Zr21Re25 Structure : A25B21_hR92_167_b2e3f_e3f

Picture of Structure; Click for Big Picture
Prototype : Re25Zr21
AFLOW prototype label : A25B21_hR92_167_b2e3f_e3f
Strukturbericht designation : None
Pearson symbol : hR92
Space group number : 167
Space group symbol : $R\bar{3}c$
AFLOW prototype command : aflow --proto=A25B21_hR92_167_b2e3f_e3f
--params=
$a$,$c/a$,$x_{2}$,$x_{3}$,$x_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$,$x_{10}$,$y_{10}$,$z_{10}$


Other compounds with this structure

  • Mg21Zn25

Rhombohedral primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac{1}{\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & - \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & 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(2b\right) & \mbox{Re I} \\ \mathbf{B}_{2} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}c \, \mathbf{\hat{z}} & \left(2b\right) & \mbox{Re I} \\ \mathbf{B}_{3} & = & x_{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{\sqrt{3}}{8}-\frac{\sqrt{3}}{2}x_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re II} \\ \mathbf{B}_{4} & = & \frac{1}{4} \, \mathbf{a}_{1} + x_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{2}\right)a \, \mathbf{\hat{x}} + \left(- \frac{\sqrt{3}}{8} +\frac{\sqrt{3}}{2}x_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re II} \\ \mathbf{B}_{5} & = & \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + x_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{2}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re II} \\ \mathbf{B}_{6} & = & -x_{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & -a\left(\frac{1}{2}x_{2}+\frac{3}{8}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{8\sqrt{3}} +\frac{\sqrt{3}}{2}x_{2}\right)a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re II} \\ \mathbf{B}_{7} & = & \frac{3}{4} \, \mathbf{a}_{1}-x_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{8}-\frac{1}{2}x_{2}\right)a \, \mathbf{\hat{x}}-a\left(\frac{\sqrt{3}}{2}x_{2}+\frac{5}{8\sqrt{3}}\right) \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re II} \\ \mathbf{B}_{8} & = & \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-x_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re II} \\ \mathbf{B}_{9} & = & x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{\sqrt{3}}{8}-\frac{\sqrt{3}}{2}x_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re III} \\ \mathbf{B}_{10} & = & \frac{1}{4} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{3}\right)a \, \mathbf{\hat{x}} + \left(- \frac{\sqrt{3}}{8} +\frac{\sqrt{3}}{2}x_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re III} \\ \mathbf{B}_{11} & = & \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re III} \\ \mathbf{B}_{12} & = & -x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & -a\left(\frac{1}{2}x_{3}+\frac{3}{8}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{8\sqrt{3}} +\frac{\sqrt{3}}{2}x_{3}\right)a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re III} \\ \mathbf{B}_{13} & = & \frac{3}{4} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{8}-\frac{1}{2}x_{3}\right)a \, \mathbf{\hat{x}}-a\left(\frac{\sqrt{3}}{2}x_{3}+\frac{5}{8\sqrt{3}}\right) \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re III} \\ \mathbf{B}_{14} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Re III} \\ \mathbf{B}_{15} & = & x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{\sqrt{3}}{8}-\frac{\sqrt{3}}{2}x_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Zr I} \\ \mathbf{B}_{16} & = & \frac{1}{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{4}\right)a \, \mathbf{\hat{x}} + \left(- \frac{\sqrt{3}}{8} +\frac{\sqrt{3}}{2}x_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Zr I} \\ \mathbf{B}_{17} & = & \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Zr I} \\ \mathbf{B}_{18} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & -a\left(\frac{1}{2}x_{4}+\frac{3}{8}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{8\sqrt{3}} +\frac{\sqrt{3}}{2}x_{4}\right)a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Zr I} \\ \mathbf{B}_{19} & = & \frac{3}{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{8}-\frac{1}{2}x_{4}\right)a \, \mathbf{\hat{x}}-a\left(\frac{\sqrt{3}}{2}x_{4}+\frac{5}{8\sqrt{3}}\right) \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Zr I} \\ \mathbf{B}_{20} & = & \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \mbox{Zr I} \\ \mathbf{B}_{21} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}-z_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{5}+\frac{1}{\sqrt{3}}y_{5}-\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{5}+y_{5}+z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{22} & = & z_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + y_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{5}+z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{5}-\frac{1}{2\sqrt{3}}y_{5}-\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{5}+y_{5}+z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{23} & = & y_{5} \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{5}-\frac{1}{2\sqrt{3}}y_{5}+\frac{1}{\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{5}+y_{5}+z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{24} & = & \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}-z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{5}-\frac{1}{\sqrt{3}}y_{5}+\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{5} - \frac{1}{3}y_{5} - \frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{25} & = & \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{5}+z_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{5}+\frac{1}{2\sqrt{3}}y_{5}+\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{5} - \frac{1}{3}y_{5} - \frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{26} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{5}+\frac{1}{2\sqrt{3}}y_{5}-\frac{1}{\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{5} - \frac{1}{3}y_{5} - \frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{27} & = & -x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{5}+z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{5}-\frac{1}{\sqrt{3}}y_{5}+\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{5}+y_{5}+z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{28} & = & -z_{5} \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2}-y_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{5}-z_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{5}+\frac{1}{2\sqrt{3}}y_{5}+\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{5}+y_{5}+z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{29} & = & -y_{5} \, \mathbf{a}_{1}-z_{5} \, \mathbf{a}_{2}-x_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{5}+\frac{1}{2\sqrt{3}}y_{5}-\frac{1}{\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{5}+y_{5}+z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{30} & = & \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{5}+z_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{5}+\frac{1}{\sqrt{3}}y_{5}-\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{5} + \frac{1}{3}y_{5} + \frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{31} & = & \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{5}-z_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{5}-\frac{1}{2\sqrt{3}}y_{5}-\frac{1}{2\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{5} + \frac{1}{3}y_{5} + \frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{32} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{5}-\frac{1}{2\sqrt{3}}y_{5}+\frac{1}{\sqrt{3}}z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{5} + \frac{1}{3}y_{5} + \frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re IV} \\ \mathbf{B}_{33} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{6}-z_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{6}+\frac{1}{\sqrt{3}}y_{6}-\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{6}+y_{6}+z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{34} & = & z_{6} \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + y_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{6}+z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{6}-\frac{1}{2\sqrt{3}}y_{6}-\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{6}+y_{6}+z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{35} & = & y_{6} \, \mathbf{a}_{1} + z_{6} \, \mathbf{a}_{2} + x_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{6}+y_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{6}-\frac{1}{2\sqrt{3}}y_{6}+\frac{1}{\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{6}+y_{6}+z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{36} & = & \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{6}-z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{6}-\frac{1}{\sqrt{3}}y_{6}+\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{6} - \frac{1}{3}y_{6} - \frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{37} & = & \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{6}+z_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{6}+\frac{1}{2\sqrt{3}}y_{6}+\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{6} - \frac{1}{3}y_{6} - \frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{6}+y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{6}+\frac{1}{2\sqrt{3}}y_{6}-\frac{1}{\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{6} - \frac{1}{3}y_{6} - \frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{39} & = & -x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{6}+z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{6}-\frac{1}{\sqrt{3}}y_{6}+\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{6}+y_{6}+z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{40} & = & -z_{6} \, \mathbf{a}_{1}-x_{6} \, \mathbf{a}_{2}-y_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{6}-z_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{6}+\frac{1}{2\sqrt{3}}y_{6}+\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{6}+y_{6}+z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{41} & = & -y_{6} \, \mathbf{a}_{1}-z_{6} \, \mathbf{a}_{2}-x_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{6}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{6}+\frac{1}{2\sqrt{3}}y_{6}-\frac{1}{\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{6}+y_{6}+z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{42} & = & \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{6}+z_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{6}+\frac{1}{\sqrt{3}}y_{6}-\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{6} + \frac{1}{3}y_{6} + \frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{43} & = & \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{6}-z_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{6}-\frac{1}{2\sqrt{3}}y_{6}-\frac{1}{2\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{6} + \frac{1}{3}y_{6} + \frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{44} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{6}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{6}-\frac{1}{2\sqrt{3}}y_{6}+\frac{1}{\sqrt{3}}z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{6} + \frac{1}{3}y_{6} + \frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re V} \\ \mathbf{B}_{45} & = & x_{7} \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{7}+\frac{1}{\sqrt{3}}y_{7}-\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{7}+y_{7}+z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{46} & = & z_{7} \, \mathbf{a}_{1} + x_{7} \, \mathbf{a}_{2} + y_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{7}+z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{7}-\frac{1}{2\sqrt{3}}y_{7}-\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{7}+y_{7}+z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{47} & = & y_{7} \, \mathbf{a}_{1} + z_{7} \, \mathbf{a}_{2} + x_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{7}+y_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{7}-\frac{1}{2\sqrt{3}}y_{7}+\frac{1}{\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{7}+y_{7}+z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{48} & = & \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{7}-\frac{1}{\sqrt{3}}y_{7}+\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{7} - \frac{1}{3}y_{7} - \frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{49} & = & \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{7}+z_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{7}+\frac{1}{2\sqrt{3}}y_{7}+\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{7} - \frac{1}{3}y_{7} - \frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{50} & = & \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{7}+y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{7}+\frac{1}{2\sqrt{3}}y_{7}-\frac{1}{\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{7} - \frac{1}{3}y_{7} - \frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{51} & = & -x_{7} \