Sublattice Melting in Na$_x$V$_2$O$_5$

We have developed efficient methods of measuring single crystal diffuse scattering, using synchrotron x-rays, that provide new insights into cation disorder in electrode materials. Large volumes in reciprocal space are transformed into 3D pair distribution functions (3D-$\Delta$PDF) that image defect-defect correlations in real space, allowing a model-independent view of short-range order. We demonstrate this with data on $\beta$-Na$_x$V$_2$O$_5$ with $x=0.2$ and 0.4 over the temperature range 100K to 500K. The sodium intercalants partially occupy sites on two-rung ladders penetrating the framework of vanadium oxide pyramids and octahedra, with no long-range order at room temperature and above. However, at $x=0.4$, the length scale of sodium-sodium correlations increases significantly below 200K with the emergence of forbidden Bragg peaks below an order-disorder transition. The 3D-$\Delta$PDF show that the sodium ions occupy alternate sites on each ladder rung, with a zig-zag configuration that is in phase with neighboring ladders. The growth in the length scale of sodium-sodium correlations with decreasing temperature is clearly seen in real space images that allow a quantitative determination of the interionic interactions that impede ionic mobility.