Cubic Pyrochlore (Eu2Ir2O7, $E8_{1}$) Structure: A2B2C7_cF88_227_c_d_af

Picture of Structure; Click for Big Picture
Prototype : Eu2Ir2O7
AFLOW prototype label : A2B2C7_cF88_227_c_d_af
Strukturbericht designation : $E8_{1}$
Pearson symbol : cF88
Space group number : 227
Space group symbol : $Fd\bar{3}m$
AFLOW prototype command : aflow --proto=A2B2C7_cF88_227_c_d_af
--params=
$a$,$x_{4}$


Other compounds with this structure

  • FNb2(Nb,Ca)2O6 (“synthetic” pyrochlore), (Nb,Ta,Ti)2(Ca,Ce,Na,K)2(F,O)7 (“natural” pyrochlore), (F,O,OH)(Nb,Fe)2(Ca,Ce,Na,K)2O6 (Koppit), (F,OH)Sb2(Ca,Mn,Na)2O6 (Roméite), (OH)Sb2(Ca,Fe,Na)2O6 (Scheebergite), (Sb,Ti)2(Ca,Fe,Mn,Na)2(O,OH)6 (Lewisite), (OH,F)(Nb,Ta,Ti)2(Ca,Fe,Na)2O6 (Pyrrhite), (OH,F)(Nb,Ta)2(Ca,Fe,Na)2O6 (Mikrolith), Sb2Pb2O7 (Bindheimite), (H2O)0.875(Al0.8125Mg0.1875)2Na0.375[F0.65,(OH)0.35]6 (Ralstonite), Sb3O6OH, BiTa2O6F, Sn2Nb2O7, and Sn2Ta2O7

  • (Hermann, 1937) and (Herrmann, 1943) use Strukturbericht $E8_{1}$ to describe the cubic pyrochlore structures. These have the general formula $R$$_{2}$$Q$$_{2}$$X$$_{7}$, where the $X$ atoms or radicals occupy the (4a) and (48f) sites, and the $R$ and $Q$ atoms occupy the (16c) and(16d) sites. In many cases the sites are only partially filled and/or have mixed chemistry. We use Eu$_{2}$Ir$_{2}$O$_{7}$ as our prototype because it represents a fully filled system.

Face-centered Cubic primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & \frac12 \, a \, \mathbf{\hat{y}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{y}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \mbox{O I} \\ \mathbf{B}_{2} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{7}{8}a \, \mathbf{\hat{x}} + \frac{7}{8}a \, \mathbf{\hat{y}} + \frac{7}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \mbox{O I} \\ \mathbf{B}_{3} & = & 0 \, \mathbf{a}_{1} + 0 \, \mathbf{a}_{2} + 0 \, \mathbf{a}_{3} & = & 0 \, \mathbf{\hat{x}} + 0 \, \mathbf{\hat{y}} + 0 \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Eu} \\ \mathbf{B}_{4} & = & \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} & \left(16c\right) & \mbox{Eu} \\ \mathbf{B}_{5} & = & \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Eu} \\ \mathbf{B}_{6} & = & \frac{1}{2} \, \mathbf{a}_{1} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16c\right) & \mbox{Eu} \\ \mathbf{B}_{7} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(16d\right) & \mbox{Ir} \\ \mathbf{B}_{8} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(16d\right) & \mbox{Ir} \\ \mathbf{B}_{9} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16d\right) & \mbox{Ir} \\ \mathbf{B}_{10} & = & \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16d\right) & \mbox{Ir} \\ \mathbf{B}_{11} & = & \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{12} & = & x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{13} & = & x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{14} & = & \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{15} & = & x_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{16} & = & \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{4}\right) \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{17} & = & \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{18} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{19} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{20} & = & \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{21} & = & -x_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \mathbf{B}_{22} & = & \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(48f\right) & \mbox{O II} \\ \end{array} \]

References

  • H. Sagayama, D. Uematsu, T. Arima, K. Sugimoto, J. J. Ishikawa, E. O'Farrell, and S. Nakatsuji, Determination of long–range all–in–all–out ordering of Ir4+ moments in a pyrochlore iridate Eu2Ir2O7 by resonant x–ray diffraction, Phys. Rev. B 87, 100403 (2013), doi:10.1103/PhysRevB.87.100403.
  • C. Hermann, O. Lohrmann, and H. Philipp, eds., Strukturbericht Band II 1928-1932 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).
  • K. Herrmann, ed., Strukturbericht Band VI 1938 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1941).

Found in

  • S. H. Chun, B. Yuan, D. Casa, J. Kim, C.–Y. Kim, Z. Tian, Y. Qiu, S. Nakatsuji, and Y.–J. Kim, Magnetic Excitations across the Metal–Insulator Transition in the Pyrochlore Iridate Eu2Ir2O7, Phys. Rev. Lett. 120, 177203 (2018), doi:10.1103/PhysRevLett.120.177203.

Geometry files


Prototype Generator

aflow --proto=A2B2C7_cF88_227_c_d_af --params=

Species:

Running:

Output: