  A leading direction in the hunt for axion dark matter is to search for its
influence on nuclear spins. The detection scheme involves polarizing a sample
of nuclei within a strong static magnetic field and then looking for a spin
precession induced by the oscillating axion field. We study the axion signal
and background contributions that arise in such experiments (a prominent
example being CASPEr), finding key differences with the existing literature.
Most importantly, in the limit where the transverse spin-relaxation time of the
material is the largest timescale of the problem, we show that the induced
signal continues to grow even beyond the coherence time of the axion field. As
a result, we find that spin-precession instruments are much more sensitive than
what has been previously estimated in a sizable range of axion masses, with
sensitivity improvement of up to a factor of 100 at an axion mass of 100 neV
using a Xenon-129 sample. This improves the detection prospects for the QCD
axion, and we estimate the experimental requirements to reach this motivated
target. Our results apply to both the axion electric and magnetic dipole moment
operators.
