In the standard picture of the crust of a neutron star, matter there is simple: a body-centered-cubic (bcc) lattice of nuclei immersed in an essentially uniform electron gas. We show that at densities above that for neutron drip ($\sim4\times10^11$) g cm$^{-3}$ or roughly one thousandth of nuclear matter density, the interstitial neutrons give rise to an attractive interaction between nuclei that renders the lattice unstable. We argue that the likely equilibrium structure is similar to that in displacive ferroelectric materials such as BaTiO_3. As a consequence, properties of matter in the inner crust are expected to be much richer than previously appreciated and we mention consequences for observable neutron star properties.
