Stoner–Wohlfarth model

The Stoner–Wohlfarth model was developed by Edmund Clifton Stoner and Erich Peter Wohlfarth and published in 1948.

[1] It included a numerical calculation of the integrated response of randomly oriented magnets.

[1] Since the magnetization in the direction of the field is Ms cos φ, these curves are usually plotted in the normalized form mh vs. h, where mh = cos φ is the component of magnetization in the direction of the field.

The red and blue stars are the stable magnetization directions, corresponding to energy minima.

At h = 0.5 the red curve appears, but for h > 0 the blue state has a lower energy because it is closer to the direction of the magnetic field.

At h = −0.5, however, the energy barrier disappears, and in more negative fields the blue state no longer exists.

Usually only the hysteresis loop is plotted; the energy maxima are only of interest if the effect of thermal fluctuations is calculated.

The shape of the hysteresis loop has a strong dependence on the angle between the magnetic field and the easy axis (Figure 3).

If the two are parallel (θ = 0), the hysteresis loop is at its biggest (with mh = hs = 1 in normalized units).

The magnetization starts parallel to the field and does not rotate until it becomes unstable and jumps to the opposite direction.

[1] An alternative way of representing the switching field solution is to divide the vector field h into a component h|| = h cos θ that is parallel to the easy axis, and a component h⊥ = h sin θ that is perpendicular.

[2] Stoner and Wohlfarth calculated the main hysteresis loop for an isotropic system of randomly oriented, identical particles.

The demagnetization is assumed to leave each particle with an equal probability of being magnetized in either of the two directions parallel to the easy axis.

[1] Some remanence calculations for randomly oriented, identical particles are shown in Figure 5.

Isothermal remanent magnetization (IRM) is acquired after demagnetizing the sample and then applying a field.

These Henkel plots are often used to display measured remanence curves of real samples and determine whether Stoner–Wohlfarth theory applies to them.

Figure 1. Illustration of the variables used in the Stoner–Wolhfarth model. The dashed line is the easy axis of the particle.
Figure 2. An example solution of the Stoner–Wolhfarth model. Both h and m h are between −1 and +1 . The solid red and blue curves are energy minima, the dashed red and blue lines are energy maxima. Energy profiles are included for three vertical profiles (insets).
Figure 3. Some hysteresis loops predicted by the Stoner–Wolhfarth model for different angles ( θ ) between the field and easy axis.
Figure 4. Main hysteresis loop for an isotropic sample with identical particles. The magnetization and field are normalized ( m h = M H / M s , h = H /2 K u ). The curve starting at the origin is the initial magnetization curve. Double arrows represent reversible change, a single arrow irreversible change.
Figure 5. Three kinds of isothermal remanence for an isotropic system of randomly oriented, identical particles. The remanences are m ir , isothermal remanent magnetization; m af , alternating field demagnetization remanence; and m df , dc demagnetization remanence.