Ca3Al2(OH)12 ($J2_{3}$) Structure : A2B3C12D12_cI232_230_a_c_h_h

Picture of Structure; Click for Big Picture
Prototype : Al2Ca3H12O12
AFLOW prototype label : A2B3C12D12_cI232_230_a_c_h_h
Strukturbericht designation : $J2_{3}$
Pearson symbol : cI232
Space group number : 230
Space group symbol : $Ia\bar{3}d$
AFLOW prototype command : aflow --proto=A2B3C12D12_cI232_230_a_c_h_h
--params=
$a$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$


  • The original determination of this structure by (Brandenberger, 1933) did not locate the hydrogen atoms, and according to (Gottfried, 1937) used the coordinates of garnet, $S1_{4}$. (Bartl, 1986) was able to locate the hydrogen atoms, and as they do not change the space group we include them in the $J2_{3}$ structure.

Body-centered Cubic primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & 0 \, \mathbf{a}_{1} + 0 \, \mathbf{a}_{2} + 0 \, \mathbf{a}_{3} & = & 0 \, \mathbf{\hat{x}} + 0 \, \mathbf{\hat{y}} + 0 \, \mathbf{\hat{z}} & \left(16a\right) & \mbox{Al} \\ \mathbf{B}_{2} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{y}} & \left(16a\right) & \mbox{Al} \\ \mathbf{B}_{3} & = & \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} & \left(16a\right) & \mbox{Al} \\ \mathbf{B}_{4} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{2}a \, \mathbf{\hat{z}} & \left(16a\right) & \mbox{Al} \\ \mathbf{B}_{5} & = & \frac{1}{2} \, \mathbf{a}_{1} & = & \frac{3}{4}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16a\right) & \mbox{Al} \\ \mathbf{B}_{6} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \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(16a\right) & \mbox{Al} \\ \mathbf{B}_{7} & = & \frac{1}{2} \, \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(16a\right) & \mbox{Al} \\ \mathbf{B}_{8} & = & \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{3}{4}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(16a\right) & \mbox{Al} \\ \mathbf{B}_{9} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{3}{8} \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{10} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{3}{8} \, \mathbf{a}_{3} & = & - \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{11} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{3}{8} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{12} & = & \frac{3}{8} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}}- \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{13} & = & \frac{3}{8} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{14} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{3}{8} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}}- \frac{1}{8}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{15} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{5}{8} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{16} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{5}{8} \, \mathbf{a}_{3} & = & \frac{5}{8}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{17} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \frac{5}{8} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{18} & = & \frac{5}{8} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{5}{8}a \, \mathbf{\hat{y}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{19} & = & \frac{5}{8} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{20} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{5}{8} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{5}{8}a \, \mathbf{\hat{z}} & \left(24c\right) & \mbox{Ca} \\ \mathbf{B}_{21} & = & \left(y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+y_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}} + z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{22} & = & \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{3}\right)a \, \mathbf{\hat{y}} + z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{23} & = & \left(y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}}-z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{24} & = & \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}}-y_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{25} & = & \left(x_{3}+y_{3}\right) \, \mathbf{a}_{1} + \left(y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & z_{3}a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} + y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{26} & = & \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & z_{3}a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{27} & = & \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{1} + \left(y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & -z_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{y}} + y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{28} & = & \left(x_{3}-y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-z_{3}\right)a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}}-y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{29} & = & \left(x_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+y_{3}\right) \, \mathbf{a}_{2} + \left(y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & y_{3}a \, \mathbf{\hat{x}} + z_{3}a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{30} & = & \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{3}\right)a \, \mathbf{\hat{x}} + z_{3}a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{31} & = & \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{2} + \left(y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & y_{3}a \, \mathbf{\hat{x}}-z_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{32} & = & \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & -y_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-z_{3}\right)a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{33} & = & \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{34} & = & \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{35} & = & \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{36} & = & \left(x_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{37} & = & \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{38} & = & \left(y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{39} & = & \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{40} & = & \left(y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{41} & = & \left(\frac{1}{2} - x_{3} + y_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{42} & = & \left(x_{3}-y_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{43} & = & \left(x_{3}+y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{44} & = & \left(\frac{1}{2} - x_{3} - y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{45} & = & \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}}-y_{3}a \, \mathbf{\hat{y}}-z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{46} & = & \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{3}\right)a \, \mathbf{\hat{y}}-z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{47} & = & \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}}-y_{3}a \, \mathbf{\hat{y}} + z_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{48} & = & \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{49} & = & \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{1} + \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -z_{3}a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}}-y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{50} & = & \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -z_{3}a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{51} & = & \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{1} + \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & z_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{y}}-y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{52} & = & \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +z_{3}\right)a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}} + y_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{53} & = & \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{2} + \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -y_{3}a \, \mathbf{\hat{x}}-z_{3}a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{54} & = & \left(x_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{3}\right)a \, \mathbf{\hat{x}}-z_{3}a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{55} & = & \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{2} + \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & -y_{3}a \, \mathbf{\hat{x}} + z_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{56} & = & \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & y_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +z_{3}\right)a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{57} & = & \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{3} & = & -\left(y_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{58} & = & \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{59} & = & \left(x_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}}-\left(z_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{60} & = & \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}}-\left(x_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{61} & = & \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{2} + \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -\left(x_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{62} & = & \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{2} + \left(x_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}}-\left(z_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{63} & = & \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{64} & = & \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{y}}-\left(y_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{65} & = & \left(\frac{1}{2} +x_{3} - y_{3}\right) \, \mathbf{a}_{1} + \left(x_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(-y_{3}-z_{3}\right) \, \mathbf{a}_{3} & = & -\left(z_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{66} & = & \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{1} + \left(-x_{3}-z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{y}}-\left(x_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{67} & = & \left(-x_{3}-y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(-y_{3}+z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{x}}-\left(y_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{68} & = & \left(\frac{1}{2} +x_{3} + y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3} + z_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{3} + z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{H} \\ \mathbf{B}_{69} & = & \left(y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}+y_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + y_{4}a \, \mathbf{\hat{y}} + z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{70} & = & \left(\frac{1}{2} - y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} - y_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{4}\right)a \, \mathbf{\hat{y}} + z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{71} & = & \left(y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} + y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{x}} + y_{4}a \, \mathbf{\hat{y}}-z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{72} & = & \left(\frac{1}{2} - y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}}-y_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{73} & = & \left(x_{4}+y_{4}\right) \, \mathbf{a}_{1} + \left(y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & z_{4}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{74} & = & \left(\frac{1}{2} - x_{4} - y_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & z_{4}a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{75} & = & \left(\frac{1}{2} - x_{4} + y_{4}\right) \, \mathbf{a}_{1} + \left(y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & -z_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{y}} + y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{76} & = & \left(x_{4}-y_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-z_{4}\right)a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}}-y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{77} & = & \left(x_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+y_{4}\right) \, \mathbf{a}_{2} + \left(y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & y_{4}a \, \mathbf{\hat{x}} + z_{4}a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{78} & = & \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} - y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{4}\right)a \, \mathbf{\hat{x}} + z_{4}a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{79} & = & \left(\frac{1}{2} - x_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} + y_{4}\right) \, \mathbf{a}_{2} + \left(y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & y_{4}a \, \mathbf{\hat{x}}-z_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{80} & = & \left(\frac{1}{2} +x_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & -y_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-z_{4}\right)a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{81} & = & \left(\frac{1}{2} +x_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}+y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{82} & = & \left(\frac{1}{2} - x_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} - y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{83} & = & \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} + y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{84} & = & \left(x_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{85} & = & \left(\frac{1}{2} - y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{86} & = & \left(y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} + y_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{87} & = & \left(\frac{1}{2} - y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} - y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{88} & = & \left(y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{89} & = & \left(\frac{1}{2} - x_{4} + y_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4} +z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{90} & = & \left(x_{4}-y_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{91} & = & \left(x_{4}+y_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{92} & = & \left(\frac{1}{2} - x_{4} - y_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{93} & = & \left(-y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}-y_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}}-y_{4}a \, \mathbf{\hat{y}}-z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{94} & = & \left(\frac{1}{2} +y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} + y_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{4}\right)a \, \mathbf{\hat{y}}-z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{95} & = & \left(-y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} - y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}}-y_{4}a \, \mathbf{\hat{y}} + z_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{96} & = & \left(\frac{1}{2} +y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + y_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{97} & = & \left(-x_{4}-y_{4}\right) \, \mathbf{a}_{1} + \left(-y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & -z_{4}a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}}-y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{98} & = & \left(\frac{1}{2} +x_{4} + y_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & -z_{4}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{99} & = & \left(\frac{1}{2} +x_{4} - y_{4}\right) \, \mathbf{a}_{1} + \left(-y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & z_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{y}}-y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{100} & = & \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +z_{4}\right)a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + y_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{101} & = & \left(-x_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-y_{4}\right) \, \mathbf{a}_{2} + \left(-y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & -y_{4}a \, \mathbf{\hat{x}}-z_{4}a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{102} & = & \left(x_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} + y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{4}\right)a \, \mathbf{\hat{x}}-z_{4}a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{103} & = & \left(\frac{1}{2} +x_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} - y_{4}\right) \, \mathbf{a}_{2} + \left(-y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & -y_{4}a \, \mathbf{\hat{x}} + z_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{104} & = & \left(\frac{1}{2} - x_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & y_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +z_{4}\right)a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{105} & = & \left(\frac{1}{2} - x_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(-y_{4}+z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}-y_{4}\right) \, \mathbf{a}_{3} & = & -\left(y_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{106} & = & \left(\frac{1}{2} +x_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} + y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{107} & = & \left(x_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(-y_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} - y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}}-\left(z_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{108} & = & \left(-x_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4} - z_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{x}}-\left(x_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{109} & = & \left(\frac{1}{2} +y_{4} - z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{2} + \left(-x_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & -\left(x_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{110} & = & \left(-y_{4}-z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} - y_{4}\right) \, \mathbf{a}_{2} + \left(x_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}}-\left(z_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{111} & = & \left(\frac{1}{2} +y_{4} + z_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} + y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{112} & = & \left(-y_{4}+z_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{y}}-\left(y_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{113} & = & \left(\frac{1}{2} +x_{4} - y_{4}\right) \, \mathbf{a}_{1} + \left(x_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(-y_{4}-z_{4}\right) \, \mathbf{a}_{3} & = & -\left(z_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{114} & = & \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{1} + \left(-x_{4}-z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{4} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{y}}-\left(x_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{115} & = & \left(-x_{4}-y_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(-y_{4}+z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{x}}-\left(y_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \mathbf{B}_{116} & = & \left(\frac{1}{2} +x_{4} + y_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4} + z_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{4} + z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(96h\right) & \mbox{O} \\ \end{array} \]

References

  • H. Bartl, Tricalciumaluminathexahydrat, Ca3[Al(OH)6]2, Bindungslängen und –valenzen aus Röntgeneinkristallmessungen, Fresen. Z. Anal. Chem. 324, 124–126 (1986), doi:10.1007/BF00473351.
  • C. Gottfried and F. Schossberger, eds., Strukturbericht Band III 1933–1935 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).
  • E. Brandenberger, Kristallstrukturelle Untersuchungen an Calciumaluminathydraten, Schweiz. Mineral. Petrog. Mitt. 13 (1933).

Geometry files


Prototype Generator

aflow --proto=A2B3C12D12_cI232_230_a_c_h_h --params=

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