Is the boundary between ferromagnetism and paramagnetism well-defined?

Question

I learnt that, when a ferromagnetic material is heated, it will become paramagnetic at a certain temperature. This temperature is called Curie point or Curie temperature. For example, Curie temperature for iron is given to be $1043\ \text{K}$. So as per my understanding at $1042\ \text K$ the material is ferromagnetic and at $1044 \ \text K$ it becomes paramagnetic. However, I don't understand what happens during this transition.

Earlier, I've learnt that both ferromagnetic and paramagnetic substances have atoms with permanent magnetic moment. A ferromagnetic substance is strongly attracted in the presence of a magnetic field, and a paramagnetic substance is weakly attracted. But "strongly" and "weakly" are relative terms. Or in other words, I think a weak ferromagnet is a strong paramagnet. If so, how could an experimenter say that a substance at a particular temperature is ferromagnetic or paramagnetic? What properties does a ferromagnetic substance loose when it gets converted to a paramagnetic material?

Is the boundary between ferromagnetism and paramagnetism well-defined? Alternatively, is the Curie point actually a single point on the temperature scale or is it a range of temperature?

Answer 1

It turns out that if you work out the physics behind the magnetism of a material, the magnetisation $M$ obeys a transcendental equation of the form $$M=\tanh\left(\frac{\alpha M+\gamma H}{T}\right)$$ where $\alpha$ and $\gamma$ are some constants, $H$ is the magnitude of the external magnetic field and $T$ is the temperature. What this says is that at a given temperature and external field, the magnetisation of the material will be given by the value which is a solution to the above equation. All well and good but how do we solve such an equation?

One way to do so is numerically (graphically) as can be seen below. In this simplified picture, I am considering no external field and only temperature $T$ varying. The solution to the equation will be given by the points of intersection between the hyperbolic tan curve and the straight line.

As you can see, the only solution for $T>1$ is at $M=0$ as that is the only intersection point. This is the behaviour of a paramagnet. However for $T<1$, we see that there are two more solutions emerging. What does this mean? This means that we have $|M|>0$, for zero external field. This is the behaviour of a ferromagnet.

So there is a clear distinction between a paramagnet and a ferromagnet.