Maghemite ($\gamma$–Fe2O3, $D5_{7}$) Structure : A2B3_cP60_212_bcd_ace

Picture of Structure; Click for Big Picture
Prototype : Fe2O3
AFLOW prototype label : A2B3_cP60_212_bcd_ace
Strukturbericht designation : $D5_{7}$
Pearson symbol : cP60
Space group number : 212
Space group symbol : $P4_{3}32$
AFLOW prototype command : aflow --proto=A2B3_cP60_212_bcd_ace
--params=
$a$,$x_{3}$,$x_{4}$,$y_{5}$,$x_{6}$,$y_{6}$,$z_{6}$


Other compounds with this structure

  • $\gamma$–Al2O3 ($\gamma$–corundum)

  • (Hermann, 1937) gives $\gamma$–Al2O3 as the prototype for Strukturbericht $D5_{7}$, but states that the data for $\gamma$–Fe2O3 is more reliable and presents the data for the later compound, which we use as the prototype. This is a rock–salt ($B1$) structure with defects. This structure can also be expressed in the enantiomorphic space group $P4_{1}32$ #213.

Simple Cubic primitive vectors:

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

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(4a\right) & \mbox{O I} \\ \mathbf{B}_{2} & = & \frac{3}{8} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{5}{8} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{7}{8}a \, \mathbf{\hat{y}} + \frac{5}{8}a \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{3} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{5}{8} \, \mathbf{a}_{2} + \frac{3}{8} \, \mathbf{a}_{3} & = & \frac{7}{8}a \, \mathbf{\hat{x}} + \frac{5}{8}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{4} & = & \frac{5}{8} \, \mathbf{a}_{1} + \frac{3}{8} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{5}{8}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{7}{8}a \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{5} & = & \frac{5}{8} \, \mathbf{a}_{1} + \frac{5}{8} \, \mathbf{a}_{2} + \frac{5}{8} \, \mathbf{a}_{3} & = & \frac{5}{8}a \, \mathbf{\hat{x}} + \frac{5}{8}a \, \mathbf{\hat{y}} + \frac{5}{8}a \, \mathbf{\hat{z}} & \left(4b\right) & \mbox{Fe I} \\ \mathbf{B}_{6} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{3}{8} \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & \frac{7}{8}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(4b\right) & \mbox{Fe I} \\ \mathbf{B}_{7} & = & \frac{3}{8} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{7}{8}a \, \mathbf{\hat{z}} & \left(4b\right) & \mbox{Fe I} \\ \mathbf{B}_{8} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{3}{8} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{7}{8}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(4b\right) & \mbox{Fe I} \\ \mathbf{B}_{9} & = & x_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Fe II} \\ \mathbf{B}_{10} & = & \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Fe II} \\ \mathbf{B}_{11} & = & -x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Fe II} \\ \mathbf{B}_{12} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Fe II} \\ \mathbf{B}_{13} & = & \left(\frac{1}{4} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} - x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Fe II} \\ \mathbf{B}_{14} & = & \left(\frac{1}{4} - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Fe II} \\ \mathbf{B}_{15} & = & \left(\frac{3}{4} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} - x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Fe II} \\ \mathbf{B}_{16} & = & \left(\frac{3}{4} - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{Fe II} \\ \mathbf{B}_{17} & = & x_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{18} & = & \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{19} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{20} & = & \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{21} & = & \left(\frac{1}{4} +x_{4}\right) \, \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{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{22} & = & \left(\frac{1}{4} - x_{4}\right) \, \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}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{23} & = & \left(\frac{3}{4} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} - x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{24} & = & \left(\frac{3}{4} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{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}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(8c\right) & \mbox{O II} \\ \mathbf{B}_{25} & = & \frac{1}{8} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{4} - y_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + y_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{5}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{26} & = & \frac{3}{8} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + \left(\frac{3}{4} - y_{5}\right) \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}}-y_{5}a \, \mathbf{\hat{y}} + \left(\frac{3}{4}-y_{5}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{27} & = & \frac{7}{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +y_{5}\right) \, \mathbf{a}_{3} & = & \frac{7}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{28} & = & \frac{5}{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +y_{5}\right) \, \mathbf{a}_{3} & = & \frac{5}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +y_{5}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{29} & = & \left(\frac{1}{4} - y_{5}\right) \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + y_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{5}\right)a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + