magnetic structure determination
In a magnetic material the atoms not only sit in an ordered crystal lattice, they also carry tiny compass needles — the magnetic moments of their unpaired electrons — and below a certain temperature these needles line up into their own ordered pattern. That pattern is the magnetic structure: all pointing the same way (ferromagnet), alternating up-down-up-down (antiferromagnet), or something more intricate like a spiral. Knowing the crystal structure tells you where the atoms are; determining the magnetic structure tells you which way each atom's spin points, and the two are genuinely separate pieces of knowledge.
The reason neutrons are the tool for this is that a neutron itself carries a magnetic moment. As it flies through the crystal it is deflected not only by the nuclei but also by the atomic magnetic moments, so an ordered array of spins diffracts neutrons much as an ordered array of atoms does. This produces magnetic Bragg peaks in addition to the ordinary nuclear ones. If the spins repeat with the same period as the atoms (as in a simple ferromagnet), the magnetic scattering just adds intensity onto the existing nuclear peaks. But antiferromagnets are the vivid case: the up-down-up-down spin pattern repeats over a distance twice the chemical cell, so it produces brand-new magnetic peaks at positions the crystal alone forbids — often at half-integer indices. Watch those extra peaks appear as you cool through the ordering (Néel or Curie) temperature and fade as you warm back through it, and you have caught magnetic order in the act.
From the positions of the magnetic peaks you deduce the repeating pattern of the spins (the magnetic unit cell), and from their intensities the directions and sizes of the moments. This is essentially the only direct microscopic probe of magnetic order — it is how the antiferromagnetism of manganese oxide was first proven, work that earned Clifford Shull a share of the Nobel Prize. One technical honesty: because the magnetic moment comes from an electron cloud with real spatial extent, the magnetic form factor falls off as the scattering angle grows (unlike the point-like nuclear scattering length, which does not), so magnetic peaks are strongest at low angles. And X-rays are, to good approximation, blind to this magnetism, which is why the neutron reigns here.
Cool antiferromagnetic manganese oxide below its ordering temperature and the neutron pattern grows extra peaks at half-integer positions like (1/2 1/2 1/2), forbidden by the chemical cell. Their appearance signals that the manganese spins have locked into an alternating up-down arrangement with a magnetic cell twice the size of the chemical one.
Antiferromagnetic order doubles the repeat, producing magnetic peaks the chemical cell forbids — the fingerprint of a spin structure.
X-rays are essentially blind to ordinary magnetism, so neutron diffraction is the standard probe of magnetic structure. Note the magnetic form factor falls off with angle (the spin cloud is spread out), unlike the point-like nuclear scattering length, so magnetic peaks dominate at low angles.