Helical strain patterns and charge ordering in YbFe$_2$O$_4$

The $R$Fe$_2$O$_4$ compounds exhibit simultaneously charge ordering (CO) of the Fe$^{2+}$ and Fe$^{3+}$ ions,\footnote{Ikeda \textit{et al}, Nature \textbf{436} 1136 (2005)} together with magnetic ordering of the Fe spins\footnote{Christianson \textit{et al}, PRL \textbf{100} 107601 (2008)} and possible multiferroic behavior.\footnote{S-W Cheong \textit{et al}, Nat. Mater. \textbf{6} 13 (2007)} Synchrotron data collected below the 3D CO transition show intensity concentrated around peaks separated by $\sim$1/3 $c$* but slightly displaced in the $(a^*, b^*)$ plane. Calculations modelling an oxygen displacement pattern are in excellent agreement with the data, suggesting an incommensurate charge ordering of the Fe ions close to the commensurate $\sqrt{3} \times \sqrt{3}$ structure, associated with a helical strain pattern. At high temperature, ordering wavevectors corresponding to many different displacement patterns are simultaneously populated by the system, leading to diffuse but highly structured features in reciprocal space.