  Muon spin spectroscopy ($\mu^+$SR) is now an established method to measure
atomic scale diffusion coefficients of ions in oxides. This is achieved via the
ion hopping rate, which causes periodic change in the local magnetic field at
the muon site(s). We present here the first systematic study on a single
crystalline sample. The highly anisotropic diffusion of Li-ions in the battery
cathode material LiFePO$_4$, combined with the extensive investigation of this
material with $\mu^+$SR and other techniques make it a perfect model compound
for this study. With this experiment we can confirm that Li diffusion in the
bulk LiFePO$_4$ is measurable with $\mu^+$SR. Hence, surface/interface effects,
which might play a crucial role in case of powders/nano crystals, are less
significant for macroscopic single crystals where bulk diffusion is in fact
present. We observe that the internal magnetic field fluctuations caused by the
diffusing Li-ions are different depending on the crystal orientation. This is
not obviously expected based on theoretical considerations. Such fluctuation
rates were used to estimate the diffusion coefficient, which agrees well with
values estimated by first principle calculations considering anisotropic
diffusion.
