Autunite {Ca[(UO2)(PO4)]2(H2O)11} Structure : AB22C23D2E2_oP200_62_c_11d_3c10d_d_d

Picture of Structure; Click for Big Picture
Prototype : CaH22O23P2U2
AFLOW prototype label : AB22C23D2E2_oP200_62_c_11d_3c10d_d_d
Strukturbericht designation : None
Pearson symbol : oP200
Space group number : 62
Space group symbol : $Pnma$
AFLOW prototype command : aflow --proto=AB22C23D2E2_oP200_62_c_11d_3c10d_d_d
--params=
$a$,$b/a$,$c/a$,$x_{1}$,$z_{1}$,$x_{2}$,$z_{2}$,$x_{3}$,$z_{3}$,$x_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$,$x_{10}$,$y_{10}$,$z_{10}$,$x_{11}$,$y_{11}$,$z_{11}$,$x_{12}$,$y_{12}$,$z_{12}$,$x_{13}$,$y_{13}$,$z_{13}$,$x_{14}$,$y_{14}$,$z_{14}$,$x_{15}$,$y_{15}$,$z_{15}$,$x_{16}$,$y_{16}$,$z_{16}$,$x_{17}$,$y_{17}$,$z_{17}$,$x_{18}$,$y_{18}$,$z_{18}$,$x_{19}$,$y_{19}$,$z_{19}$,$x_{20}$,$y_{20}$,$z_{20}$,$x_{21}$,$y_{21}$,$z_{21}$,$x_{22}$,$y_{22}$,$z_{22}$,$x_{23}$,$y_{23}$,$z_{23}$,$x_{24}$,$y_{24}$,$z_{24}$,$x_{25}$,$y_{25}$,$z_{25}$,$x_{26}$,$y_{26}$,$z_{26}$,$x_{27}$,$y_{27}$,$z_{27}$


  • Autunite, Ca(UO2)2(PO4)2·$n$H2O, is found in three varieties: naturally occurring Autunite, with $n \gtrsim 10$, and meta–autunite (I), which is partially dehydrated, $6 \gtrsim n \gtrsim 10$. Further dehydration in the laboratory produces meta–autunite (II).
  • The original determination of the autunite structure designated $H5_{9}$ by (Herrmann, 1941) was a tetragonal structure, and none of the positions of the oxygen atom or water molecules were determined. (Locock, 2003) find a pseudo–tetragonal ($a ≈ 2c$) unit cell which doubles the size of the original cell. They were able to locate all of the atoms in the structure.
  • (Locock, 2003) found the ($4c$) calcium site to be occupied only 86% of the time.

Simple Orthorhombic primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & x_{1} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & x_{1}a \, \mathbf{\hat{x}} + \frac{1}{4}b \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{Ca} \\ \mathbf{B}_{2} & = & \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{1}\right)a \, \mathbf{\hat{x}} + \frac{3}{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{Ca} \\ \mathbf{B}_{3} & = & -x_{1} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-z_{1} \, \mathbf{a}_{3} & = & -x_{1}a \, \mathbf{\hat{x}} + \frac{3}{4}b \, \mathbf{\hat{y}}-z_{1}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{Ca} \\ \mathbf{B}_{4} & = & \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{1}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{1}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{Ca} \\ \mathbf{B}_{5} & = & x_{2} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + \frac{1}{4}b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O I} \\ \mathbf{B}_{6} & = & \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{2}\right)a \, \mathbf{\hat{x}} + \frac{3}{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O I} \\ \mathbf{B}_{7} & = & -x_{2} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-z_{2} \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}} + \frac{3}{4}b \, \mathbf{\hat{y}}-z_{2}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O I} \\ \mathbf{B}_{8} & = & \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{2}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{2}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O I} \\ \mathbf{B}_{9} & = & x_{3} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + \frac{1}{4}b \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O II} \\ \mathbf{B}_{10} & = & \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{x}} + \frac{3}{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O II} \\ \mathbf{B}_{11} & = & -x_{3} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \frac{3}{4}b \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O II} \\ \mathbf{B}_{12} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{3}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O II} \\ \mathbf{B}_{13} & = & x_{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \frac{1}{4}b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O III} \\ \mathbf{B}_{14} & = & \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{x}} + \frac{3}{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O III} \\ \mathbf{B}_{15} & = & -x_{4} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \frac{3}{4}b \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O III} \\ \mathbf{B}_{16} & = & \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{4}\right)c \, \mathbf{\hat{z}} & \left(4c\right) & \mbox{O III} \\ \mathbf{B}_{17} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{18} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{19} & = & -x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)b \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{20} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{5}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{5}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{21} & = & -x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{22} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{5}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{23} & = & x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{5}\right)b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{24} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H I} \\ \mathbf{B}_{25} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{26} & = & \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}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{27} & = & -x_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)b \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{28} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{6}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{29} & = & -x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{30} & = & \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}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{31} & = & x_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{6}\right)b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{32} & = & \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H II} \\ \mathbf{B}_{33} & = & x_{7} \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + y_{7}b \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{34} & = & \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{7}\right)a \, \mathbf{\hat{x}}-y_{7}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{35} & = & -x_{7} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{7}\right)b \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{36} & = & \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{7}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{37} & = & -x_{7} \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2}-z_{7} \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}}-y_{7}b \, \mathbf{\hat{y}}-z_{7}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} +x_{7}\right) \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{7}\right)a \, \mathbf{\hat{x}} + y_{7}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{39} & = & x_{7} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{7}\right) \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{7}\right)b \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{40} & = & \left(\frac{1}{2} - x_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{7}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H III} \\ \mathbf{B}_{41} & = & x_{8} \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + y_{8}b \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{42} & = & \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{8}\right)a \, \mathbf{\hat{x}}-y_{8}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{43} & = & -x_{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{8}\right)b \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{44} & = & \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{8}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{45} & = & -x_{8} \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2}-z_{8} \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}}-y_{8}b \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{46} & = & \left(\frac{1}{2} +x_{8}\right) \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{8}\right)a \, \mathbf{\hat{x}} + y_{8}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{47} & = & x_{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{8}\right) \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{8}\right)b \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{48} & = & \left(\frac{1}{2} - x_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{8}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IV} \\ \mathbf{B}_{49} & = & x_{9} \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}} + y_{9}b \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{50} & = & \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{1}-y_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{9}\right)a \, \mathbf{\hat{x}}-y_{9}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{51} & = & -x_{9} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{9}\right)b \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{52} & = & \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{9}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{53} & = & -x_{9} \, \mathbf{a}_{1}-y_{9} \, \mathbf{a}_{2}-z_{9} \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}}-y_{9}b \, \mathbf{\hat{y}}-z_{9}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{54} & = & \left(\frac{1}{2} +x_{9}\right) \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{9}\right)a \, \mathbf{\hat{x}} + y_{9}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{55} & = & x_{9} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{9}\right) \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{9}\right)b \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{56} & = & \left(\frac{1}{2} - x_{9}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{9}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H V} \\ \mathbf{B}_{57} & = & x_{10} \, \mathbf{a}_{1} + y_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & x_{10}a \, \mathbf{\hat{x}} + y_{10}b \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{58} & = & \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{1}-y_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{10}\right)a \, \mathbf{\hat{x}}-y_{10}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{59} & = & -x_{10} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & -x_{10}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{10}\right)b \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{60} & = & \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{10}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{61} & = & -x_{10} \, \mathbf{a}_{1}-y_{10} \, \mathbf{a}_{2}-z_{10} \, \mathbf{a}_{3} & = & -x_{10}a \, \mathbf{\hat{x}}-y_{10}b \, \mathbf{\hat{y}}-z_{10}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{62} & = & \left(\frac{1}{2} +x_{10}\right) \, \mathbf{a}_{1} + y_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{10}\right)a \, \mathbf{\hat{x}} + y_{10}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{63} & = & x_{10} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{10}\right) \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & x_{10}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{10}\right)b \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{64} & = & \left(\frac{1}{2} - x_{10}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{10}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{10}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VI} \\ \mathbf{B}_{65} & = & x_{11} \, \mathbf{a}_{1} + y_{11} \, \mathbf{a}_{2} + z_{11} \, \mathbf{a}_{3} & = & x_{11}a \, \mathbf{\hat{x}} + y_{11}b \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{66} & = & \left(\frac{1}{2} - x_{11}\right) \, \mathbf{a}_{1}-y_{11} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{11}\right)a \, \mathbf{\hat{x}}-y_{11}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{67} & = & -x_{11} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{11}\right) \, \mathbf{a}_{2}-z_{11} \, \mathbf{a}_{3} & = & -x_{11}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{11}\right)b \, \mathbf{\hat{y}}-z_{11}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{68} & = & \left(\frac{1}{2} +x_{11}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{11}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{11}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{11}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{69} & = & -x_{11} \, \mathbf{a}_{1}-y_{11} \, \mathbf{a}_{2}-z_{11} \, \mathbf{a}_{3} & = & -x_{11}a \, \mathbf{\hat{x}}-y_{11}b \, \mathbf{\hat{y}}-z_{11}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{70} & = & \left(\frac{1}{2} +x_{11}\right) \, \mathbf{a}_{1} + y_{11} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{11}\right)a \, \mathbf{\hat{x}} + y_{11}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{11}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{71} & = & x_{11} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{11}\right) \, \mathbf{a}_{2} + z_{11} \, \mathbf{a}_{3} & = & x_{11}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{11}\right)b \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{72} & = & \left(\frac{1}{2} - x_{11}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{11}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{11}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VII} \\ \mathbf{B}_{73} & = & x_{12} \, \mathbf{a}_{1} + y_{12} \, \mathbf{a}_{2} + z_{12} \, \mathbf{a}_{3} & = & x_{12}a \, \mathbf{\hat{x}} + y_{12}b \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{74} & = & \left(\frac{1}{2} - x_{12}\right) \, \mathbf{a}_{1}-y_{12} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{12}\right)a \, \mathbf{\hat{x}}-y_{12}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{75} & = & -x_{12} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{12}\right) \, \mathbf{a}_{2}-z_{12} \, \mathbf{a}_{3} & = & -x_{12}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{12}\right)b \, \mathbf{\hat{y}}-z_{12}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{76} & = & \left(\frac{1}{2} +x_{12}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{12}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{12}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{12}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{77} & = & -x_{12} \, \mathbf{a}_{1}-y_{12} \, \mathbf{a}_{2}-z_{12} \, \mathbf{a}_{3} & = & -x_{12}a \, \mathbf{\hat{x}}-y_{12}b \, \mathbf{\hat{y}}-z_{12}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{78} & = & \left(\frac{1}{2} +x_{12}\right) \, \mathbf{a}_{1} + y_{12} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{12}\right)a \, \mathbf{\hat{x}} + y_{12}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{12}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{79} & = & x_{12} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{12}\right) \, \mathbf{a}_{2} + z_{12} \, \mathbf{a}_{3} & = & x_{12}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{12}\right)b \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{80} & = & \left(\frac{1}{2} - x_{12}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{12}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{12}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H VIII} \\ \mathbf{B}_{81} & = & x_{13} \, \mathbf{a}_{1} + y_{13} \, \mathbf{a}_{2} + z_{13} \, \mathbf{a}_{3} & = & x_{13}a \, \mathbf{\hat{x}} + y_{13}b \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IX} \\ \mathbf{B}_{82} & = & \left(\frac{1}{2} - x_{13}\right) \, \mathbf{a}_{1}-y_{13} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{13}\right)a \, \mathbf{\hat{x}}-y_{13}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IX} \\ \mathbf{B}_{83} & = & -x_{13} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{13}\right) \, \mathbf{a}_{2}-z_{13} \, \mathbf{a}_{3} & = & -x_{13}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{13}\right)b \, \mathbf{\hat{y}}-z_{13}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IX} \\ \mathbf{B}_{84} & = & \left(\frac{1}{2} +x_{13}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{13}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{13}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{13}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IX} \\ \mathbf{B}_{85} & = & -x_{13} \, \mathbf{a}_{1}-y_{13} \, \mathbf{a}_{2}-z_{13} \, \mathbf{a}_{3} & = & -x_{13}a \, \mathbf{\hat{x}}-y_{13}b \, \mathbf{\hat{y}}-z_{13}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IX} \\ \mathbf{B}_{86} & = & \left(\frac{1}{2} +x_{13}\right) \, \mathbf{a}_{1} + y_{13} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{13}\right)a \, \mathbf{\hat{x}} + y_{13}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{13}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IX} \\ \mathbf{B}_{87} & = & x_{13} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{13}\right) \, \mathbf{a}_{2} + z_{13} \, \mathbf{a}_{3} & = & x_{13}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{13}\right)b \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IX} \\ \mathbf{B}_{88} & = & \left(\frac{1}{2} - x_{13}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{13}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{13}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H IX} \\ \mathbf{B}_{89} & = & x_{14} \, \mathbf{a}_{1} + y_{14} \, \mathbf{a}_{2} + z_{14} \, \mathbf{a}_{3} & = & x_{14}a \, \mathbf{\hat{x}} + y_{14}b \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H X} \\ \mathbf{B}_{90} & = & \left(\frac{1}{2} - x_{14}\right) \, \mathbf{a}_{1}-y_{14} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{14}\right)a \, \mathbf{\hat{x}}-y_{14}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H X} \\ \mathbf{B}_{91} & = & -x_{14} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{14}\right) \, \mathbf{a}_{2}-z_{14} \, \mathbf{a}_{3} & = & -x_{14}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{14}\right)b \, \mathbf{\hat{y}}-z_{14}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H X} \\ \mathbf{B}_{92} & = & \left(\frac{1}{2} +x_{14}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{14}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{14}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{14}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H X} \\ \mathbf{B}_{93} & = & -x_{14} \, \mathbf{a}_{1}-y_{14} \, \mathbf{a}_{2}-z_{14} \, \mathbf{a}_{3} & = & -x_{14}a \, \mathbf{\hat{x}}-y_{14}b \, \mathbf{\hat{y}}-z_{14}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H X} \\ \mathbf{B}_{94} & = & \left(\frac{1}{2} +x_{14}\right) \, \mathbf{a}_{1} + y_{14} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{14}\right)a \, \mathbf{\hat{x}} + y_{14}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{14}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H X} \\ \mathbf{B}_{95} & = & x_{14} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{14}\right) \, \mathbf{a}_{2} + z_{14} \, \mathbf{a}_{3} & = & x_{14}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{14}\right)b \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H X} \\ \mathbf{B}_{96} & = & \left(\frac{1}{2} - x_{14}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{14}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{14}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H X} \\ \mathbf{B}_{97} & = & x_{15} \, \mathbf{a}_{1} + y_{15} \, \mathbf{a}_{2} + z_{15} \, \mathbf{a}_{3} & = & x_{15}a \, \mathbf{\hat{x}} + y_{15}b \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H XI} \\ \mathbf{B}_{98} & = & \left(\frac{1}{2} - x_{15}\right) \, \mathbf{a}_{1}-y_{15} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{15}\right)a \, \mathbf{\hat{x}}-y_{15}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H XI} \\ \mathbf{B}_{99} & = & -x_{15} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{15}\right) \, \mathbf{a}_{2}-z_{15} \, \mathbf{a}_{3} & = & -x_{15}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{15}\right)b \, \mathbf{\hat{y}}-z_{15}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H XI} \\ \mathbf{B}_{100} & = & \left(\frac{1}{2} +x_{15}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{15}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{15}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{15}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H XI} \\ \mathbf{B}_{101} & = & -x_{15} \, \mathbf{a}_{1}-y_{15} \, \mathbf{a}_{2}-z_{15} \, \mathbf{a}_{3} & = & -x_{15}a \, \mathbf{\hat{x}}-y_{15}b \, \mathbf{\hat{y}}-z_{15}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H XI} \\ \mathbf{B}_{102} & = & \left(\frac{1}{2} +x_{15}\right) \, \mathbf{a}_{1} + y_{15} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{15}\right)a \, \mathbf{\hat{x}} + y_{15}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{15}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H XI} \\ \mathbf{B}_{103} & = & x_{15} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{15}\right) \, \mathbf{a}_{2} + z_{15} \, \mathbf{a}_{3} & = & x_{15}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{15}\right)b \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H XI} \\ \mathbf{B}_{104} & = & \left(\frac{1}{2} - x_{15}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{15}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{15}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{H XI} \\ \mathbf{B}_{105} & = & x_{16} \, \mathbf{a}_{1} + y_{16} \, \mathbf{a}_{2} + z_{16} \, \mathbf{a}_{3} & = & x_{16}a \, \mathbf{\hat{x}} + y_{16}b \, \mathbf{\hat{y}} + z_{16}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IV} \\ \mathbf{B}_{106} & = & \left(\frac{1}{2} - x_{16}\right) \, \mathbf{a}_{1}-y_{16} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{16}\right)a \, \mathbf{\hat{x}}-y_{16}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IV} \\ \mathbf{B}_{107} & = & -x_{16} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{16}\right) \, \mathbf{a}_{2}-z_{16} \, \mathbf{a}_{3} & = & -x_{16}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{16}\right)b \, \mathbf{\hat{y}}-z_{16}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IV} \\ \mathbf{B}_{108} & = & \left(\frac{1}{2} +x_{16}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{16}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{16}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{16}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{16}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IV} \\ \mathbf{B}_{109} & = & -x_{16} \, \mathbf{a}_{1}-y_{16} \, \mathbf{a}_{2}-z_{16} \, \mathbf{a}_{3} & = & -x_{16}a \, \mathbf{\hat{x}}-y_{16}b \, \mathbf{\hat{y}}-z_{16}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IV} \\ \mathbf{B}_{110} & = & \left(\frac{1}{2} +x_{16}\right) \, \mathbf{a}_{1} + y_{16} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{16}\right)a \, \mathbf{\hat{x}} + y_{16}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{16}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IV} \\ \mathbf{B}_{111} & = & x_{16} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{16}\right) \, \mathbf{a}_{2} + z_{16} \, \mathbf{a}_{3} & = & x_{16}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{16}\right)b \, \mathbf{\hat{y}} + z_{16}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IV} \\ \mathbf{B}_{112} & = & \left(\frac{1}{2} - x_{16}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{16}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{16}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{16}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IV} \\ \mathbf{B}_{113} & = & x_{17} \, \mathbf{a}_{1} + y_{17} \, \mathbf{a}_{2} + z_{17} \, \mathbf{a}_{3} & = & x_{17}a \, \mathbf{\hat{x}} + y_{17}b \, \mathbf{\hat{y}} + z_{17}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O V} \\ \mathbf{B}_{114} & = & \left(\frac{1}{2} - x_{17}\right) \, \mathbf{a}_{1}-y_{17} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{17}\right)a \, \mathbf{\hat{x}}-y_{17}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O V} \\ \mathbf{B}_{115} & = & -x_{17} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{17}\right) \, \mathbf{a}_{2}-z_{17} \, \mathbf{a}_{3} & = & -x_{17}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{17}\right)b \, \mathbf{\hat{y}}-z_{17}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O V} \\ \mathbf{B}_{116} & = & \left(\frac{1}{2} +x_{17}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{17}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{17}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{17}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{17}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O V} \\ \mathbf{B}_{117} & = & -x_{17} \, \mathbf{a}_{1}-y_{17} \, \mathbf{a}_{2}-z_{17} \, \mathbf{a}_{3} & = & -x_{17}a \, \mathbf{\hat{x}}-y_{17}b \, \mathbf{\hat{y}}-z_{17}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O V} \\ \mathbf{B}_{118} & = & \left(\frac{1}{2} +x_{17}\right) \, \mathbf{a}_{1} + y_{17} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{17}\right)a \, \mathbf{\hat{x}} + y_{17}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{17}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O V} \\ \mathbf{B}_{119} & = & x_{17} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{17}\right) \, \mathbf{a}_{2} + z_{17} \, \mathbf{a}_{3} & = & x_{17}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{17}\right)b \, \mathbf{\hat{y}} + z_{17}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O V} \\ \mathbf{B}_{120} & = & \left(\frac{1}{2} - x_{17}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{17}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{17}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{17}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O V} \\ \mathbf{B}_{121} & = & x_{18} \, \mathbf{a}_{1} + y_{18} \, \mathbf{a}_{2} + z_{18} \, \mathbf{a}_{3} & = & x_{18}a \, \mathbf{\hat{x}} + y_{18}b \, \mathbf{\hat{y}} + z_{18}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VI} \\ \mathbf{B}_{122} & = & \left(\frac{1}{2} - x_{18}\right) \, \mathbf{a}_{1}-y_{18} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{18}\right)a \, \mathbf{\hat{x}}-y_{18}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VI} \\ \mathbf{B}_{123} & = & -x_{18} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{18}\right) \, \mathbf{a}_{2}-z_{18} \, \mathbf{a}_{3} & = & -x_{18}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{18}\right)b \, \mathbf{\hat{y}}-z_{18}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VI} \\ \mathbf{B}_{124} & = & \left(\frac{1}{2} +x_{18}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{18}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{18}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{18}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{18}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VI} \\ \mathbf{B}_{125} & = & -x_{18} \, \mathbf{a}_{1}-y_{18} \, \mathbf{a}_{2}-z_{18} \, \mathbf{a}_{3} & = & -x_{18}a \, \mathbf{\hat{x}}-y_{18}b \, \mathbf{\hat{y}}-z_{18}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VI} \\ \mathbf{B}_{126} & = & \left(\frac{1}{2} +x_{18}\right) \, \mathbf{a}_{1} + y_{18} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{18}\right)a \, \mathbf{\hat{x}} + y_{18}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{18}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VI} \\ \mathbf{B}_{127} & = & x_{18} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{18}\right) \, \mathbf{a}_{2} + z_{18} \, \mathbf{a}_{3} & = & x_{18}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{18}\right)b \, \mathbf{\hat{y}} + z_{18}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VI} \\ \mathbf{B}_{128} & = & \left(\frac{1}{2} - x_{18}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{18}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{18}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{18}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VI} \\ \mathbf{B}_{129} & = & x_{19} \, \mathbf{a}_{1} + y_{19} \, \mathbf{a}_{2} + z_{19} \, \mathbf{a}_{3} & = & x_{19}a \, \mathbf{\hat{x}} + y_{19}b \, \mathbf{\hat{y}} + z_{19}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VII} \\ \mathbf{B}_{130} & = & \left(\frac{1}{2} - x_{19}\right) \, \mathbf{a}_{1}-y_{19} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{19}\right)a \, \mathbf{\hat{x}}-y_{19}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VII} \\ \mathbf{B}_{131} & = & -x_{19} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{19}\right) \, \mathbf{a}_{2}-z_{19} \, \mathbf{a}_{3} & = & -x_{19}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{19}\right)b \, \mathbf{\hat{y}}-z_{19}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VII} \\ \mathbf{B}_{132} & = & \left(\frac{1}{2} +x_{19}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{19}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{19}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{19}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{19}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VII} \\ \mathbf{B}_{133} & = & -x_{19} \, \mathbf{a}_{1}-y_{19} \, \mathbf{a}_{2}-z_{19} \, \mathbf{a}_{3} & = & -x_{19}a \, \mathbf{\hat{x}}-y_{19}b \, \mathbf{\hat{y}}-z_{19}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VII} \\ \mathbf{B}_{134} & = & \left(\frac{1}{2} +x_{19}\right) \, \mathbf{a}_{1} + y_{19} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{19}\right)a \, \mathbf{\hat{x}} + y_{19}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{19}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VII} \\ \mathbf{B}_{135} & = & x_{19} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{19}\right) \, \mathbf{a}_{2} + z_{19} \, \mathbf{a}_{3} & = & x_{19}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{19}\right)b \, \mathbf{\hat{y}} + z_{19}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VII} \\ \mathbf{B}_{136} & = & \left(\frac{1}{2} - x_{19}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{19}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{19}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{19}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VII} \\ \mathbf{B}_{137} & = & x_{20} \, \mathbf{a}_{1} + y_{20} \, \mathbf{a}_{2} + z_{20} \, \mathbf{a}_{3} & = & x_{20}a \, \mathbf{\hat{x}} + y_{20}b \, \mathbf{\hat{y}} + z_{20}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VIII} \\ \mathbf{B}_{138} & = & \left(\frac{1}{2} - x_{20}\right) \, \mathbf{a}_{1}-y_{20} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{20}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{20}\right)a \, \mathbf{\hat{x}}-y_{20}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{20}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VIII} \\ \mathbf{B}_{139} & = & -x_{20} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{20}\right) \, \mathbf{a}_{2}-z_{20} \, \mathbf{a}_{3} & = & -x_{20}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{20}\right)b \, \mathbf{\hat{y}}-z_{20}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VIII} \\ \mathbf{B}_{140} & = & \left(\frac{1}{2} +x_{20}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{20}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{20}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{20}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{20}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{20}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VIII} \\ \mathbf{B}_{141} & = & -x_{20} \, \mathbf{a}_{1}-y_{20} \, \mathbf{a}_{2}-z_{20} \, \mathbf{a}_{3} & = & -x_{20}a \, \mathbf{\hat{x}}-y_{20}b \, \mathbf{\hat{y}}-z_{20}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VIII} \\ \mathbf{B}_{142} & = & \left(\frac{1}{2} +x_{20}\right) \, \mathbf{a}_{1} + y_{20} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{20}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{20}\right)a \, \mathbf{\hat{x}} + y_{20}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{20}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VIII} \\ \mathbf{B}_{143} & = & x_{20} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{20}\right) \, \mathbf{a}_{2} + z_{20} \, \mathbf{a}_{3} & = & x_{20}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{20}\right)b \, \mathbf{\hat{y}} + z_{20}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VIII} \\ \mathbf{B}_{144} & = & \left(\frac{1}{2} - x_{20}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{20}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{20}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{20}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{20}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{20}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O VIII} \\ \mathbf{B}_{145} & = & x_{21} \, \mathbf{a}_{1} + y_{21} \, \mathbf{a}_{2} + z_{21} \, \mathbf{a}_{3} & = & x_{21}a \, \mathbf{\hat{x}} + y_{21}b \, \mathbf{\hat{y}} + z_{21}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IX} \\ \mathbf{B}_{146} & = & \left(\frac{1}{2} - x_{21}\right) \, \mathbf{a}_{1}-y_{21} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{21}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{21}\right)a \, \mathbf{\hat{x}}-y_{21}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{21}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IX} \\ \mathbf{B}_{147} & = & -x_{21} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{21}\right) \, \mathbf{a}_{2}-z_{21} \, \mathbf{a}_{3} & = & -x_{21}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{21}\right)b \, \mathbf{\hat{y}}-z_{21}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IX} \\ \mathbf{B}_{148} & = & \left(\frac{1}{2} +x_{21}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{21}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{21}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{21}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{21}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{21}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IX} \\ \mathbf{B}_{149} & = & -x_{21} \, \mathbf{a}_{1}-y_{21} \, \mathbf{a}_{2}-z_{21} \, \mathbf{a}_{3} & = & -x_{21}a \, \mathbf{\hat{x}}-y_{21}b \, \mathbf{\hat{y}}-z_{21}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IX} \\ \mathbf{B}_{150} & = & \left(\frac{1}{2} +x_{21}\right) \, \mathbf{a}_{1} + y_{21} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{21}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{21}\right)a \, \mathbf{\hat{x}} + y_{21}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{21}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IX} \\ \mathbf{B}_{151} & = & x_{21} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{21}\right) \, \mathbf{a}_{2} + z_{21} \, \mathbf{a}_{3} & = & x_{21}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{21}\right)b \, \mathbf{\hat{y}} + z_{21}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IX} \\ \mathbf{B}_{152} & = & \left(\frac{1}{2} - x_{21}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{21}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{21}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{21}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{21}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{21}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O IX} \\ \mathbf{B}_{153} & = & x_{22} \, \mathbf{a}_{1} + y_{22} \, \mathbf{a}_{2} + z_{22} \, \mathbf{a}_{3} & = & x_{22}a \, \mathbf{\hat{x}} + y_{22}b \, \mathbf{\hat{y}} + z_{22}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O X} \\ \mathbf{B}_{154} & = & \left(\frac{1}{2} - x_{22}\right) \, \mathbf{a}_{1}-y_{22} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{22}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{22}\right)a \, \mathbf{\hat{x}}-y_{22}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{22}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O X} \\ \mathbf{B}_{155} & = & -x_{22} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{22}\right) \, \mathbf{a}_{2}-z_{22} \, \mathbf{a}_{3} & = & -x_{22}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{22}\right)b \, \mathbf{\hat{y}}-z_{22}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O X} \\ \mathbf{B}_{156} & = & \left(\frac{1}{2} +x_{22}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{22}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{22}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{22}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{22}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{22}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O X} \\ \mathbf{B}_{157} & = & -x_{22} \, \mathbf{a}_{1}-y_{22} \, \mathbf{a}_{2}-z_{22} \, \mathbf{a}_{3} & = & -x_{22}a \, \mathbf{\hat{x}}-y_{22}b \, \mathbf{\hat{y}}-z_{22}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O X} \\ \mathbf{B}_{158} & = & \left(\frac{1}{2} +x_{22}\right) \, \mathbf{a}_{1} + y_{22} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{22}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{22}\right)a \, \mathbf{\hat{x}} + y_{22}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{22}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O X} \\ \mathbf{B}_{159} & = & x_{22} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{22}\right) \, \mathbf{a}_{2} + z_{22} \, \mathbf{a}_{3} & = & x_{22}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{22}\right)b \, \mathbf{\hat{y}} + z_{22}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O X} \\ \mathbf{B}_{160} & = & \left(\frac{1}{2} - x_{22}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{22}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{22}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{22}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{22}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{22}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O X} \\ \mathbf{B}_{161} & = & x_{23} \, \mathbf{a}_{1} + y_{23} \, \mathbf{a}_{2} + z_{23} \, \mathbf{a}_{3} & = & x_{23}a \, \mathbf{\hat{x}} + y_{23}b \, \mathbf{\hat{y}} + z_{23}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XI} \\ \mathbf{B}_{162} & = & \left(\frac{1}{2} - x_{23}\right) \, \mathbf{a}_{1}-y_{23} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{23}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{23}\right)a \, \mathbf{\hat{x}}-y_{23}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{23}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XI} \\ \mathbf{B}_{163} & = & -x_{23} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{23}\right) \, \mathbf{a}_{2}-z_{23} \, \mathbf{a}_{3} & = & -x_{23}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{23}\right)b \, \mathbf{\hat{y}}-z_{23}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XI} \\ \mathbf{B}_{164} & = & \left(\frac{1}{2} +x_{23}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{23}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{23}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{23}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{23}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{23}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XI} \\ \mathbf{B}_{165} & = & -x_{23} \, \mathbf{a}_{1}-y_{23} \, \mathbf{a}_{2}-z_{23} \, \mathbf{a}_{3} & = & -x_{23}a \, \mathbf{\hat{x}}-y_{23}b \, \mathbf{\hat{y}}-z_{23}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XI} \\ \mathbf{B}_{166} & = & \left(\frac{1}{2} +x_{23}\right) \, \mathbf{a}_{1} + y_{23} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{23}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{23}\right)a \, \mathbf{\hat{x}} + y_{23}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{23}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XI} \\ \mathbf{B}_{167} & = & x_{23} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{23}\right) \, \mathbf{a}_{2} + z_{23} \, \mathbf{a}_{3} & = & x_{23}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{23}\right)b \, \mathbf{\hat{y}} + z_{23}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XI} \\ \mathbf{B}_{168} & = & \left(\frac{1}{2} - x_{23}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{23}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{23}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{23}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{23}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{23}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XI} \\ \mathbf{B}_{169} & = & x_{24} \, \mathbf{a}_{1} + y_{24} \, \mathbf{a}_{2} + z_{24} \, \mathbf{a}_{3} & = & x_{24}a \, \mathbf{\hat{x}} + y_{24}b \, \mathbf{\hat{y}} + z_{24}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XII} \\ \mathbf{B}_{170} & = & \left(\frac{1}{2} - x_{24}\right) \, \mathbf{a}_{1}-y_{24} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{24}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{24}\right)a \, \mathbf{\hat{x}}-y_{24}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{24}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XII} \\ \mathbf{B}_{171} & = & -x_{24} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{24}\right) \, \mathbf{a}_{2}-z_{24} \, \mathbf{a}_{3} & = & -x_{24}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{24}\right)b \, \mathbf{\hat{y}}-z_{24}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XII} \\ \mathbf{B}_{172} & = & \left(\frac{1}{2} +x_{24}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{24}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{24}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{24}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{24}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{24}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XII} \\ \mathbf{B}_{173} & = & -x_{24} \, \mathbf{a}_{1}-y_{24} \, \mathbf{a}_{2}-z_{24} \, \mathbf{a}_{3} & = & -x_{24}a \, \mathbf{\hat{x}}-y_{24}b \, \mathbf{\hat{y}}-z_{24}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XII} \\ \mathbf{B}_{174} & = & \left(\frac{1}{2} +x_{24}\right) \, \mathbf{a}_{1} + y_{24} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{24}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{24}\right)a \, \mathbf{\hat{x}} + y_{24}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{24}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XII} \\ \mathbf{B}_{175} & = & x_{24} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{24}\right) \, \mathbf{a}_{2} + z_{24} \, \mathbf{a}_{3} & = & x_{24}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{24}\right)b \, \mathbf{\hat{y}} + z_{24}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XII} \\ \mathbf{B}_{176} & = & \left(\frac{1}{2} - x_{24}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{24}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{24}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{24}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{24}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{24}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XII} \\ \mathbf{B}_{177} & = & x_{25} \, \mathbf{a}_{1} + y_{25} \, \mathbf{a}_{2} + z_{25} \, \mathbf{a}_{3} & = & x_{25}a \, \mathbf{\hat{x}} + y_{25}b \, \mathbf{\hat{y}} + z_{25}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XIII} \\ \mathbf{B}_{178} & = & \left(\frac{1}{2} - x_{25}\right) \, \mathbf{a}_{1}-y_{25} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{25}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{25}\right)a \, \mathbf{\hat{x}}-y_{25}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{25}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XIII} \\ \mathbf{B}_{179} & = & -x_{25} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{25}\right) \, \mathbf{a}_{2}-z_{25} \, \mathbf{a}_{3} & = & -x_{25}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{25}\right)b \, \mathbf{\hat{y}}-z_{25}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XIII} \\ \mathbf{B}_{180} & = & \left(\frac{1}{2} +x_{25}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{25}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{25}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{25}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{25}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{25}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XIII} \\ \mathbf{B}_{181} & = & -x_{25} \, \mathbf{a}_{1}-y_{25} \, \mathbf{a}_{2}-z_{25} \, \mathbf{a}_{3} & = & -x_{25}a \, \mathbf{\hat{x}}-y_{25}b \, \mathbf{\hat{y}}-z_{25}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XIII} \\ \mathbf{B}_{182} & = & \left(\frac{1}{2} +x_{25}\right) \, \mathbf{a}_{1} + y_{25} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{25}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{25}\right)a \, \mathbf{\hat{x}} + y_{25}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{25}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XIII} \\ \mathbf{B}_{183} & = & x_{25} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{25}\right) \, \mathbf{a}_{2} + z_{25} \, \mathbf{a}_{3} & = & x_{25}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{25}\right)b \, \mathbf{\hat{y}} + z_{25}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XIII} \\ \mathbf{B}_{184} & = & \left(\frac{1}{2} - x_{25}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{25}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{25}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{25}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{25}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{25}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{O XIII} \\ \mathbf{B}_{185} & = & x_{26} \, \mathbf{a}_{1} + y_{26} \, \mathbf{a}_{2} + z_{26} \, \mathbf{a}_{3} & = & x_{26}a \, \mathbf{\hat{x}} + y_{26}b \, \mathbf{\hat{y}} + z_{26}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{P} \\ \mathbf{B}_{186} & = & \left(\frac{1}{2} - x_{26}\right) \, \mathbf{a}_{1}-y_{26} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{26}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{26}\right)a \, \mathbf{\hat{x}}-y_{26}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{26}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{P} \\ \mathbf{B}_{187} & = & -x_{26} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{26}\right) \, \mathbf{a}_{2}-z_{26} \, \mathbf{a}_{3} & = & -x_{26}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{26}\right)b \, \mathbf{\hat{y}}-z_{26}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{P} \\ \mathbf{B}_{188} & = & \left(\frac{1}{2} +x_{26}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{26}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{26}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{26}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{26}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{26}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{P} \\ \mathbf{B}_{189} & = & -x_{26} \, \mathbf{a}_{1}-y_{26} \, \mathbf{a}_{2}-z_{26} \, \mathbf{a}_{3} & = & -x_{26}a \, \mathbf{\hat{x}}-y_{26}b \, \mathbf{\hat{y}}-z_{26}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{P} \\ \mathbf{B}_{190} & = & \left(\frac{1}{2} +x_{26}\right) \, \mathbf{a}_{1} + y_{26} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{26}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{26}\right)a \, \mathbf{\hat{x}} + y_{26}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{26}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{P} \\ \mathbf{B}_{191} & = & x_{26} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{26}\right) \, \mathbf{a}_{2} + z_{26} \, \mathbf{a}_{3} & = & x_{26}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{26}\right)b \, \mathbf{\hat{y}} + z_{26}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{P} \\ \mathbf{B}_{192} & = & \left(\frac{1}{2} - x_{26}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{26}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{26}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{26}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{26}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{26}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{P} \\ \mathbf{B}_{193} & = & x_{27} \, \mathbf{a}_{1} + y_{27} \, \mathbf{a}_{2} + z_{27} \, \mathbf{a}_{3} & = & x_{27}a \, \mathbf{\hat{x}} + y_{27}b \, \mathbf{\hat{y}} + z_{27}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{U} \\ \mathbf{B}_{194} & = & \left(\frac{1}{2} - x_{27}\right) \, \mathbf{a}_{1}-y_{27} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{27}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{27}\right)a \, \mathbf{\hat{x}}-y_{27}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{27}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{U} \\ \mathbf{B}_{195} & = & -x_{27} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{27}\right) \, \mathbf{a}_{2}-z_{27} \, \mathbf{a}_{3} & = & -x_{27}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{27}\right)b \, \mathbf{\hat{y}}-z_{27}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{U} \\ \mathbf{B}_{196} & = & \left(\frac{1}{2} +x_{27}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{27}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{27}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{27}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{27}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{27}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{U} \\ \mathbf{B}_{197} & = & -x_{27} \, \mathbf{a}_{1}-y_{27} \, \mathbf{a}_{2}-z_{27} \, \mathbf{a}_{3} & = & -x_{27}a \, \mathbf{\hat{x}}-y_{27}b \, \mathbf{\hat{y}}-z_{27}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{U} \\ \mathbf{B}_{198} & = & \left(\frac{1}{2} +x_{27}\right) \, \mathbf{a}_{1} + y_{27} \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{27}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{27}\right)a \, \mathbf{\hat{x}} + y_{27}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{27}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{U} \\ \mathbf{B}_{199} & = & x_{27} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{27}\right) \, \mathbf{a}_{2} + z_{27} \, \mathbf{a}_{3} & = & x_{27}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{27}\right)b \, \mathbf{\hat{y}} + z_{27}c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{U} \\ \mathbf{B}_{200} & = & \left(\frac{1}{2} - x_{27}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{27}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{27}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{27}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{27}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{27}\right)c \, \mathbf{\hat{z}} & \left(8d\right) & \mbox{U} \\ \end{array} \]

References

  • A. J. Locock and P. C. Burns, The crystal structure of synthetic autunite, Ca[(UO2)(PO4)]2(H2O)11, Am. Mineral. 88, 240–244 (2003), doi:10.2138/am-2003-0128.
  • K. Herrmann, ed., Strukturbericht Band VI 1938 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1941).

Found in

  • R. T. Downs and M. Hall–Wallace, The American Mineralogist Crystal Structure Database, Am. Mineral. 88, 247–250 (2003).

Geometry files


Prototype Generator

aflow --proto=AB22C23D2E2_oP200_62_c_11d_3c10d_d_d --params=

Species:

Running:

Output: