$\beta$–B (R–105) Structure: A_hR105_166_bc9h4i

Picture of Structure; Click for Big Picture
Prototype : $\beta$–B
AFLOW prototype label : A_hR105_166_bc9h4i
Strukturbericht designation : None
Pearson symbol : hR105
Space group number : 166
Space group symbol : $\mbox{R}\bar{3}\mbox{m}$
AFLOW prototype command : aflow --proto=A_hR105_166_bc9h4i [--hex]
--params=
$a$,$c/a$,$x_{2}$,$x_{3}$,$z_{3}$,$x_{4}$,$z_{4}$,$x_{5}$,$z_{5}$,$x_{6}$,$z_{6}$,$x_{7}$,$z_{7}$,$x_{8}$,$z_{8}$,$x_{9}$,$z_{9}$,$x_{10}$,$z_{10}$,$x_{11}$,$z_{11}$,$ x_{12}$,$y_{12}$,$z_{12}$,$x_{13}$,$y_{13}$,$z_{13}$,$x_{14}$,$y_{14}$,$z_{14}$,$x_{15}$,$y_{15}$,$z_{15}$


  • This is apparently the ground state of boron, with 105 atoms in the unit cell. Note the relationship between the icosahedra in this structure, $\alpha$–B and T–50 B. (Donohue, 1982) gives two possible sets of internal coordinates for the atoms on page 64. We use the second set (Geist, 1970), as it has no partially filled sites. Hexagonal settings of this structure can be obtained with the option ––hex.

