Na4Ge9O20 Structure : A9B4C20_tI132_88_a2f_f_5f

Picture of Structure; Click for Big Picture
Prototype : Ge9Na4O20
AFLOW prototype label : A9B4C20_tI132_88_a2f_f_5f
Strukturbericht designation : None
Pearson symbol : tI132
Space group number : 88
Space group symbol : $I4_{1}/a$
AFLOW prototype command : aflow --proto=A9B4C20_tI132_88_a2f_f_5f
--params=
$a$,$c/a$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$


Body-centered Tetragonal primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & \frac{3}{8} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{8}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Ge I} \\ \mathbf{B}_{2} & = & \frac{5}{8} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{3}{8}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Ge I} \\ \mathbf{B}_{3} & = & \left(y_{2}+z_{2}\right) \, \mathbf{a}_{1} + \left(x_{2}+z_{2}\right) \, \mathbf{a}_{2} + \left(x_{2}+y_{2}\right) \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + y_{2}a \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge II} \\ \mathbf{B}_{4} & = & \left(\frac{1}{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{1} + \left(-x_{2}+z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{2} - y_{2}\right) \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{2}\right)a \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge II} \\ \mathbf{B}_{5} & = & \left(\frac{1}{2} +x_{2} + z_{2}\right) \, \mathbf{a}_{1} + \left(-y_{2}+z_{2}\right) \, \mathbf{a}_{2} + \left(x_{2}-y_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge II} \\ \mathbf{B}_{6} & = & \left(\frac{1}{2} - x_{2} + z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{2} + z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{2} + y_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge II} \\ \mathbf{B}_{7} & = & \left(-y_{2}-z_{2}\right) \, \mathbf{a}_{1} + \left(-x_{2}-z_{2}\right) \, \mathbf{a}_{2} + \left(-x_{2}-y_{2}\right) \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}}-y_{2}a \, \mathbf{\hat{y}}-z_{2}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge II} \\ \mathbf{B}_{8} & = & \left(\frac{1}{2} +y_{2} - z_{2}\right) \, \mathbf{a}_{1} + \left(x_{2}-z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{2} + y_{2}\right) \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{2}\right)a \, \mathbf{\hat{y}}-z_{2}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge II} \\ \mathbf{B}_{9} & = & \left(\frac{1}{2} - x_{2} - z_{2}\right) \, \mathbf{a}_{1} + \left(y_{2}-z_{2}\right) \, \mathbf{a}_{2} + \left(-x_{2}+y_{2}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{2}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge II} \\ \mathbf{B}_{10} & = & \left(\frac{1}{2} +x_{2} - z_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{2} - y_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{2}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge II} \\ \mathbf{B}_{11} & = & \left(y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+y_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge III} \\ \mathbf{B}_{12} & = & \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{3}\right)a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge III} \\ \mathbf{B}_{13} & = & \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge III} \\ \mathbf{B}_{14} & = & \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge III} \\ \mathbf{B}_{15} & = & \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}}-y_{3}a \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge III} \\ \mathbf{B}_{16} & = & \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{3}\right)a \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge III} \\ \mathbf{B}_{17} & = & \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{3}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge III} \\ \mathbf{B}_{18} & = & \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{3}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Ge III} \\ \mathbf{B}_{19} & = & \left(y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}+y_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + y_{4}a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Na} \\ \mathbf{B}_{20} & = & \left(\frac{1}{2} - y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} - y_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{4}\right)a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Na} \\ \mathbf{B}_{21} & = & \left(\frac{1}{2} +x_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(-y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Na} \\ \mathbf{B}_{22} & = & \left(\frac{1}{2} - x_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} + y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Na} \\ \mathbf{B}_{23} & = & \left(-y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}-y_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}}-y_{4}a \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Na} \\ \mathbf{B}_{24} & = & \left(\frac{1}{2} +y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} + y_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{4}\right)a \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Na} \\ \mathbf{B}_{25} & = & \left(\frac{1}{2} - x_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{4}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Na} \\ \mathbf{B}_{26} & = & \left(\frac{1}{2} +x_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} - y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{4}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Na} \\ \mathbf{B}_{27} & = & \left(y_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(x_{5}+y_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + y_{5}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O I} \\ \mathbf{B}_{28} & = & \left(\frac{1}{2} - y_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5} - y_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O I} \\ \mathbf{B}_{29} & = & \left(\frac{1}{2} +x_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(-y_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(x_{5}-y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O I} \\ \mathbf{B}_{30} & = & \left(\frac{1}{2} - x_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5} + y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O I} \\ \mathbf{B}_{31} & = & \left(-y_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(-x_{5}-y_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-y_{5}a \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O I} \\ \mathbf{B}_{32} & = & \left(\frac{1}{2} +y_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5} + y_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O I} \\ \mathbf{B}_{33} & = & \left(\frac{1}{2} - x_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(y_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{5}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O I} \\ \mathbf{B}_{34} & = & \left(\frac{1}{2} +x_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5} - y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{5}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O I} \\ \mathbf{B}_{35} & = & \left(y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}+y_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O II} \\ \mathbf{B}_{36} & = & \left(\frac{1}{2} - y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6} - y_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{6}\right)a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O II} \\ \mathbf{B}_{37} & = & \left(\frac{1}{2} +x_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(-y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}-y_{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{1}{4} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O II} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} - x_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6} + y_{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)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O II} \\ \mathbf{B}_{39} & = & \left(-y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}-y_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O II} \\ \mathbf{B}_{40} & = & \left(\frac{1}{2} +y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6} + y_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O II} \\ \mathbf{B}_{41} & = & \left(\frac{1}{2} - x_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}+y_{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)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O II} \\ \mathbf{B}_{42} & = & \left(\frac{1}{2} +x_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6} - y_{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)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O II} \\ \mathbf{B}_{43} & = & \left(y_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}+y_{7}\right) \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + y_{7}a \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O III} \\ \mathbf{B}_{44} & = & \left(\frac{1}{2} - y_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{7} - y_{7}\right) \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{7}\right)a \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O III} \\ \mathbf{B}_{45} & = & \left(\frac{1}{2} +x_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(-y_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}-y_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O III} \\ \mathbf{B}_{46} & = & \left(\frac{1}{2} - x_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{7} + y_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O III} \\ \mathbf{B}_{47} & = & \left(-y_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}-y_{7}\right) \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}}-y_{7}a \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O III} \\ \mathbf{B}_{48} & = & \left(\frac{1}{2} +y_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{7} + y_{7}\right) \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{7}\right)a \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O III} \\ \mathbf{B}_{49} & = & \left(\frac{1}{2} - x_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(y_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}+y_{7}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O III} \\ \mathbf{B}_{50} & = & \left(\frac{1}{2} +x_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{7} - y_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O III} \\ \mathbf{B}_{51} & = & \left(y_{8}+z_{8}\right) \, \mathbf{a}_{1} + \left(x_{8}+z_{8}\right) \, \mathbf{a}_{2} + \left(x_{8}+y_{8}\right) \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + y_{8}a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O IV} \\ \mathbf{B}_{52} & = & \left(\frac{1}{2} - y_{8} + z_{8}\right) \, \mathbf{a}_{1} + \left(-x_{8}+z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{8} - y_{8}\right) \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{8}\right)a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O IV} \\ \mathbf{B}_{53} & = & \left(\frac{1}{2} +x_{8} + z_{8}\right) \, \mathbf{a}_{1} + \left(-y_{8}+z_{8}\right) \, \mathbf{a}_{2} + \left(x_{8}-y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O IV} \\ \mathbf{B}_{54} & = & \left(\frac{1}{2} - x_{8} + z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{8} + z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{8} + y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O IV} \\ \mathbf{B}_{55} & = & \left(-y_{8}-z_{8}\right) \, \mathbf{a}_{1} + \left(-x_{8}-z_{8}\right) \, \mathbf{a}_{2} + \left(-x_{8}-y_{8}\right) \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}}-y_{8}a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O IV} \\ \mathbf{B}_{56} & = & \left(\frac{1}{2} +y_{8} - z_{8}\right) \, \mathbf{a}_{1} + \left(x_{8}-z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{8} + y_{8}\right) \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{8}\right)a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O IV} \\ \mathbf{B}_{57} & = & \left(\frac{1}{2} - x_{8} - z_{8}\right) \, \mathbf{a}_{1} + \left(y_{8}-z_{8}\right) \, \mathbf{a}_{2} + \left(-x_{8}+y_{8}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{4} +y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O IV} \\ \mathbf{B}_{58} & = & \left(\frac{1}{2} +x_{8} - z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{8} - z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{8} - y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O IV} \\ \mathbf{B}_{59} & = & \left(y_{9}+z_{9}\right) \, \mathbf{a}_{1} + \left(x_{9}+z_{9}\right) \, \mathbf{a}_{2} + \left(x_{9}+y_{9}\right) \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}} + y_{9}a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O V} \\ \mathbf{B}_{60} & = & \left(\frac{1}{2} - y_{9} + z_{9}\right) \, \mathbf{a}_{1} + \left(-x_{9}+z_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{9} - y_{9}\right) \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{9}\right)a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O V} \\ \mathbf{B}_{61} & = & \left(\frac{1}{2} +x_{9} + z_{9}\right) \, \mathbf{a}_{1} + \left(-y_{9}+z_{9}\right) \, \mathbf{a}_{2} + \left(x_{9}-y_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O V} \\ \mathbf{B}_{62} & = & \left(\frac{1}{2} - x_{9} + z_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{9} + z_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{9} + y_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O V} \\ \mathbf{B}_{63} & = & \left(-y_{9}-z_{9}\right) \, \mathbf{a}_{1} + \left(-x_{9}-z_{9}\right) \, \mathbf{a}_{2} + \left(-x_{9}-y_{9}\right) \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}}-y_{9}a \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O V} \\ \mathbf{B}_{64} & = & \left(\frac{1}{2} +y_{9} - z_{9}\right) \, \mathbf{a}_{1} + \left(x_{9}-z_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{9} + y_{9}\right) \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{9}\right)a \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O V} \\ \mathbf{B}_{65} & = & \left(\frac{1}{2} - x_{9} - z_{9}\right) \, \mathbf{a}_{1} + \left(y_{9}-z_{9}\right) \, \mathbf{a}_{2} + \left(-x_{9}+y_{9}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{4} +y_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O V} \\ \mathbf{B}_{66} & = & \left(\frac{1}{2} +x_{9} - z_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{9} - z_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{9} - y_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{O V} \\ \end{array} \]

References

Geometry files


Prototype Generator

aflow --proto=A9B4C20_tI132_88_a2f_f_5f --params=

Species:

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Output: