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I am getting confused about different length scales in magnetic materials. I understand that the correlation length for a ferromagnetic materials is defined as <(s(x)−<(s(x))>)(s(y)−<(s(y))>)>=exp(−|x−y|/ξ), and should diverge near Tc and becomes very small at T->0.

On the other hand, the ferromagnetic materials have long-range magnetic order below Tc, which manifests as magnetic domains, whose size can be evaluated by <(s(x)s(y)>.

These two concepts work fine for normal ferromagnetic materials and seem to not have direct relation, as the former is evaluating the correlation of "fluctuation" and the other is evaluating the correlation of the "average".

(Please let me know if any of the above points is wrong).

The problem really shows up when one considers the case of ferromagnetic nanoparticle, i.e. in the "superparamagnetic" state. Basically the magnetization will switch randomly between two opposite (assuming single axis anisotropy) directions due to thermal fluctuation. In this case, <(s(x))> = <(s(y))> = 0. Then it seems that the definition of the correlation length and the domain size becomes the same. Does this make sense? For example, one often says that a "single-domain" nanoparticle switches coherently like a superspin. This seems to either require or (implicitly) assume that the correlation length is larger than the particle size. However, a correlation length of a typical nanoparticle size (tens of nm) is already too long to be taken for granted. Is there anything wrong in my understanding or am I missing some important points?

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