AFLOW Prototype: A8B5_hR26_160_a3bc_a3b
Prototype | : | Cr5Al8 |
AFLOW prototype label | : | A8B5_hR26_160_a3bc_a3b |
Strukturbericht designation | : | $D8_{10}$ |
Pearson symbol | : | hR26 |
Space group number | : | 160 |
Space group symbol | : | $R3m$ |
AFLOW prototype command | : | aflow --proto=A8B5_hR26_160_a3bc_a3b [--hex] --params=$a$,$c/a$,$x_{1}$,$x_{2}$,$x_{3}$,$z_{3}$,$x_{4}$,$z_{4}$,$x_{5}$,$z_{5}$,$x_{6}$,$z_{6}$,$x_{7}$,$z_{7}$,$x_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$ |
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} + x_{1} \, \mathbf{a}_{2} + x_{1} \, \mathbf{a}_{3} & = & x_{1}c \, \mathbf{\hat{z}} & \left(1a\right) & \text{Al 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(1a\right) & \text{Cr I} \\ \mathbf{B}_{3} & = & x_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Al II} \\ \mathbf{B}_{4} & = & z_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Al II} \\ \mathbf{B}_{5} & = & x_{3} \, \mathbf{a}_{1} + z_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Al II} \\ \mathbf{B}_{6} & = & x_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{4}-z_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{4}-z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{4}+\frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Al III} \\ \mathbf{B}_{7} & = & z_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{4}+z_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{4}-z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{4}+\frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Al III} \\ \mathbf{B}_{8} & = & x_{4} \, \mathbf{a}_{1} + z_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{4}+z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{4}+\frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Al III} \\ \mathbf{B}_{9} & = & x_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}-z_{5}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{5}-z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{5}+\frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Al IV} \\ \mathbf{B}_{10} & = & z_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{5}+z_{5}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{5}-z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{5}+\frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Al IV} \\ \mathbf{B}_{11} & = & x_{5} \, \mathbf{a}_{1} + z_{5} \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{5}+z_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{5}+\frac{1}{3}z_{5}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Al IV} \\ \mathbf{B}_{12} & = & x_{6} \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{6}-z_{6}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{6}-z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{6}+\frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Cr II} \\ \mathbf{B}_{13} & = & z_{6} \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + x_{6} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{6}+z_{6}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{6}-z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{6}+\frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Cr II} \\ \mathbf{B}_{14} & = & x_{6} \, \mathbf{a}_{1} + z_{6} \, \mathbf{a}_{2} + x_{6} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{6}+z_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{6}+\frac{1}{3}z_{6}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Cr II} \\ \mathbf{B}_{15} & = & x_{7} \, \mathbf{a}_{1} + x_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{7}+\frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Cr III} \\ \mathbf{B}_{16} & = & z_{7} \, \mathbf{a}_{1} + x_{7} \, \mathbf{a}_{2} + x_{7} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{7}-z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{7}+\frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Cr III} \\ \mathbf{B}_{17} & = & x_{7} \, \mathbf{a}_{1} + z_{7} \, \mathbf{a}_{2} + x_{7} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{7}+z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{7}+\frac{1}{3}z_{7}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Cr III} \\ \mathbf{B}_{18} & = & x_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{8}+\frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Cr IV} \\ \mathbf{B}_{19} & = & z_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + x_{8} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{8}+z_{8}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{8}-z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{8}+\frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Cr IV} \\ \mathbf{B}_{20} & = & x_{8} \, \mathbf{a}_{1} + z_{8} \, \mathbf{a}_{2} + x_{8} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{8}+z_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{8}+\frac{1}{3}z_{8}\right)c \, \mathbf{\hat{z}} & \left(3b\right) & \text{Cr IV} \\ \mathbf{B}_{21} & = & x_{9} \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{9}-z_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{9}+\frac{1}{\sqrt{3}}y_{9}-\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \text{Al V} \\ \mathbf{B}_{22} & = & z_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + y_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-y_{9}+z_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{9}-\frac{1}{2\sqrt{3}}y_{9}-\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \text{Al V} \\ \mathbf{B}_{23} & = & y_{9} \, \mathbf{a}_{1} + z_{9} \, \mathbf{a}_{2} + x_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{9}+y_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{9}-\frac{1}{2\sqrt{3}}y_{9}+\frac{1}{\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \text{Al V} \\ \mathbf{B}_{24} & = & z_{9} \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + x_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{9}+z_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{9}+\frac{1}{\sqrt{3}}y_{9}-\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \text{Al V} \\ \mathbf{B}_{25} & = & y_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(y_{9}-z_{9}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{\sqrt{3}}x_{9}-\frac{1}{2\sqrt{3}}y_{9}-\frac{1}{2\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \text{Al V} \\ \mathbf{B}_{26} & = & x_{9} \, \mathbf{a}_{1} + z_{9} \, \mathbf{a}_{2} + y_{9} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{9}-y_{9}\right)a \, \mathbf{\hat{x}} + \left(-\frac{1}{2\sqrt{3}}x_{9}-\frac{1}{2\sqrt{3}}y_{9}+\frac{1}{\sqrt{3}}z_{9}\right)a \, \mathbf{\hat{y}} + \frac{1}{3}\left(x_{9}+y_{9}+z_{9}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \text{Al V} \\ \end{array} \]