Can we measure the angular momentum of a magnetic domain using precession? The amount of angular momentum of a single iron atom is small, just $\hbar / 2$. In a single magnetic domain, though, all of the iron atoms have their spins aligned. Presumably, it should be possible to cause a magnetic domain balanced on a point to precess like a gyroscope. Has an experiment like this ever been performed?
To make it concrete, say we have a cylinder of iron that contains a single magnetic domain (radius $R$ length $L$). Now, suppose we balance it horizontally on a point a distance $h \le L/2$ from its center. Assuming we perfectly cancel the Earth's magnetic field, with what angular frequency will the cylinder precess? Are there lengths for which this experiment produces observable angular frequencies (the requirement of a single magnetic domain means $R$ and $L\ll 1$ meter, but I don't know how much)? If there are, has it been done? If not, what's the limitation blocking it (e.g. can't cancel Earth's magnetic field perfectly enough, can't produce a grain of metal that has a single domain, can't produce a tip both small enough to balance and strong enough to hold the domain, can't place the domain on the tip, etc)?
 A: Given the cylinder model, the angular frequency of the precession is given by:
$$\omega_p  = \frac{\tau}{L}.$$
The torque supplied by gravity will be $Mgh = \rho \pi R^2\ell gh$ (with $\rho$ the mass density of the material, and changing the cylinder's length to $\ell$), and the angular momentum will be $\frac{\hbar}{2}N =  \frac{\hbar}{2} \frac{\rho}{m} \pi R^2 \ell$ with $m$ the mass of a single atom (or unit cell, in the case of an alloy). Thus
$$\omega_p = \frac{2 m g h}{\hbar}.$$
If we, naively, put in iron ($m= 9.3\times10^{-26}\operatorname{kg}$) we get that $\omega_p = 1.7\times 10^{10} \operatorname{m}^{-1}\operatorname{s}^{-1} h$, which is stupidly large for pretty much any value of $h$.
To get an idea of whether precession can even occur or not, we need to compare the angular frequency of precession to the ratio of $\omega_s = L/I_s$ (the classical angular frequency of the spin). If $\omega_s$ is not much greater than $\omega$, precession cannot occur (the object will just fall). For a rod rotating about its center, $I_s = \frac{1}{2} M R^2 = \frac{\rho \pi R^4 \ell}{2}$. Thus
$$\omega_s = \frac{\hbar}{m R^2}.$$
The requirement that $\omega_s / \omega_p \gg 1$ leads to
$$\frac{\hbar^2}{2 m^2 g R^2h} \gg 1. \tag1$$
The Wikipedia article on single magnetic domains suggests 10 to 100 nanometers as the bounds for single domain grain sizes. Assuming $h\approx R$ and solving for $R$ produces $R \ll 400 \operatorname{nm}$, which isn't too different. So, if $R=10\operatorname{nm}$, $\ell=90\operatorname{nm}$ and $h=1\operatorname{nm}$ then $\omega_p=17 \operatorname{rad}\operatorname{s}^{-1}$ and $\omega_s = 5.6\times 10^6 \operatorname{rad}\operatorname{s}^{-1}$.
So, in principle, it is possible (setting $h \approx 0.2 \operatorname{nm}$, just a couple of atoms, would be even better), with the experiment working best on an absolutely minimal sized grain ($\omega_s \propto R^{-2}$, and we want that as big as possible). Thus, the experiment would be a very difficult one involving nanoscience, and all of the literally sticky difficulties that entails.