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{7}-\frac{1}{\sqrt{3}}y_{7}+\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{7}+y_{7}+z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{52} & = & -z_{7} \, \mathbf{a}_{1}-x_{7} \, \mathbf{a}_{2}-y_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{7}-z_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{7}+\frac{1}{2\sqrt{3}}y_{7}+\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{7}+y_{7}+z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{53} & = & -y_{7} \, \mathbf{a}_{1}-z_{7} \, \mathbf{a}_{2}-x_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{7}-y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{7}+\frac{1}{2\sqrt{3}}y_{7}-\frac{1}{\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{7}+y_{7}+z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{54} & = & \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{7}+\frac{1}{\sqrt{3}}y_{7}-\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{7} + \frac{1}{3}y_{7} + \frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{55} & = & \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{7}-z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{7}-\frac{1}{2\sqrt{3}}y_{7}-\frac{1}{2\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{7} + \frac{1}{3}y_{7} + \frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{56} & = & \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{7}-y_{7}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{7}-\frac{1}{2\sqrt{3}}y_{7}+\frac{1}{\sqrt{3}}z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{7} + \frac{1}{3}y_{7} + \frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Re VI} \\ \mathbf{B}_{57} & = & x_{8} \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{8}+\frac{1}{\sqrt{3}}y_{8}-\frac{1}{2\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{8}+y_{8}+z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{58} & = & z_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + y_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{8}+z_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{8}-\frac{1}{2\sqrt{3}}y_{8}-\frac{1}{2\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{8}+y_{8}+z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{59} & = & y_{8} \, \mathbf{a}_{1} + z_{8} \, \mathbf{a}_{2} + x_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{8}+y_{8}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{8}-\frac{1}{2\sqrt{3}}y_{8}+\frac{1}{\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{8}+y_{8}+z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{60} & = & \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{8}-\frac{1}{\sqrt{3}}y_{8}+\frac{1}{2\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{8} - \frac{1}{3}y_{8} - \frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{61} & = & \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{8}+z_{8}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{8}+\frac{1}{2\sqrt{3}}y_{8}+\frac{1}{2\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{8} - \frac{1}{3}y_{8} - \frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{62} & = & \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{8}+y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{8}+\frac{1}{2\sqrt{3}}y_{8}-\frac{1}{\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{8} - \frac{1}{3}y_{8} - \frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{63} & = & -x_{8} \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{8}+z_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{8}-\frac{1}{\sqrt{3}}y_{8}+\frac{1}{2\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{8}+y_{8}+z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{64} & = & -z_{8} \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2}-y_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{8}-z_{8}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{8}+\frac{1}{2\sqrt{3}}y_{8}+\frac{1}{2\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{8}+y_{8}+z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{65} & = & -y_{8} \, \mathbf{a}_{1}-z_{8} \, \mathbf{a}_{2}-x_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{8}-y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{8}+\frac{1}{2\sqrt{3}}y_{8}-\frac{1}{\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{8}+y_{8}+z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{66} & = & \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{8}+z_{8}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{8}+\frac{1}{\sqrt{3}}y_{8}-\frac{1}{2\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{8} + \frac{1}{3}y_{8} + \frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{67} & = & \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{8}-z_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{8}-\frac{1}{2\sqrt{3}}y_{8}-\frac{1}{2\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{8} + \frac{1}{3}y_{8} + \frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{68} & = & \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{8}-y_{8}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{8}-\frac{1}{2\sqrt{3}}y_{8}+\frac{1}{\sqrt{3}}z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{8} + \frac{1}{3}y_{8} + \frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr II} \\ \mathbf{B}_{69} & = & x_{9} \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{9}-z_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{9}+\frac{1}{\sqrt{3}}y_{9}-\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{70} & = & z_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + