Commensurate structural modulation in the charge- and orbitally-ordered   phase of the quadruple perovskite (NaMn$_3$)Mn$_4$O$_{12}$
  By means of synchrotron x-ray and electron diffraction, we studied the structural changes at the charge order transition $T_{CO}$=176 K in the mixed-valence quadruple perovskite (NaMn$_3$)Mn$_4$O$_{12}$. Below $T_{CO}$ we find satellite peaks indicating a commensurate structural modulation with the same propagation vector q =(1/2,0,-1/2) of the CE magnetic order that appears at low temperature, similarly to the case of simple perovskites like La$_{0.5}$Ca$_{0.5}$MnO$_3$. In the present case, the modulated structure together with the observation of a large entropy change at $T_{CO}$ gives evidence of a rare case of full Mn$^{3+}$/Mn$^{4+}$ charge and orbital order consistent with the Goodenough-Kanamori model. 
A. Prodi, A. Daoud-Aladine, F. Gozzo, B. Schmitt, O. Lebedev, G. van   Tendeloo, E. Gilioli, F. Bolzoni, H. Aruga-Katori, H....
cond-mat.str-el cond-mat.mtrl-sci