Al(PO3)3 ($G5_{2}$) Structure : AB9C3_cI208_220_c_3e_e

Picture of Structure; Click for Big Picture
Prototype : AlO9P3
AFLOW prototype label : AB9C3_cI208_220_c_3e_e
Strukturbericht designation : $G5_{2}$
Pearson symbol : cI208
Space group number : 220
Space group symbol : $I\bar{4}3d$
AFLOW prototype command : aflow --proto=AB9C3_cI208_220_c_3e_e
--params=
$a$,$x_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$


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

References

  • L. Pauling and J. Sherman, The Crystal Structure of Aluminum Metaphosphate, Al(PO3)3, Zeitschrift für Kristallographie – Crystalline Materials 96, 481–487 (1937), doi:10.1524/zkri.1937.96.1.481.

Found in

  • H. van der Meer, The crystal structure of a monoclinic form of aluminium metaphosphate, Al(PO3)3, Acta Crystallogr. Sect. B Struct. Sci. 32, 2423–2426 (1976), doi:10.1107/S0567740876007899.

Geometry files


Prototype Generator

aflow --proto=AB9C3_cI208_220_c_3e_e --params=

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