  Nanostructured rare-earth-ion doped materials are increasingly being
investigated for on-chip implementations of quantum information processing
protocols as well as commercial applications such as fluorescent lighting.
However, achieving high-quality and optimized materials at the nanoscale is
still challenging. Here we present a detailed study of the restriction of
phonon processes in the transition from bulk crystals to small ($\le$ 40 nm)
nanocrystals by observing the relaxation dynamics between crystal-field levels
of Tb$^{3+}$:Y$_3$Al$_5$O$_{12}$. We find that population relaxation dynamics
are modified as the particle size is reduced, consistent with our simulations
of inhibited relaxation through a modified vibrational density of states and
hence modified phonon emission. However, our experiments also indicate that
non-radiative processes not driven by phonons are also present in the smaller
particles, causing transitions and rapid thermalization between the levels on a
timescale of $<$100 ns.
