Stoner–Wohlfarth Switching Astroid

Worked Example

Let us calculate the applied field components, residual, and critical values for a system where the applied field angle is 45 degrees, and the field magnitude is 1 (in normalized units relative to the anisotropy field).

1. Input Parameters:
   Field angle (alpha) = 45 degrees
   Field magnitude (h) = 1

2. Calculate applied field vector (hx, hy):
   alpha = 45 * (pi / 180) = 0.7854 rad
   hx = h * cos(alpha) = 1 * cos(0.7854) = 0.7071067811865476
   hy = h * sin(alpha) = 1 * sin(0.7854) = 0.7071067811865475

3. Calculate Astroid residual to determine metastability regime:
   Residual = |hx|^(2/3) + |hy|^(2/3) - 1.0
   Residual = (0.7071067811865476)^(2/3) + (0.7071067811865475)^(2/3) - 1.0
   Residual = 0.7937005259840997 + 0.7937005259840997 - 1.0
   Residual = 1.5874010519681994 - 1.0 = 0.5874010519681994

4. Calculate Switching field magnitude (Hswitch) at this angle:
   Hswitch = (cos(alpha)^(2/3) + sin(alpha)^(2/3))^(-3/2)
   Hswitch = (0.7937005259840997 + 0.7937005259840997)^(-3/2)
   Hswitch = (1.5874010519681994)^(-3/2)
   Hswitch = 0.5

5. Calculate Critical magnetization angle (theta_c) at this angle:
   theta_c = pi - atan(tan(alpha)^(1/3))
   tan(45 deg) = 1
   tan(45 deg)^(1/3) = 1
   theta_c = pi - atan(1) = pi - pi/4 = 3*pi/4 rad
   Critical magnetization angle = 135.0 degrees

The astroid residual is positive (0.5874010519681994 > 0), meaning this applied field configuration lies outside the switching astroid, in a regime where the metastable state does not exist (Metastability regime: Outside). The values correspond exactly to independent calculations.

Experiment

You can interactively explore the astroid boundaries and magnetization regimes. Navigate to the interactive Stoner–Wohlfarth Switching Astroid module and try the following:

1. Change the "Field angle degrees" select control to "30".
2. Set the "Field magnitude" select control to "0.5".
3. Observe the visual plot updates. The applied vector (blue dot) falls inside the boundary curve, yielding a negative residual and indicating the "Inside" regime.
4. Increase the "Field magnitude" to "1". The vector now falls outside the boundary (positive residual), moving to the "Outside" regime where the original metastable state has collapsed.

Limitations

This module considers an ideal zero-temperature single-domain particle undergoing coherent magnetization. It assumes the energy is purely given by the uniaxial anisotropy and the external field. It does not account for thermal fluctuations or multi-domain behavior such as domain-wall motion, both of which can lead to switching at fields lower than the theoretical coercive field dictated by the astroid cusp. Therefore, the boundary field here is not a universal macroscopic coercive field claim about practical magnetic materials.

References

The following source was used to build this module.

• A Review of the Stoner-Wohlfarth Model - An outline and detailed exposition of the single-domain astroid and switching mechanics.