y_{5}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{30} & = & \left(\frac{3}{4} - y_{5}\right) \, \mathbf{a}_{1} + \frac{3}{8} \, \mathbf{a}_{2}-y_{5} \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{5}\right)a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}}-y_{5}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{31} & = & \left(\frac{1}{4} +y_{5}\right) \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{x}} + \frac{7}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{32} & = & \left(\frac{3}{4} +y_{5}\right) \, \mathbf{a}_{1} + \frac{5}{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +y_{5}\right)a \, \mathbf{\hat{x}} + \frac{5}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{33} & = & y_{5} \, \mathbf{a}_{1} + \left(\frac{1}{4} - y_{5}\right) \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & y_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{34} & = & -y_{5} \, \mathbf{a}_{1} + \left(\frac{3}{4} - y_{5}\right) \, \mathbf{a}_{2} + \frac{3}{8} \, \mathbf{a}_{3} & = & -y_{5}a \, \mathbf{\hat{x}} + \left(\frac{3}{4}-y_{5}\right)a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{35} & = & \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +y_{5}\right) \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{y}} + \frac{7}{8}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{36} & = & \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +y_{5}\right) \, \mathbf{a}_{2} + \frac{5}{8} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +y_{5}\right)a \, \mathbf{\hat{y}} + \frac{5}{8}a \, \mathbf{\hat{z}} & \left(12d\right) & \mbox{Fe III} \\ \mathbf{B}_{37} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}} + z_{6}a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{6}\right)a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{39} & = & -x_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{40} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{6}\right)a \, \mathbf{\hat{y}}-z_{6}a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{41} & = & z_{6} \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + y_{6} \, \mathbf{a}_{3} & = & z_{6}a \, \mathbf{\hat{x}} + x_{6}a \, \mathbf{\hat{y}} + y_{6}a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{42} & = & \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{2}-y_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{6}\right)a \, \mathbf{\hat{y}}-y_{6}a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{43} & = & \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{1}-x_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-z_{6}\right)a \, \mathbf{\hat{x}}-x_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{44} & = & -z_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{3} & = & -z_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-y_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{45} & = & y_{6} \, \mathbf{a}_{1} + z_{6} \, \mathbf{a}_{2} + x_{6} \, \mathbf{a}_{3} & = & y_{6}a \, \mathbf{\hat{x}} + z_{6}a \, \mathbf{\hat{y}} + x_{6}a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{46} & = & -y_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{3} & = & -y_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{47} & = & \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{2}-x_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-z_{6}\right)a \, \mathbf{\hat{y}}-x_{6}a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{48} & = & \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{1}-z_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{6}\right)a \, \mathbf{\hat{x}}-z_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{49} & = & \left(\frac{1}{4} +y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4}-z_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{50} & = & \left(\frac{1}{4} - y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{51} & = & \left(\frac{3}{4} +y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} - x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4}-x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{52} & = & \left(\frac{3}{4} - y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{53} & = & \left(\frac{1}{4} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} - y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4}-y_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{54} & = & \left(\frac{3}{4} - x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +y_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{55} & = & \left(\frac{1}{4} - x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{56} & = & \left(\frac{3}{4} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4}-z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{57} & = & \left(\frac{1}{4} +z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} - x_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4}-x_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{58} & = & \left(\frac{3}{4} +z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} - y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4}-y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{59} & = & \left(\frac{3}{4} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +x_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \mathbf{B}_{60} & = & \left(\frac{1}{4} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{6}\right)a \, \mathbf{\hat{z}} & \left(24e\right) & \mbox{O III} \\ \end{array} \]

References

Found in

  • C. Hermann, O. Lohrmann, and H. Philipp, eds., Strukturbericht Band II 1928–1932 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).

Geometry files


Prototype Generator

aflow --proto=A2B3_cP60_212_bcd_ace --params=

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