# Definition of Magnetisation

I have seen several definitions of the magnetization, and I want to know if all are equivalent or they are valid in different cases.

So if we have a ferromagnetic element, which means we can have magnetization without the need of an external magnetic field and that depending from Temperature, we can have different magnetic moments, in contrast to when an external field is applied and the spins are either parallel or anti-parallel to it.

Definition 1:

"Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume."

Personal opinion: To me this doesn't make much sense. In simple terms 1 magnetic moment (of spin)= 1 electron. This means that the number of magnetic moments per unit volume is equal to the number of electrons per unit volume (we disregard spin of neutron protons), which means it is a constant value for a ferromagnetic object. That means that the magnetization is a fixed value. But this isn't the case since we know that magnetization changes with location and time. So for me this definition is not correct.

Definition 2:

"Spontaneous magnetization is the appearance of an ordered spin state (magnetization) at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or T_C."

Personal opinion: This makes more sense. Because now the magnetization is something that varies. So it't the nr. of ordered magnetic moments (electron spins). What I can't understand is, what it's meant with ordered? Every electron magnetic moment has a direction, it's ordered. So how do we define ordering, which represents the magnetization? If I can give a very classical and naive example on how I understand it, it would be: If we have 100 electrons per unit volume,100 magnetic moments, and 40 from them point at the same directions, which is the highest nr. of electrons that point in the same direction, then this should be the magnetization. Of course there might be a cluster of electrons, whose spin points in other direction, but their numbers is smaller then 40. So we do not "count" them in the group of the ordered spins. Is this naive analogy somehow acceptable ?

Definition 3:

This is a definition which I found out solving exercises regarding magnetization. Basically magnetization is proportional to the difference between the nr. of electrons with spin up and those with spin down. Of course this is the case of an applied external field on an object made out of some sort of para/ferromagnetic element. But if we have no external field acting on a ferromagnetic element, then how do we define the magnetization. Making a spin up spin down difference here doesn't make sense, because the electron spins point in random directions.

So which is the correct definition?