I'd been studying about simple theories for ferromagnetic material. In simple spin-wave theory, low-temperature spontaneous magnetization is expected to follow Bloch $T^{3/2}$ law, and several experiments were taken in order to check validity of this model.

The question I'd encountered is that how one can perform quantitative measurements on this spontaneous magnetization. In principle, spontaneous magnetization occurs in the absence of external field, but nonzero magnetization must cancel in macroscopic scale due to random orientation of domain structure. Further investigation showed that original work of Weiss & Forrer (1926) extrapolated high-field magnetization value in linear region(or saturation magnetization) to zero-field strength in order to acquire spontaneous magnetization. It seems that nowadays this method is still used as qualitative measure of spontaneous magnetization (I found it on https://doi.org/10.1143/jpsj.73.2539), and some introductory pdf reference even treat spontaneous magnetization and saturation magnetization as the same concept.

How can this view can be physically justified? If this method don't have rigorous theoretical ground, then is there any modern technique that directly measures spontaneous magnetization?


2 Answers 2


nonzero magnetization must cancel in macroscopic scale due to random orientation of domain structure

It tends to cancel, but it doesn't have to: it all depends what's more energetically favorable.

The Weiss theory of ferromagnetism that you mentioned explains the spontaneous magnetization by an effective "molecular field" (since then we know it's not actually a field, it's a consequence of the Coulomb repulsion + the Pauli principle) — atoms magnetized along a direction influence neighboring atoms to align along the same direction. In metals it can also be carried by conduction electrons, which pick up the magnetization of the core electrons at one location and transmit it to core electrons further away. That's why the common ferromagnets are metals.

With this you get smaller or bigger domains, and if the energetical cost of the domain walls (where neighboring atoms are not perfectly aligned) is smaller that the energetically cost of the stray field the material would emit if it was magnetized, then it's more favorable to have randomly aligned domains with no macroscopic magnetization. That's why soft ferromagnetic materials tend to demagnetize over time.

But then the papers you read were probably about hard ferromagnets. Something makes it more favorable for the whole material to be magnetized along a specific direction, typically an asymmetry of the crystal lattice or of the sample itself. In the case of an ideal hard ferromagnet, the lowest energy state is to be mono-domain. Then its hysteresis curve is perfectly square like the left one in this paper, and its spontaneous magnetization (its remanence) is the same as its saturation magnetization. As you can also see in the (b) figure on the last link, in a realistic material they are not exactly equal. For example around sample edges, the stray field might be more costly than creating a domain. But it's usually close enough, and theory tends to use idealized materials to make things easier. So yes, for hard magnets this is indeed a justified theory.

About your second question, spontaneous magnetization can be measured like any magnetization. I'm mostly familiar with the magneto-optic techniques: since photons also carry an angular momentum, they are influenced differently by electrons carrying the same or the opposite angular momentum. In practice, laser light reflected on or transmitted through a magnetized material changes its polarization. You can exploit this to just measure the overall polarization change and deduct the total magnetization of the material (that's how you can get the hysteresis curves), or you can look at domains themselves with Kerr microscopes for example.


As explained by @elyuku, even at DC magnetization doesn't have to cancel if the large cost of an external magnetic stray field is counteracted by "gained" magnetic energy, such as in terms of magnetic anisotropy energy.

If there is macroscopic magnetization, the most accurate and direct method is to pull the object through a pickup coil and measure the induced voltage. This is the essence of Vibrating Sample Magnetometers.

However, this method is typically restricted to macroscopic objects.

For nano-objects, you can use sensitive local probes for the local stray field. These include in no particular orcer:

  • Atomic force miscroscope with magnetic tips
  • Lorentz TEM
  • Hall Probe magnetometry
  • Nitrogen vacancy scanning probe

All of these probe the magnetic stray field. But there are some methods which are sensitive to magnetization and don't rely on the stray field. These methods interact with the electronic system directly, so are also sensitive to stray-field-free types of magnetism such as antiferromagnetism:

  • X-ray magnetic dichroism
  • Magnetooptical Kerr and Voigt effects (optical method)
  • Anomalous Hall effect (electric transport)
  • Polarized neutron diffraction/absorption

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