Is there a minimum field required in order to start magnetizing an initially unmagnetized ferromagnet?

Question

The question Would a compass with unmagnetized needle work? never seems to have been resolved, due to getting bogged down in a number of sub-issues where different people made seemingly contradictory claims. The purpose of the present question is to isolate one of the issues for separate discussion: Is there a minimum field required in order to start magnetizing a ferromagnet?

Physika says:

Then the question is whether the Earth's magnetic field is strong enough to realign the domains in an iron needle. The Landau Free Energy is used to determine this, as the domains will align in whatever way minimizes this energy. Parameters that determine this energy include things like: size and shape of the needle, material (in this case iron), and external field strength.

If the external field is strong enough to cause magnetization, the direction of the induced magnetization will be in a direction that minimizes the anisotropy energy and is pre-determined by the dimensions of the needle. The dimensions give rise to an "easy" axis [...] Anyway, I haven't done the calculation, but I don't think the Earth's field is strong enough to cause realignment of the domains.

Pieter says:

A small field can move domain walls. Not in steel, but we use a pipe of weak iron to shield the Earth's field. There is also mu-metal. And there is Metglass, amorphous iron-boron used in transformers.

I'm not sure, but both of these statements seem to me like they might not be right. Physika's description would seem to imply that the initial hysteresis curve of iron is horizontal, and yet the hysteresis curves I've seen don't look like that. Pieter may be saying that this is the case for hard ferromagnetic materials but not soft ones, but my understanding is that these are both idealizations. Pieter's statements about shielding also seem inconclusive to me, since the shielding works simply because these materials are highly permeable, not because they're ferromagnetic.

The following material from comments may also be relevant:

Magicsowon:

I think even at very small fields, there is magnetization induced in ferromagnets. The domains parallel to the field would get slightly bigger, and the ones antiparallel would get slightly smaller. But, the effect is too small to make a difference.

Physika:

By "there is magnetization induced in ferromagnets," do you mean induced in unmagnetized ferromagnets like in the question? Since ferromagnets are spontaneously magnetic, by definition, nothing needs to induce the magnetization. Regarding the domains: I guess the shortest way I can phrase this is that there is an effective shielding due to the exchange interaction at the boundaries, so the domains inside the bulk of the needle will be unaffected. Domain wall motion resembles a wave, so changes due to external stimuli start on the surface: youtube.com/watch?v=erzydYEDXIE

I'm also getting confused by the issue of whether this is a bulk property or not, and perhaps the resolution of the seeming contradiction is that one thing is true in bulk, the other for finite samples...?

Answer 1

Mobile domain walls are the cause of the large permeability of a soft ferromagnetic material. Materials for magnetic shielding must be like that, mu-metal works in the same way.

There is always some hysteresis, some remanence, and a coercive field, even in amorphous ferromagnets. But those can be very small, smaller than the Earth's magnetic field.

As to whether this is a bulk property or due to finite samples - a truly infinite ferromagnet could be single domain, I suppose. The reason why domains arise is that they minimize the energy in the external field at zero applied field. This is also the cause of shape anisotropy.

See also this animation of how a domain wall can move in an applied field. There may be barriers as in this animation, but in soft magnets one tries to minimize them.

So to address the quote from Physika, one could look at that figure with two domains, and forget about the inclusion. Without an applied field, the two domains would be equally large. Any applied field, however small, would lower the energy of one domain and raise the energy of the other one. The domain wall would move a bit and the sample would acquire a net magnetization. Which of course gives rise to an external field, so there will be a balance. In such a system, there is no threshold to overcome. The sample can then be described as having a relative permeability, which could be very large (like $10^4$).