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I'm trying to understand the direction of magnetic moments in a ferromagnetic material after cooling down below it's Curie temperature.

A permanent magnet of ferromagnetic material will loose it's order and become paramagnetic above the it's Curie temparature.

As far as I understand, all information about the direction of it's previous magnetisation is lost here - is that correct?

I understand that spontaneous magnetisation happens when cooling down.

What's unclear to me is what influences the direction of the spontaneous magnetisation:

  • It could be random, depending on some initial cluster of parallel magnetic moments that gets dominant
  • it could be determined by the external magnetic field - for example of the earth, if there is no stronger one.
  • or does it somehow relate to the previous magnetisation? Maybe based on contminations with material of higher curie temperature?
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If there is a magnetic field present, it will dictate the direction of the magnetization, as you anticipated and as @user3683367 said. This is then not referred to as spontaneous symmetry breaking but the external magnetic field breaks the symmetry explicitly.

In the absence of an external magnetic field, the alignment is indeed random. You intuition that one of the clusters or domains will become dominant is actually quite right. This can be observed very nicely in Monte Carlo-simulations. As the temperature approaches the critical temperature from above, the domain sizes grow larger and larger but there is no long-range-order (LRO). Upon crossing $T_c$, one domain will eventually take up the whole (simulation) volume, establishing a definite LRO. While the temperature is still above zero, there will be fluctuations on top of that (opposite domains inside the big one) but they are rather short-ranged and short-lived.

The information on alignment of the magnetic moments before the temperature is raised above $T_c$ should be lost entirely, at least in theory. This would require that the magnet is heated homogeneously such that LRO is destroyed throughout the bulk.

Contaminations a.k.a. impurities will in general complicate the story from a theoretical point of view. Keep in mind that the Curie temperature or critical temperature of a material is only meaningful as a many-body property, i.e. it tells you something about the interactions of the electrons in iron with each other. The critical temperature for iron will not really tell you what will happen if you dope another material with a different critical temperature with iron.

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  • $\begingroup$ I would expect that there is no such thing like "no exterenal field" even assuming the field of earth etc are blocked as good as feasible. Is there a practical minimum value where the magnetic field is no longer relevant? (like because it's effects get weaker that thermal noise?) $\endgroup$ – Volker Siegel Sep 26 '14 at 14:33
  • $\begingroup$ In principle, any finite magnetic field, regardless how small, will lead to breaking of symmetry in that direction. This can be understood also within the framework set by renormalization group theory (which I cannot elaborate on in a comment). This is of course assuming the that the thermodynamic limit holds, i.e. the system is sufficiently large and is given time to reach equilibrium. If you go from a really high temperature (i.e. higher than the energy scale of the magnetic field) to $T=0$ instantly, I suppose you'd freeze out the fluctuations. $\endgroup$ – Jonas Greitemann Sep 26 '14 at 15:11
  • $\begingroup$ So, in summary: I have a magnet, heat it up, cool it down. The magnet was strong magnetized and pointing upwards with north. Now, it's magnetisation is pointing to north, while the magnet did not more or rotate. And the magnetisation is weaker than before, but stronger than the field of Earth? $\endgroup$ – Volker Siegel Sep 26 '14 at 15:22
  • $\begingroup$ However, at criticality, i.e. close to the critical temperature, one indeed observes a crossover behavior as you anticipated. The relevant quantity is $h/|t|^\Delta$ where $h$ is the magnetic field & $t$ is the reduced temperature $t=(T-T_c)/T_c$ (i.e. $t=0$) is the critical point. $\Delta$ is called the gap exponent and depends on the model. For $h/|t|^\Delta\ll 1$ one observes the critical behavior expected for the system without external field. For $h/|t|^\Delta\gg 1$ the critical behavior is changed due to the field. $\endgroup$ – Jonas Greitemann Sep 26 '14 at 15:22
  • $\begingroup$ In an ideal magnet, the magnitude of the magnetization does not depend on the strength of the field in the ferromagnetic state. Then, the strength or the magnet after heating & cooling will be the same as before. In reality, there is also the phenomenon of hysteresis which would indeed lead to the behavior you described. Hysteresis is kind of difficult to capture theoretically as it is related to disorder in the magnetic material. $\endgroup$ – Jonas Greitemann Sep 26 '14 at 15:35
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As far as I understand it is determined by an external field. The ferromagnetic system has two ground states for its magnetization both with equal energy (degenerate ground states). If you distort the system with an external magnetic field you make one of those states the preferred one so that your system magnetizes in this direction.

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