If a ferromagnet is cooled to below its Curie temperature under a magnetic field, what is the minimum field that could make the magnet saturated?
If the ferromagnet is a meso- or macroscopic piece of material and being cooled through TC without external field, then magnetic domains will form. The field required to remove the domain walls depends much on mechanisms that hinder their motion (they can e.g. be trapped at defects).
i suppose that cooling under a magnetic field leads to less domainwalls occurring from the beginning, depending on the magnitude of the field. They still will require extra energy to be expelled, bringing us back to the same mechanisms and energies.
This said, even if the cooling occurs under sufficiently high fields to have no domains from the outset, the demagnetizing field (dipolar interaction) will normally be inhomogeneous (unless the body is a sphere or an ellipsoid) and lead to a non-uniform magnetization in the material. Full saturation then takes larger fields in excess of the demagnetizing field (might be a sizeable fraction of M to align these regions, and will depend on direction of the applied field with respect to the shape of the body).
Have no better answer than that, sorry.
So, the answer is that the minimum magnetic field is infinity.
Even if one would remove any magnetic imperfections causing noncollinearity of the spins in the ferromagnet, there still will be a place for magnons which will decrease magnetic order parameter at finite temperature and make the experimentally measured magnetization to be depended on the applied magnetic field. Also magnetization saturation cannot be achieved if one would magnetize the anisotropic ferromagnet in the intermediate directions (i.e. neither at easy nor at hard direction) - it just follows from a simple vector analysis.
Perhaps, one should set a term of technical magnetization saturation parameter and the respective saturation field for that. Then, a more determined (and more useful from the practical view of the problem) answer/receipt could arise.
As indicated by Kai's answer, some assumptions have to be made. You may get a more specific answer if you give some more specific details. What is material? What is it's size and shape?
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