  While NMR measurements of nuclear energy spectra are routinely used to
characterize the static properties of quantum magnets, the dynamical
information locked in NMR 1/T1 relaxation rates remains notoriously difficult
to interpret. The difficulty arises from the fact that information about all
possible low-energy spin excitations of the electrons, and their coupling to
the nuclear moments, is folded into a single number, 1/T1. Here we develop a
quantitative theory of the NMR 1/T1 relaxation rate in a collinear
antiferromagnet, focusing on the specific example of BaFe2As2. One of the most
striking features of magnetism in BaFe2As2 is a strong dependence of 1/T1 on
the orientation of the applied magnetic field. By careful analysis of the
coupling between the nuclear and electronic moments, we show how this
anisotropy arises from the "filtering" of spin fluctuations by the form-factor
for transferred hyperfine interactions. This allows us to make convincing,
quantitative, fits to experimental 1/T1 data for BaFe2As2, for different field
orientations. We go on to show how a quantitative, angle-dependent theory for
the relaxation rate leads to new ways of measuring the dynamical parameters of
magnetic systems, in particular the spin wave velocities.
