How the neutron magnetic moment was measured? 
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*How was the neutron magnetic moment measured? 

*Was the antineutron magnetic moment measured too?
 A: The best measurement of the neutron magnetic moment was reported in 1979 by Greene and collaborators.
That measurement used nuclear magnetic resonance (NMR) to measure the rate at which the spins of polarized neutrons precess around a magnetic field.  The magnetic field was measured by flowing water through the same volume as the neutrons and also performing NMR on the protons in the water, so the experiment gave a high-quality (0.25 parts per million uncertainty) measurement for the ratio $μ_\mathrm{n}/μ_\mathrm{p}$ of the magnetic moments of the neutron and proton.  That measured ratio is now one of the many inputs to the biannually-updated CODATA fit of all the known measurements of fundamental constants.
Contrary to occasional comments here, the neutron magnetic moment cannot be measured using Stern-Gerlach deflection; the magnetic moment is small enough that the deflection is negligible in laboratory fields except for ultra-cold neutrons.
I don't believe there has ever been a direct measurement of the antineutron magnetic moment. It's just too hard to make antineutrons.  For example, the Greene et al. measurement took place on a cold neutron beamline with a rate of about $10^5\,\mathrm{n/s}$, which was extracted from a reactor with a (hot) neutron flux of about $10^{14}\,\mathrm{n\,cm^{-2}\,s^{-1}}$.  This sort of fabulous inefficiency is pretty typical for cold neutron sources.  The very first measurement of the neutron magnetic moment (Alvarez and Bloch, 1940) was based on NMR measurements on a population of a few million (hot) neutrons, which seems like a reasonable floor for the number that you'd need.  By contrast, the best current measurement of the neutron mass (Cresti et al, 1986) is based on 59 antineutron events from a sample of a few thousand candidates.  So the antineutron population in a top-notch antineutron mass experiment thousands of times smaller than the neutron population in a minimal neutron magnetic moment experiment. Furthermore, if you tried to produce an antineutron beam for doing NMR, that beam would contain at least half neutrons (since every $\bar n$ is created with an $n$, and they cannot be separated by their charge). 
Without a compelling reason to think that the antineutron magnetic moment is different from the neutron's, I don't think such a measurement would be a good use of a bright antiproton beam facility.
