Commensurate and incommensurate charge order in Fe$_2$OBO$_3$

Charge order CO in the form of a Wigner crystal had been proposed by Attfield {\em et al}. [Nature {\bf 396}, 655 (1998)] based on measurements, particularly M\"{o}ssbauer spectroscopy, on polycrystalline Fe$_2$OBO$_3$, but no superstructure due to the CO had been detected. We have grown the first single crystals of pure Fe$_2$OBO$_3$, and resistivity and thermal analysis indicate not one, but two transitions associated with CO. To elucidate the nature of these two transitions a synchrotron study was performed. At low $T$ a superstructure corresponding to a doubling of the $a$ axis was observed for the first time. The phase between the two phase transitions, in contrast, exhibits an incommensurate modulation with propagation vector ($\frac{1}{2}$,$0$,$\tau$), $\tau$ increasing with $T$ towards $\frac{1}{2}$. Resonances in the energy-dependence of the scattered intensity around the Fe $K$ edge suggest that the modulations in both phases are indeed associated with CO.