# Difference between domain size and correlation length in ferromagnetic materials?

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?

Assume an Ising chain (1d, say for the sake of the argument) of length $$L$$. In the ferromagnetic phase this has only one big domain.
For the proper statistical physics description (universality, renormalization group...) to be valid, one must assume $$a\ll \xi \ll L$$, where $$a$$ is the lattice parameter. In a finite system the latter constraint is violated close to $$T_c$$ and in this case one should be very careful (such considerations are not useless though, e.g. for a technique called finite-size scaling).
Sufficiently deep in the ferromagnetic phase, the length scales are well- separated, and flipping of a $$\xi$$ domain is a different process (occuring much faster, polynomically) than flipping the "average" of size $$L$$ (a process which is exponentially suppressed in the thermodynamic limit $$L\rightarrow\infty$$).