Rhombohedral primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =&\frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\frac12 \, c \, \mathbf{\hat{z}}& \left(1b\right) & \mbox{B I} \\ \mathbf{B}_{2} & =&x_{2} \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}+ x_{2} \, \mathbf{a}_{3}& =&x_{2} \, c \, \mathbf{\hat{z}}& \left(2c\right) & \mbox{B II} \\ \mathbf{B}_{3} & =&- x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}& =&- x_{2} \, c \, \mathbf{\hat{z}}& \left(2c\right) & \mbox{B II} \\ \mathbf{B}_{4} & =&x_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{3} - z_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{3} - z_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{5} & =&z_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{3} - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{3} - z_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{6} & =&x_{3} \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(z_{3} - x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{7} & =&- x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{3} - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{3} - x_{3}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{8} & =&- z_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{3} - z_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{3} - x_{3}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{9} & =&- x_{3} \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(x_{3} - z_{3}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{10} & =&x_{4} \, \mathbf{a}_{1}+ x_{4} \, \mathbf{a}_{2}+ z_{4} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{4} - z_{4}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{4} - z_{4}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{4} + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IV} \\ \mathbf{B}_{11} & =&z_{4} \, \mathbf{a}_{1}+ x_{4} \, \mathbf{a}_{2}+ x_{4} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{4} - x_{4}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{4} - z_{4}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{4} + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IV} \\ \mathbf{B}_{12} & =&x_{4} \, \mathbf{a}_{1}+ z_{4} \, \mathbf{a}_{2}+ x_{4} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(z_{4} - x_{4}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{4} + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IV} \\ \mathbf{B}_{13} & =&- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{4} - x_{4}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{4} - x_{4}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{4} + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IV} \\ \mathbf{B}_{14} & =&- z_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{4} - z_{4}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{4} - x_{4}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{4} + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IV} \\ \mathbf{B}_{15} & =&- x_{4} \, \mathbf{a}_{1}- z_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(x_{4} - z_{4}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{4} + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IV} \\ \mathbf{B}_{16} & =&x_{5} \, \mathbf{a}_{1}+ x_{5} \, \mathbf{a}_{2}+ z_{5} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{5} - z_{5}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{5} - z_{5}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{5} + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B V} \\ \mathbf{B}_{17} & =&z_{5} \, \mathbf{a}_{1}+ x_{5} \, \mathbf{a}_{2}+ x_{5} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{5} - x_{5}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{5} - z_{5}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{5} + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B V} \\ \mathbf{B}_{18} & =&x_{5} \, \mathbf{a}_{1}+ z_{5} \, \mathbf{a}_{2}+ x_{5} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(z_{5} - x_{5}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{5} + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B V} \\ \mathbf{B}_{19} & =&- x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{5} - x_{5}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{5} - x_{5}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{5} + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B V} \\ \mathbf{B}_{20} & =&- z_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{5} - z_{5}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{5} - x_{5}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{5} + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B V} \\ \mathbf{B}_{21} & =&- x_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(x_{5} - z_{5}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{5} + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B V} \\ \mathbf{B}_{22} & =&x_{6} \, \mathbf{a}_{1}+ x_{6} \, \mathbf{a}_{2}+ z_{6} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{6} - z_{6}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{6} - z_{6}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{6} + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VI} \\ \mathbf{B}_{23} & =&z_{6} \, \mathbf{a}_{1}+ x_{6} \, \mathbf{a}_{2}+ x_{6} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{6} - x_{6}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{6} - z_{6}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{6} + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VI} \\ \mathbf{B}_{24} & =&x_{6} \, \mathbf{a}_{1}+ z_{6} \, \mathbf{a}_{2}+ x_{6} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(z_{6} - x_{6}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{6} + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VI} \\ \mathbf{B}_{25} & =&- x_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{6} - x_{6}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{6} - x_{6}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{6} + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VI} \\ \mathbf{B}_{26} & =&- z_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{6} - z_{6}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{6} - x_{6}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{6} + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VI} \\ \mathbf{B}_{27} & =&- x_{6} \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(x_{6} - z_{6}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{6} + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VI} \\ \mathbf{B}_{28} & =&x_{7} \, \mathbf{a}_{1}+ x_{7} \, \mathbf{a}_{2}+ z_{7} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{7} - z_{7}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{7} - z_{7}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{7} + z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VII} \\ \mathbf{B}_{29} & =&z_{7} \, \mathbf{a}_{1}+ x_{7} \, \mathbf{a}_{2}+ x_{7} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{7} - x_{7}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{7} - z_{7}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{7} + z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VII} \\ \mathbf{B}_{30} & =&x_{7} \, \mathbf{a}_{1}+ z_{7} \, \mathbf{a}_{2}+ x_{7} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(z_{7} - x_{7}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{7} + z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VII} \\ \mathbf{B}_{31} & =&- x_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{7} - x_{7}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{7} - x_{7}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{7} + z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VII} \\ \mathbf{B}_{32} & =&- z_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{7} - z_{7}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{7} - x_{7}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{7} + z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VII} \\ \mathbf{B}_{33} & =&- x_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(x_{7} - z_{7}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{7} + z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VII} \\ \mathbf{B}_{34} & =&x_{8} \, \mathbf{a}_{1}+ x_{8} \, \mathbf{a}_{2}+ z_{8} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{8} - z_{8}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{8} - z_{8}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{8} + z_{8}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VIII} \\ \mathbf{B}_{35} & =&z_{8} \, \mathbf{a}_{1}+ x_{8} \, \mathbf{a}_{2}+ x_{8} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{8} - x_{8}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{8} - z_{8}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{8} + z_{8}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VIII} \\ \mathbf{B}_{36} & =&x_{8} \, \mathbf{a}_{1}+ z_{8} \, \mathbf{a}_{2}+ x_{8} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(z_{8} - x_{8}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{8} + z_{8}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VIII} \\ \mathbf{B}_{37} & =&- x_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{8} - x_{8}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{8} - x_{8}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{8} + z_{8}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VIII} \\ \mathbf{B}_{38} & =&- z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{8} - z_{8}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{8} - x_{8}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{8} + z_{8}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VIII} \\ \mathbf{B}_{39} & =&- x_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(x_{8} - z_{8}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{8} + z_{8}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B VIII} \\ \mathbf{B}_{40} & =&x_{9} \, \mathbf{a}_{1}+ x_{9} \, \mathbf{a}_{2}+ z_{9} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{9} - z_{9}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{9} - z_{9}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{9} + z_{9}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IX} \\ \mathbf{B}_{41} & =&z_{9} \, \mathbf{a}_{1}+ x_{9} \, \mathbf{a}_{2}+ x_{9} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{9} - x_{9}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{9} - z_{9}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{9} + z_{9}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IX} \\ \mathbf{B}_{42} & =&x_{9} \, \mathbf{a}_{1}+ z_{9} \, \mathbf{a}_{2}+ x_{9} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(z_{9} - x_{9}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{9} + z_{9}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IX} \\ \mathbf{B}_{43} & =&- x_{9} \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{9} - x_{9}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{9} - x_{9}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{9} + z_{9}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IX} \\ \mathbf{B}_{44} & =&- z_{9} \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}- x_{9} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{9} - z_{9}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{9} - x_{9}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{9} + z_{9}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IX} \\ \mathbf{B}_{45} & =&- x_{9} \, \mathbf{a}_{1}- z_{9} \, \mathbf{a}_{2}- x_{9} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(x_{9} - z_{9}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{9} + z_{9}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B IX} \\ \mathbf{B}_{46} & =&x_{10} \, \mathbf{a}_{1}+ x_{10} \, \mathbf{a}_{2}+ z_{10} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{10} - z_{10}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{10} - z_{10}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{10} + z_{10}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B X} \\ \mathbf{B}_{47} & =&z_{10} \, \mathbf{a}_{1}+ x_{10} \, \mathbf{a}_{2}+ x_{10} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{10} - x_{10}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{10} - z_{10}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{10} + z_{10}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B X} \\ \mathbf{B}_{48} & =&x_{10} \, \mathbf{a}_{1}+ z_{10} \, \mathbf{a}_{2}+ x_{10} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(z_{10} - x_{10}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{10} + z_{10}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B X} \\ \mathbf{B}_{49} & =&- x_{10} \, \mathbf{a}_{1}- x_{10} \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{10} - x_{10}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{10} - x_{10}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{10} + z_{10}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B X} \\ \mathbf{B}_{50} & =&- z_{10} \, \mathbf{a}_{1}- x_{10} \, \mathbf{a}_{2}- x_{10} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{10} - z_{10}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{10} - x_{10}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{10} + z_{10}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B X} \\ \mathbf{B}_{51} & =&- x_{10} \, \mathbf{a}_{1}- z_{10} \, \mathbf{a}_{2}- x_{10} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(x_{10} - z_{10}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{10} + z_{10}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B X} \\ \mathbf{B}_{52} & =&x_{11} \, \mathbf{a}_{1}+ x_{11} \, \mathbf{a}_{2}+ z_{11} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{11} - z_{11}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{11} - z_{11}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{11} + z_{11}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B XI} \\ \mathbf{B}_{53} & =&z_{11} \, \mathbf{a}_{1}+ x_{11} \, \mathbf{a}_{2}+ x_{11} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{11} - x_{11}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(x_{11} - z_{11}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{11} + z_{11}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B XI} \\ \mathbf{B}_{54} & =&x_{11} \, \mathbf{a}_{1}+ z_{11} \, \mathbf{a}_{2}+ x_{11} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(z_{11} - x_{11}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(2 x_{11} + z_{11}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B XI} \\ \mathbf{B}_{55} & =&- x_{11} \, \mathbf{a}_{1}- x_{11} \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{11} - x_{11}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{11} - x_{11}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{11} + z_{11}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B XI} \\ \mathbf{B}_{56} & =&- z_{11} \, \mathbf{a}_{1}- x_{11} \, \mathbf{a}_{2}- x_{11} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{11} - z_{11}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \left(z_{11} - x_{11}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{11} + z_{11}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B XI} \\ \mathbf{B}_{57} & =&- x_{11} \, \mathbf{a}_{1}- z_{11} \, \mathbf{a}_{2}- x_{11} \, \mathbf{a}_{3}& =&\frac1{\sqrt3} \left(x_{11} - z_{11}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(2 x_{11} + z_{11}\right) \, c \, \mathbf{\hat{z}}& \left(6h\right) & \mbox{B XI} \\ \mathbf{B}_{58} & =&x_{12} \, \mathbf{a}_{1}+ y_{12} \, \mathbf{a}_{2}+ z_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{12} - z_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 y_{12} - x_{12} - z_{12}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{59} & =&z_{12} \, \mathbf{a}_{1}+ x_{12} \, \mathbf{a}_{2}+ y_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{12} - y_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 x_{12} - y_{12} - z_{12}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{60} & =&y_{12} \, \mathbf{a}_{1}+ z_{12} \, \mathbf{a}_{2}+ x_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{12} - x_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 z_{12} - x_{12} - y_{12}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{61} & =&- y_{12} \, \mathbf{a}_{1}- x_{12} \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{12} - y_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(y_{12} + z_{12} - 2 x_{12}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{62} & =&- x_{12} \, \mathbf{a}_{1}- z_{12} \, \mathbf{a}_{2}- y_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{12} - x_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{12} + y_{12} - 2 z_{12}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{63} & =&- z_{12} \, \mathbf{a}_{1}- y_{12} \, \mathbf{a}_{2}- x_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{12} - z_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{12} + z_{12} - 2 y_{12}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{64} & =&- x_{12} \, \mathbf{a}_{1}- y_{12} \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{12} - x_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{12} + z_{12} - 2 y_{12}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{65} & =&- z_{12} \, \mathbf{a}_{1}- x_{12} \, \mathbf{a}_{2}- y_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{12} - z_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(y_{12} + z_{12} - 2 x_{12}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{66} & =&- y_{12} \, \mathbf{a}_{1}- z_{12} \, \mathbf{a}_{2}- x_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{12} - y_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{12} + y_{12} - 2 z_{12}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{67} & =&y_{12} \, \mathbf{a}_{1}+ x_{12} \, \mathbf{a}_{2}+ z_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{12} - z_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 x_{12} - y_{12} - z_{12}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{68} & =&x_{12} \, \mathbf{a}_{1}+ z_{12} \, \mathbf{a}_{2}+ y_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{12} - y_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 z_{12} - x_{12} - y_{12}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{69} & =&z_{12} \, \mathbf{a}_{1}+ y_{12} \, \mathbf{a}_{2}+ x_{12} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{12} - x_{12}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 y_{12} - x_{12} - z_{12}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{12} + y_{12} + z_{12}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XII} \\ \mathbf{B}_{70} & =&x_{13} \, \mathbf{a}_{1}+ y_{13} \, \mathbf{a}_{2}+ z_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{13} - z_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 y_{13} - x_{13} - z_{13}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{71} & =&z_{13} \, \mathbf{a}_{1}+ x_{13} \, \mathbf{a}_{2}+ y_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{13} - y_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 x_{13} - y_{13} - z_{13}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{72} & =&y_{13} \, \mathbf{a}_{1}+ z_{13} \, \mathbf{a}_{2}+ x_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{13} - x_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 z_{13} - x_{13} - y_{13}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{73} & =&- y_{13} \, \mathbf{a}_{1}- x_{13} \, \mathbf{a}_{2}- z_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{13} - y_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(y_{13} + z_{13} - 2 x_{13}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{74} & =&- x_{13} \, \mathbf{a}_{1}- z_{13} \, \mathbf{a}_{2}- y_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{13} - x_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{13} + y_{13} - 2 z_{13}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{75} & =&- z_{13} \, \mathbf{a}_{1}- y_{13} \, \mathbf{a}_{2}- x_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{13} - z_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{13} + z_{13} - 2 y_{13}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{76} & =&- x_{13} \, \mathbf{a}_{1}- y_{13} \, \mathbf{a}_{2}- z_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{13} - x_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{13} + z_{13} - 2 y_{13}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{77} & =&- z_{13} \, \mathbf{a}_{1}- x_{13} \, \mathbf{a}_{2}- y_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{13} - z_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(y_{13} + z_{13} - 2 x_{13}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{78} & =&- y_{13} \, \mathbf{a}_{1}- z_{13} \, \mathbf{a}_{2}- x_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{13} - y_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{13} + y_{13} - 2 z_{13}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{79} & =&y_{13} \, \mathbf{a}_{1}+ x_{13} \, \mathbf{a}_{2}+ z_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{13} - z_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 x_{13} - y_{13} - z_{13}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{80} & =&x_{13} \, \mathbf{a}_{1}+ z_{13} \, \mathbf{a}_{2}+ y_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{13} - y_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 z_{13} - x_{13} - y_{13}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{81} & =&z_{13} \, \mathbf{a}_{1}+ y_{13} \, \mathbf{a}_{2}+ x_{13} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{13} - x_{13}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 y_{13} - x_{13} - z_{13}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{13} + y_{13} + z_{13}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIII} \\ \mathbf{B}_{82} & =&x_{14} \, \mathbf{a}_{1}+ y_{14} \, \mathbf{a}_{2}+ z_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{14} - z_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 y_{14} - x_{14} - z_{14}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{83} & =&z_{14} \, \mathbf{a}_{1}+ x_{14} \, \mathbf{a}_{2}+ y_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{14} - y_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 x_{14} - y_{14} - z_{14}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{84} & =&y_{14} \, \mathbf{a}_{1}+ z_{14} \, \mathbf{a}_{2}+ x_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{14} - x_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 z_{14} - x_{14} - y_{14}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{85} & =&- y_{14} \, \mathbf{a}_{1}- x_{14} \, \mathbf{a}_{2}- z_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{14} - y_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(y_{14} + z_{14} - 2 x_{14}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{86} & =&- x_{14} \, \mathbf{a}_{1}- z_{14} \, \mathbf{a}_{2}- y_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{14} - x_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{14} + y_{14} - 2 z_{14}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{87} & =&- z_{14} \, \mathbf{a}_{1}- y_{14} \, \mathbf{a}_{2}- x_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{14} - z_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{14} + z_{14} - 2 y_{14}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{88} & =&- x_{14} \, \mathbf{a}_{1}- y_{14} \, \mathbf{a}_{2}- z_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{14} - x_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{14} + z_{14} - 2 y_{14}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{89} & =&- z_{14} \, \mathbf{a}_{1}- x_{14} \, \mathbf{a}_{2}- y_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{14} - z_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(y_{14} + z_{14} - 2 x_{14}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{90} & =&- y_{14} \, \mathbf{a}_{1}- z_{14} \, \mathbf{a}_{2}- x_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{14} - y_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{14} + y_{14} - 2 z_{14}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{91} & =&y_{14} \, \mathbf{a}_{1}+ x_{14} \, \mathbf{a}_{2}+ z_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{14} - z_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 x_{14} - y_{14} - z_{14}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{92} & =&x_{14} \, \mathbf{a}_{1}+ z_{14} \, \mathbf{a}_{2}+ y_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{14} - y_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 z_{14} - x_{14} - y_{14}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{93} & =&z_{14} \, \mathbf{a}_{1}+ y_{14} \, \mathbf{a}_{2}+ x_{14} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{14} - x_{14}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 y_{14} - x_{14} - z_{14}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{14} + y_{14} + z_{14}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XIV} \\ \mathbf{B}_{94} & =&x_{15} \, \mathbf{a}_{1}+ y_{15} \, \mathbf{a}_{2}+ z_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{15} - z_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 y_{15} - x_{15} - z_{15}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{95} & =&z_{15} \, \mathbf{a}_{1}+ x_{15} \, \mathbf{a}_{2}+ y_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{15} - y_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 x_{15} - y_{15} - z_{15}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{96} & =&y_{15} \, \mathbf{a}_{1}+ z_{15} \, \mathbf{a}_{2}+ x_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{15} - x_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 z_{15} - x_{15} - y_{15}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{97} & =&- y_{15} \, \mathbf{a}_{1}- x_{15} \, \mathbf{a}_{2}- z_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{15} - y_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(y_{15} + z_{15} - 2 x_{15}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{98} & =&- x_{15} \, \mathbf{a}_{1}- z_{15} \, \mathbf{a}_{2}- y_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{15} - x_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{15} + y_{15} - 2 z_{15}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{99} & =&- z_{15} \, \mathbf{a}_{1}- y_{15} \, \mathbf{a}_{2}- x_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{15} - z_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{15} + z_{15} - 2 y_{15}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{100} & =&- x_{15} \, \mathbf{a}_{1}- y_{15} \, \mathbf{a}_{2}- z_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{15} - x_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{15} + z_{15} - 2 y_{15}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{101} & =&- z_{15} \, \mathbf{a}_{1}- x_{15} \, \mathbf{a}_{2}- y_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{15} - z_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(y_{15} + z_{15} - 2 x_{15}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{102} & =&- y_{15} \, \mathbf{a}_{1}- z_{15} \, \mathbf{a}_{2}- x_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{15} - y_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(x_{15} + y_{15} - 2 z_{15}\right) \, a \, \mathbf{\hat{y}}- \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{103} & =&y_{15} \, \mathbf{a}_{1}+ x_{15} \, \mathbf{a}_{2}+ z_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(y_{15} - z_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 x_{15} - y_{15} - z_{15}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{104} & =&x_{15} \, \mathbf{a}_{1}+ z_{15} \, \mathbf{a}_{2}+ y_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(x_{15} - y_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 z_{15} - x_{15} - y_{15}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \mathbf{B}_{105} & =&z_{15} \, \mathbf{a}_{1}+ y_{15} \, \mathbf{a}_{2}+ x_{15} \, \mathbf{a}_{3}& =&\frac12 \, \left(z_{15} - x_{15}\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, \left(2 y_{15} - x_{15} - z_{15}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \, \left(x_{15} + y_{15} + z_{15}\right) \, c \, \mathbf{\hat{z}}& \left(12i\right) & \mbox{B XV} \\ \end{array} \]

References

  • D. Geist, R. Kloss, and H. Follner, Verfeinerung des beta–rhomboedrischen Bors, Acta Crystallogr. Sect. B Struct. Sci. 26, 1800–1802 (1970), doi:10.1107/S0567740870004910.

Found in

  • J. Donohue, The Structure of the Elements (Robert E. Krieger Publishing Company, Malabar, Florida, 1982)., pp. 61-78.

Geometry files


Prototype Generator

aflow --proto=A_hR105_166_bc9h4i --params=

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