AFLOW Prototype: ABC3_hR10_161_a_a_b
Prototype | : | LiNbO3 |
AFLOW prototype label | : | ABC3_hR10_161_a_a_b |
Strukturbericht designation | : | None |
Pearson symbol | : | hR10 |
Space group number | : | 161 |
Space group symbol | : | $\text{R}3\text{c}$ |
AFLOW prototype command | : | aflow --proto=ABC3_hR10_161_a_a_b [--hex] --params=$a$,$c/a$,$x_{1}$,$x_{2}$,$x_{3}$,$y_{3}$,$z_{3}$ |
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(2a\right) & \text{Li} \\ \mathbf{B}_{2} & =&\left(\frac12 + x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{1}\right) \, c \, \mathbf{\hat{z}}& \left(2a\right) & \text{Li} \\ \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(2a\right) & \text{Nb} \\ \mathbf{B}_{4} & =&\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{2}\right) \, c \, \mathbf{\hat{z}}& \left(2a\right) & \text{Nb} \\ \mathbf{B}_{5} & =&x_{3} \, \mathbf{a}_{1}+ y_{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} - 2 y_{3} + z_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \left(x_{3} + y_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \text{O} \\ \mathbf{B}_{6} & =&z_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}+ y_{3} \, \mathbf{a}_{3}& =&\frac12 \left(z_{3} - y_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(z_{3} - 2 x_{3} + y_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \left(x_{3} + y_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \text{O} \\ \mathbf{B}_{7} & =&y_{3} \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& =&\frac12 \left(y_{3} - x_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(y_{3} - 2 z_{3} + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac13 \left(x_{3} + y_{3} + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \text{O} \\ \mathbf{B}_{8} & =&\left(\frac12 + y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =&\frac12 \left(y_{3} - z_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(z_{3} - 2 x_{3} + y_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac16 \left(3 + 2 x_{3} + 2 y_{3} + 2 z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \text{O} \\ \mathbf{B}_{9} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{3}& =&\frac12 \left(x_{3} - y_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(y_{3} - 2 z_{3} + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac16 \left(3 + 2 x_{3} + 2 y_{3} + 2 z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \text{O} \\ \mathbf{B}_{10} & =&\left(\frac12 + z_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{3}& =&\frac12 \left(z_{3} - x_{3}\right) \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \left(x_{3} - 2 y_{3} + z_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac16 \left(3 + 2 x_{3} + 2 y_{3} + 2 z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(6b\right) & \text{O} \\ \end{array} \]