AFLOW Prototype: ABC2_hR24_167_e_e_2e
Prototype | : | KBO2 |
AFLOW prototype label | : | ABC2_hR24_167_e_e_2e |
Strukturbericht designation | : | $F5_{13}$ |
Pearson symbol | : | hR24 |
Space group number | : | 167 |
Space group symbol | : | $R\bar{3}c$ |
AFLOW prototype command | : | aflow --proto=ABC2_hR24_167_e_e_2e [--hex] --params=$a$,$c/a$,$x_{1}$,$x_{2}$,$x_{3}$,$x_{4}$ |
Basis vectors:
\[ \begin{array}{ccccccc} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1} & = & x_{1} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{2}x_{1}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{B} \\ \mathbf{B}_{2} & = & \frac{1}{4} \, \mathbf{a}_{1} + x_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{1}\right)a \, \mathbf{\hat{x}} + \left(- \frac{\sqrt{3}}{8} +\frac{\sqrt{3}}{2}x_{1}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{B} \\ \mathbf{B}_{3} & = & \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + x_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{1}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{B} \\ \mathbf{B}_{4} & = & -x_{1} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & -a\left(\frac{1}{2}x_{1}+\frac{3}{8}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{8\sqrt{3}} +\frac{\sqrt{3}}{2}x_{1}\right)a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{B} \\ \mathbf{B}_{5} & = & \frac{3}{4} \, \mathbf{a}_{1}-x_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{8} - \frac{1}{2}x_{1}\right)a \, \mathbf{\hat{x}}-a\left(\frac{\sqrt{3}}{2}x_{1}+\frac{5}{8\sqrt{3}}\right) \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{B} \\ \mathbf{B}_{6} & = & \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-x_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{1}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{B} \\ \mathbf{B}_{7} & = & x_{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{2}x_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{K} \\ \mathbf{B}_{8} & = & \frac{1}{4} \, \mathbf{a}_{1} + x_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{2}\right)a \, \mathbf{\hat{x}} + \left(- \frac{\sqrt{3}}{8} +\frac{\sqrt{3}}{2}x_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{K} \\ \mathbf{B}_{9} & = & \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + x_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{2}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{K} \\ \mathbf{B}_{10} & = & -x_{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & -a\left(\frac{1}{2}x_{2}+\frac{3}{8}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{8\sqrt{3}} +\frac{\sqrt{3}}{2}x_{2}\right)a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{K} \\ \mathbf{B}_{11} & = & \frac{3}{4} \, \mathbf{a}_{1}-x_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{8} - \frac{1}{2}x_{2}\right)a \, \mathbf{\hat{x}}-a\left(\frac{\sqrt{3}}{2}x_{2}+\frac{5}{8\sqrt{3}}\right) \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{K} \\ \mathbf{B}_{12} & = & \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-x_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{K} \\ \mathbf{B}_{13} & = & x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{2}x_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O I} \\ \mathbf{B}_{14} & = & \frac{1}{4} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{3}\right)a \, \mathbf{\hat{x}} + \left(- \frac{\sqrt{3}}{8} +\frac{\sqrt{3}}{2}x_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O I} \\ \mathbf{B}_{15} & = & \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O I} \\ \mathbf{B}_{16} & = & -x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & -a\left(\frac{1}{2}x_{3}+\frac{3}{8}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{8\sqrt{3}} +\frac{\sqrt{3}}{2}x_{3}\right)a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O I} \\ \mathbf{B}_{17} & = & \frac{3}{4} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{8} - \frac{1}{2}x_{3}\right)a \, \mathbf{\hat{x}}-a\left(\frac{\sqrt{3}}{2}x_{3}+\frac{5}{8\sqrt{3}}\right) \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O I} \\ \mathbf{B}_{18} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O I} \\ \mathbf{B}_{19} & = & x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{2}x_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O II} \\ \mathbf{B}_{20} & = & \frac{1}{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{8} +\frac{1}{2}x_{4}\right)a \, \mathbf{\hat{x}} + \left(- \frac{\sqrt{3}}{8} +\frac{\sqrt{3}}{2}x_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O II} \\ \mathbf{B}_{21} & = & \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - x_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O II} \\ \mathbf{B}_{22} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & -a\left(\frac{1}{2}x_{4}+\frac{3}{8}\right) \, \mathbf{\hat{x}} + \left(\frac{1}{8\sqrt{3}} +\frac{\sqrt{3}}{2}x_{4}\right)a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O II} \\ \mathbf{B}_{23} & = & \frac{3}{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{8} - \frac{1}{2}x_{4}\right)a \, \mathbf{\hat{x}}-a\left(\frac{\sqrt{3}}{2}x_{4}+\frac{5}{8\sqrt{3}}\right) \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O II} \\ \mathbf{B}_{24} & = & \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}a \, \mathbf{\hat{y}} + \frac{5}{12}c \, \mathbf{\hat{z}} & \left(6e\right) & \text{O II} \\ \end{array} \]