Synchrotron x-ray single-crystal structure analysis of a spinel oxide FeV$_{2}$O$_{4}$ with spin and orbital degrees of freedom

It has been reported that FeV$_{2}$O$_{4}$, which has orbital and spin degrees of freedom both in tetrahedral Fe$^{2+}$(d$^{6})$ sites and octahedral V$^{3+}$(d$^{2})$ sites, exhibits successive structural phase transitions, accompanying a ferrimagnetic transition. The origin of the phase transitions is supposed to be a cooperation and/or competition between the orbital and spin degrees of freedom both in Fe$^{2+}$ and V$^{3+}$. By a synchrotron x-ray single-crystal structure analysis, we determined the space group and atomic coordinate of each phase (cubic- HT-tetra.- HT-ortho.- LT-tetra.). The results suggest that the HT-tetra. (a$>$c) and HT-ortho. phases should be ascribed to the FeO$_{4}$ local compression, whereas VO$_{6}$ elongation should be responsible for the LT-tetra. (c$>$a) phase. We also discuss the orbital ordering (OO) pattern assuming strong electron-lattice coupling. A conceivable OO pattern of V$^{3+}$ at the LT-tetra. (c$>$a) is \textit{ferroic} one with \textit{yz} and \textit{zx} orbitals occupied, which is unique among spinel-type vanadates.