anisotropy term EA:

E = EH + EA

### Zeeman-energy

The energy due to interaction with the external magnetic field is given by:
EH = - MS H cos(θ-φ)
MS: saturation magnetization
θ: angle between the easy axis and the magnetization direction
φ: angle between the easy axis and the field direction

Uniaxial Anisotropy:
EA = K1 sin2(θ) + K2 sin4(θ) + ...

Biaxial Anisotropy:
EA = K1 cos2(θ) sin2(θ) + ...

## Uniaxial system

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 K1 the free energy of the particle can be rewritten in terms of dimensionless quantities as follows:
e = 1/2 sin2(θ) - hxsin(θ) - hycos(θ)
with
e = E/(2 K1)
HK = 2K1/MS : anisotropy field
hx = Hx/HK sin(φ) : normalized field component perpendicular to the easy axis
hy = Hy/HK cos(φ) : normalized field component parallel to the easy axis

## The Stoner-Wohlfarth astroid

Many basic features of the magnetization process can be understood by analyzing the energy minima in the hx-hy-plane. For applied fields smaller than a critical value two minima will exist where the magnetization is pointing along the easy axis. These minima can be stable or metastable. For applied fields larger than the critical value the energy landscape will only have one minimum. The Stoner-Wohlfarth astroid curve is the curve that separates regions with two minima of the free energy density from those with only one energy minimum. This curve is of particular importance as discontinuous changes of the magnetization can take place when crossing it.

### Tangents to the astroid

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 hx-hy-plane by drawing tangents to the astroid which go trough this field point. In the applet below only the tangents that lead to stable magnetization directions, i.e. locally minimize the energy, are shown.

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= cos3φ
y= sin3φ
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.

## Stoner-Wohlfarth astroid Applet

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 sin2(θ) - hxsin(θ) - hycos(θ)

Unidirectional and uniaxial anisotropy:
e = 1/2 sin2(θ+θ0) + 1/2 k cos(θ) - hxsin(θ) - hycos(θ)
where k denotes the ratio of the unidirectional and uniaxial anisotropy constant.

Biaxial anisotropy:
e = 1/8 sin2(2 θ) - hxsin(θ) - hycos(θ)

Uniaxial and biaxial anisotropy:
e = 1/4 sin2(θ+θ0)+ 1/16 k sin2(2 θ) - hxsin(θ) - hycos(θ)
where k denotes the ratio of the biaxial and uniaxial anisotropy constant.

Triaxial anisotropy:
e = -1/9 cos2(3 θ) - hxsin(θ) - hycos(θ)

## Acknowledgements

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

## References

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