Charge Localization and near zero in-plane thermal expansion in layered Na$_{x}$CoO$_{2}$, $x\sim\frac{1}{2}$

We have used neutron powder diffraction and X-ray single crystal diffraction to investigate the lattice response to charge localization and magnetic ordering in samples with stochiometry close to Na$_{0.5}$CoO$_{2}$ over the temperature range of 2 to 600K. Our measurements show there presence of two crystallographical distinguishable Co sites but with essentially identical bond valences. This indicates only a marginal charge disproportionation between Co$^{3}$ and Co$^{4}$. Our measurements show Na-ordering achieved by de-intercalation imposes a constraint on the charge and spin-state degrees of freedom that acts as to reduce the in-plane thermal expansion to near zero values between 2 to 300K. At higher temperatures we find a phase non-reversable transition to a phase with a $3a \times\space 3b \times\space c$ cell, wtr to the orthorhombic P$nmm$ low temperature phase. This phase does not exhibit any of the anomalies observed at 52 and 87K in the magnetic susceptibility of the as made phase.