  We report on Magnetic Resonance Force Microscopy measurements of the
Boltzmann polarization of the nuclear spins in copper by detecting the
frequency shift of a soft cantilever. We use the time-dependent solution of the
Bloch equations to derive a concise equation describing the effect of rf
magnetic fields on both on- and off-resonant spins in high magnetic field
gradients. We then apply this theory to saturation experiments performed on a
100 nm thick layer of copper, where we use the higher modes of the cantilever
as source of the rf field. We demonstrate a detection volume sensitivity of
only (40 nm)$^3$, corresponding to about 1.6$\cdot 10^4$ polarized copper
nuclear spins. We propose an experiment on protons where, with the appropriate
technical improvements, frequency-shift based magnetic resonance imaging with a
resolution better than (10 nm)$^3$ could be possible. Achieving this resolution
would make imaging based on the Boltzmann polarization competitive with the
more traditional stochastic spin-fluctuation based imaging, with the
possibility to work at milliKelvin temperatures.
