Dodecatungstophosphoric Acid Hexahydrate [H3PW12O40·6H2O] Structure : A27B52CD12_cP184_224_dl_eh3k_a_k

Picture of Structure; Click for Big Picture
Prototype : H15O46PW12
AFLOW prototype label : A27B52CD12_cP184_224_dl_eh3k_a_k
Strukturbericht designation : None
Pearson symbol : cP184
Space group number : 224
Space group symbol : $Pn\bar{3}m$
AFLOW prototype command : aflow --proto=A27B52CD12_cP184_224_dl_eh3k_a_k
--params=
$a$,$x_{3}$,$x_{4}$,$x_{5}$,$z_{5}$,$x_{6}$,$z_{6}$,$x_{7}$,$z_{7}$,$x_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$


  • (Brown, 1977) presents this as an improvement on the $H4_{16}$ structure, H3PW12O40·5H2O. The primary difference is the addition of a sixth water molecule and the location of the hydrogen molecules not directly attached to a water molecule.
  • The water molecules are formed by the H–II and O–II atoms, and the ($24h$) (O–II) and ($48l$) (H–II) Wyckoff sites are only occupied half of the time. Presumably this means that the nearly flat H–O molecular ions in this structure actually consist of one water molecule, the central hydrogen atom (H–I), and a water molecule on the other side of the central hydrogen, with the other water positions empty. Exactly which water molecules are occupied on around each H–I atom is completely up to chance. (Brown, 1977) state that the molecule has a positive charge, and write it as H5O2+.
  • We use the neutron data from (Brown, 1977) to locate the non–hydrogen atoms.
  • This structure is a partially dehydrated form of H3PW12O40·29H2O ($H4_{21}$). Further dehydration produces the H3PW12O40·3H2O structure.

Simple Cubic primitive vectors:

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

Basis vectors:

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

References

  • G. M. Brown, M.–R. Noe–Spirlet, W. R. Busing, and H. A. Levy, Dodecatungstophosphoric acid hexahydrate, (H5O2+)3(PW12O403–). The true structure of Keggin's 'pentahydrate' from single–crystal X–ray and neutron diffraction data, Acta Crystallogr. Sect. B Struct. Sci. 33, 1038–1046 (1977), doi:10.1107/S0567740877005330.

Geometry files


Prototype Generator

aflow --proto=A27B52CD12_cP184_224_dl_eh3k_a_k --params=

Species:

Running:

Output: