Non-monoexponential diffusion appears to be a real feature of the niobium tungsten oxide crystallographic shear phases; the fit to the data with a single-component fit was poor and it is clear from the decay curves that a biexponential fit is a far better representation of the data (Supplementary Figure ). This phenomenon was also observed in LixNb16W5O55 whereas the bronze phase LixNb18W16O93 was well-described by a single-component fit. The biexponential may represent two diffusion processes or it may capture a distribution. The data are insufficient to explore more complex models; however, the current fit does capture the distribution of correlations and rapid transport for all populations of ions -even the slow ones. The structural origin of this complex diffusion may be related to the diversity of tunnels and of sites within the tunnels: there are three classes of tunnels in the material, four per block that border two perpendicular shear planes (i.e., at the corners of the blocks), eight on the edge that border one shear plane, and in the center of the block that do not border a shear plane. Each tunnel may contain multiple Li local minima. High activation barriers are reported, at least for related phases , for hops between blocks, so that transport is essentially one-dimensional along the c axis. Additional distributions in correlation times could also result from differences in the potential energy maps due to tungsten vs. niobium substitution in the metal octahedra that form the walls of the tunnels. A characteristic quadrupolar lineshape is observed in the static spectra, even at the highest temperature measured (453K) which reflects the anisotropic environment sampled by the Li ions as they move rapidly down the tunnels (Supplementary Figure ). At lower temperatures, the signal broadens indicating a slowing down of the motion; furthermore the line shape does not appear to arise from a single environment (i.e., is not consistent with a single second-rank tensor) consistent with the multi-exponential behavior discussed above. The relatively sharp and featureless central transition resonance indicates that exchange between the lithium environments is rapid on the timescale determined by the 7 Li chemical shift. A single resonance was also observed in high-resolution magic-angle-spinning Li measurements. Non-monoexponential PFG behavior with sharp 1D spectra suggest that lithium ions sample many environments within a block (local hopping) but diffuse down the different tunnels (net long-range diffusion) with a distribution of diffusion coefficients. While a more detailed analysis of the effect of different types of distributions of correlation times on both the static lineshapes, and on the PFG measurements, will be a subject of future work, the analysis presented here clearly indicates Li diffusivities of between 10 -10 and 10 - 12 m 2 •s -1 .