Corundum (Al2O3, $D5_{1}$) Structure: A2B3_hR10_167_c_e

Picture of Structure; Click for Big Picture
Prototype : Al2O3
AFLOW prototype label : A2B3_hR10_167_c_e
Strukturbericht designation : $D5_{1}$
Pearson symbol : hR10
Space group number : 167
Space group symbol : $\mbox{R}\bar{3}\mbox{c}$
AFLOW prototype command : aflow --proto=A2B3_hR10_167_c_e [--hex]
--params=
$a$,$c/a$,$x_{1}$,$x_{2}$


Other compounds with this structure

  • Cr2O3 (eskolaite), Fe2O3 (hematite), (Fe,Ti)2O3 (ilmenite), Ti2O3 (tistarite), V2O3 (karelianite)

  • The aluminum atoms can be replaced by two different species of atoms stacked in alternating layers along the $c$-axis, forming the ilmenite structure. Hexagonal settings of this structure can be obtained with the option ––hex. Alloying with Fe and Ti produces sapphire (a blue crystal), and alloying with Cr produces ruby (a red crystal).

Rhombohedral primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac1{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac1{\sqrt3} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & - \frac12 \, a \, \mathbf{\hat{x}} - \frac1{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = &x_{1} \, \mathbf{a}_{1}+ x_{1} \, \mathbf{a}_{2}+ x_{1} \, \mathbf{a}_{3}& =& x_{1} c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Al} \\ \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(4c\right) & \mbox{Al} \\ \mathbf{B}_{3} & = &- x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}& = &- x_{1} c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Al} \\ \mathbf{B}_{4} & = &\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(4c\right) & \mbox{Al} \\ \mathbf{B}_{5} & = &x_{2} \, \mathbf{a}_{1}+ \left(\frac12 - x_{2}\right) \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& = &\frac18 \left(4 \, x_{2} - 1\right) \, a \, \mathbf{\hat{x}}+ \frac{\sqrt{3}}{8} \left(1 - 4 x_{2}\right) \, a \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B}_{6} & = &\frac14 \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}+ \left(\frac12 - x_{2}\right) \, \mathbf{a}_{3}& = &\frac18 \left(4 x_{2} - 1\right) \, a \, \mathbf{\hat{x}}- \frac{\sqrt{3}}{8} \left(1 - 4 x_{2}\right) \, a \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B}_{7} & = &\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ x_{2} \, \mathbf{a}_{3}& = &- \frac14 \left(4 x_{2} - 1\right) \, a \, \mathbf{\hat{x}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B}_{8} & = &- x_{2} \, \mathbf{a}_{1}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& = &- \frac18 \left(4 x_{2} + 3\right) \, a \, \mathbf{\hat{x}}+ \frac1{8\sqrt3} \left(1 + 12 x_{2}\right) \, a \mathbf{\hat{y}}+ \frac5{12} \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B}_{9} & = &\frac34 \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{3}& = &- \frac18 \left(4 x_{2} - 1\right) \, a \, \mathbf{\hat{x}}- \frac1{8\sqrt3} \left(5 + 12 x_{2}\right) \, a \mathbf{\hat{y}}+ \frac5{12} \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \mathbf{B_{{10}}} & = &\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}& = &\frac14 \left(4 x_{2} + 1\right) \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, a \mathbf{\hat{y}}+ \frac5{12} \, c \, \mathbf{\hat{z}}& \left(6e\right) & \mbox{O} \\ \end{array} \]

References

  • L. W. Finger and R. M. Hazen, Crystal structure and compression of ruby to 46 kbar, J. Appl. Phys. 49, 5823–5826 (1978), doi:10.1063/1.324598.

Geometry files


Prototype Generator

aflow --proto=A2B3_hR10_167_c_e --params=

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