y_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{9}+z_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{9}-\frac{1}{2\sqrt{3}}y_{9}-\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{71} & = & y_{9} \, \mathbf{a}_{1} + z_{9} \, \mathbf{a}_{2} + x_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{9}+y_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{9}-\frac{1}{2\sqrt{3}}y_{9}+\frac{1}{\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{72} & = & \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{9}-z_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{9}-\frac{1}{\sqrt{3}}y_{9}+\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{9} - \frac{1}{3}y_{9} - \frac{1}{3}z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{73} & = & \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{9}+z_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{9}+\frac{1}{2\sqrt{3}}y_{9}+\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{9} - \frac{1}{3}y_{9} - \frac{1}{3}z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{74} & = & \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{9}+y_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{9}+\frac{1}{2\sqrt{3}}y_{9}-\frac{1}{\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{9} - \frac{1}{3}y_{9} - \frac{1}{3}z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{75} & = & -x_{9} \, \mathbf{a}_{1}-y_{9} \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{9}+z_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{9}-\frac{1}{\sqrt{3}}y_{9}+\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{76} & = & -z_{9} \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2}-y_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{9}-z_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{9}+\frac{1}{2\sqrt{3}}y_{9}+\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{77} & = & -y_{9} \, \mathbf{a}_{1}-z_{9} \, \mathbf{a}_{2}-x_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{9}-y_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{9}+\frac{1}{2\sqrt{3}}y_{9}-\frac{1}{\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{78} & = & \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{9}+z_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{9}+\frac{1}{\sqrt{3}}y_{9}-\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{9} + \frac{1}{3}y_{9} + \frac{1}{3}z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{79} & = & \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{9}-z_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{9}-\frac{1}{2\sqrt{3}}y_{9}-\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{9} + \frac{1}{3}y_{9} + \frac{1}{3}z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{80} & = & \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{9}-y_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{9}-\frac{1}{2\sqrt{3}}y_{9}+\frac{1}{\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{9} + \frac{1}{3}y_{9} + \frac{1}{3}z_{9}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr III} \\ \mathbf{B}_{81} & = & x_{10} \, \mathbf{a}_{1} + y_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{10}-z_{10}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{10}+\frac{1}{\sqrt{3}}y_{10}-\frac{1}{2\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{10}+y_{10}+z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{82} & = & z_{10} \, \mathbf{a}_{1} + x_{10} \, \mathbf{a}_{2} + y_{10} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{10}+z_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{10}-\frac{1}{2\sqrt{3}}y_{10}-\frac{1}{2\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{10}+y_{10}+z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{83} & = & y_{10} \, \mathbf{a}_{1} + z_{10} \, \mathbf{a}_{2} + x_{10} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{10}+y_{10}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{10}-\frac{1}{2\sqrt{3}}y_{10}+\frac{1}{\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{10}+y_{10}+z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{84} & = & \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{10}-z_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{10}-\frac{1}{\sqrt{3}}y_{10}+\frac{1}{2\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{10} - \frac{1}{3}y_{10} - \frac{1}{3}z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{85} & = & \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{10}+z_{10}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{10}+\frac{1}{2\sqrt{3}}y_{10}+\frac{1}{2\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{10} - \frac{1}{3}y_{10} - \frac{1}{3}z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{86} & = & \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{10}+y_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{10}+\frac{1}{2\sqrt{3}}y_{10}-\frac{1}{\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - \frac{1}{3}x_{10} - \frac{1}{3}y_{10} - \frac{1}{3}z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{87} & = & -x_{10} \, \mathbf{a}_{1}-y_{10} \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{10}+z_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{10}-\frac{1}{\sqrt{3}}y_{10}+\frac{1}{2\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{10}+y_{10}+z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{88} & = & -z_{10} \, \mathbf{a}_{1}-x_{10} \, \mathbf{a}_{2}-y_{10} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{10}-z_{10}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{\sqrt{3}}x_{10}+\frac{1}{2\sqrt{3}}y_{10}+\frac{1}{2\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{10}+y_{10}+z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{89} & = & -y_{10} \, \mathbf{a}_{1}-z_{10} \, \mathbf{a}_{2}-x_{10} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{10}-y_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2\sqrt{3}}x_{10}+\frac{1}{2\sqrt{3}}y_{10}-\frac{1}{\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}}-\frac{1}{3}\left(x_{10}+y_{10}+z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{90} & = & \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{10}+z_{10}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{10}+\frac{1}{\sqrt{3}}y_{10}-\frac{1}{2\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{10} + \frac{1}{3}y_{10} + \frac{1}{3}z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{91} & = & \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{10}-z_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{10}-\frac{1}{2\sqrt{3}}y_{10}-\frac{1}{2\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{10} + \frac{1}{3}y_{10} + \frac{1}{3}z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \mathbf{B}_{92} & = & \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{10}-y_{10}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{10}-\frac{1}{2\sqrt{3}}y_{10}+\frac{1}{\sqrt{3}}z_{10}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +\frac{1}{3}x_{10} + \frac{1}{3}y_{10} + \frac{1}{3}z_{10}\right)c \, \mathbf{\hat{z}} & \left(12f\right) & \mbox{Zr IV} \\ \end{array} \]

References

  • K. Cenzual, E. Parthé, and R. M. Waterstrat, Zr21Re25, a new rhombohedral structure type containing 12 Å–thick infinite MgZn2(Laves)–type columns, Acta Crystallogr. C 42, 261–266 (1986), doi:10.1107/S0108270186096555.

Found in

Geometry files


Prototype Generator

aflow --proto=A25B21_hR92_167_b2e3f_e3f --params=

Species:

Running:

Output: