Stoner-Wohlfarth model, remanent magnetization

[frimidis]

Homework Statement

" Plot remanent magnetization versus the sample angle for the sample. Derive an
equation for the remanent magnetization using the definition of a hysteresis loop and
Stoner-Wohlfarth model. "

$H$ magnetic field
$K_u$ anisotropy constant
$M_s$ saturation magnetization.
The angles become are defined in a picture down below.

In this case the sample angle that is in the question is probably $\varphi-\theta$. For simplification we can assume that the sample angle is initially parallel to the magnetic field.

Homework Equations

Stoner-Wohlfarth model

$U=K_u\sin(\varphi)^2-\mu_0M_sH\cos(\varphi-\theta)$

$m=\cos(\varphi-\theta)$ , reduced units, projection

After differentiation with respect to $\varphi$ and using reduced units $H_k=2K/\mu_0M_s$ and $h=H/H_k$ and then solving for the maximum/minimum points

$h=\dfrac{\sin(\varphi)\cos(\varphi)}{\cos(\varphi-\theta)}$

Angles illustrated in this figure. e.a in the figure is the easy axis.

https://imgur.com/TcLI1kS

The Attempt at a Solution

I think I know the answer and it's simple. In the answer for the plotting and equation is given by $\cos(\alpha)$ where $\alpha$ is the angle between the magnetization and magnetic field. And that's the conclusion I get from inspecting the values of the remanent magnetization when $h=0$ in this picture.

https://imgur.com/CiO3gtH

However at that point I'm totally confused. Why are these values simply $\cos(\alpha) \cdot 1$. If $h=0$ and then the remanent magnetization is somehow a projection of it, why is it nonzero? I'm lost in the equations. Does the equation that is solved for $h=0$ come into play in this one?

I'm aware of the physics here. In a ferromagnetic sample there's supposed to be a nonzero remanent magnetization when the field is 0. But how does that show in the equations?

So the answers are as far as I know, the values of $m$ in the figure are given by $\cos(\alpha)$ but I have no idea why. I understand the physical situation what's going on. I just want to know how these values are obtained when $h=0$

Basically I know what the answer is, but I don't know how it's obtained. I have been stuck on this for days and have hardly process, it's part of a project. Some clarification is much appreciated, thanks![/B]

 

[mark726]

Hi there,

Thank you for your forum post. Let me try to help clarify the equations and how the values for the remanent magnetization are obtained.

First, let's start with the Stoner-Wohlfarth model. This model is often used to describe the behavior of ferromagnetic materials, specifically in the presence of an external magnetic field. In this model, the energy of the system is given by the equation:

$U=K_u\sin(\varphi)^2-\mu_0M_sH\cos(\varphi-\theta)$

Where $K_u$ is the anisotropy constant, $\varphi$ is the angle between the magnetization and the easy axis, $M_s$ is the saturation magnetization, $H$ is the applied magnetic field, and $\theta$ is the angle between the applied field and the easy axis.

Now, let's take a look at the equation for the remanent magnetization, which is given by:

$m=\cos(\varphi-\theta)$

This equation represents the projection of the magnetization onto the easy axis. When the sample is initially aligned with the magnetic field, $\varphi$ is equal to $\theta$, and therefore the remanent magnetization is equal to 1 (since $\cos(0)=1$). This is why the remanent magnetization is nonzero when the applied field is 0.

Now, let's take a look at the equation you provided for $h$, which is given by:

$h=\dfrac{\sin(\varphi)\cos(\varphi)}{\cos(\varphi-\theta)}$

This equation is used to find the maximum/minimum points of the Stoner-Wohlfarth energy curve, which is described by the first equation. When we set $h=0$, we are essentially finding the points where the derivative of the energy curve is equal to 0, which corresponds to the maximum/minimum points. When we solve for $h=0$, we get the following equation:

$0=\dfrac{\sin(\varphi)\cos(\varphi)}{\cos(\varphi-\theta)}$

This equation can be simplified to:

$\sin(\varphi)\cos(\varphi)=0$

Which has solutions at $\varphi=0$ and $\varphi=\pi/2$. These correspond to the maximum