Corundum (α-alumina, Al$_{2}$O$_{3}$, $D5_{1}$) Structure: A2B3_hR10_167_c_e-001
| Prototype | Al$_{2}$O$_{3}$ |
| AFLOW prototype label | A2B3_hR10_167_c_e-001 |
| Strukturbericht designation | $D5_{1}$ |
| Mineral name | corundum |
| ICSD | 9770 |
| Pearson symbol | hR10 |
| Space group number | 167 |
| Space group symbol | $R\overline{3}c$ |
| AFLOW prototype command |
aflow --proto=A2B3_hR10_167_c_e-001
--params=$a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak x_{2}$ |
Other compounds with this structure
$\gamma$-Al$_{2}$S$_{3}$, Cr$_{2}$O$_{3}$ (Eskolaite), Fe$_{2}$O$_{3}$ (Hematite), $\alpha$-Ga$_{2}$O$_{3}$, Lu$_{2}$S$_{3}$, Rh$_{2}$O$_{3}$, Ti$_{2}$O$_{3}$ (Tistarite), V$_{2}$O$_{3}$ (Karelianite), Yb$_{2}$S$_{3}$
- Alumina comes in a variety of forms. In the Encyclopedia we have:
- Corundum, or $\alpha$-alumina ($D5_{1}$) (this structure) is the mineral usual found in nature.
- $\beta$-alumina ($D5_{6}$)
- We describe $\gamma$-alumina ($D5_{7}$) using Fe$_{2}$O$_{3}$ as the prototype.
- $\delta$-alumina is a tetragonal distortion of the spinel structure. It is found in nature as deltalumite.
- $\kappa$–Al$_{2}$O$_{3}$.
- In corundum the aluminum atoms can be replaced by two different species of atoms stacked in alternating layers along the $c$-axis, forming the ilmenite structure.
- Alloying with Fe and Ti produces sapphire (a blue crystal), and alloying with Cr produces ruby (a red crystal).
- Hexagonal settings of rhombohedral structures can be obtained with the option
--hex.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{\sqrt{3}}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $c x_{1} \,\mathbf{\hat{z}}$ | (4c) | Al I |
| $\mathbf{B_{2}}$ | = | $- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- c \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | Al I |
| $\mathbf{B_{3}}$ | = | $- x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $- c x_{1} \,\mathbf{\hat{z}}$ | (4c) | Al I |
| $\mathbf{B_{4}}$ | = | $\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $c \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | Al I |
| $\mathbf{B_{5}}$ | = | $x_{2} \, \mathbf{a}_{1}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \left(4 x_{2} - 1\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{8}a \left(4 x_{2} - 1\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (6e) | O I |
| $\mathbf{B_{6}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \left(4 x_{2} - 1\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{8}a \left(4 x_{2} - 1\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (6e) | O I |
| $\mathbf{B_{7}}$ | = | $- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $- a \left(x_{2} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (6e) | O I |
| $\mathbf{B_{8}}$ | = | $- x_{2} \, \mathbf{a}_{1}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{8}a \left(4 x_{2} + 3\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{24}a \left(12 x_{2} + 1\right) \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ | (6e) | O I |
| $\mathbf{B_{9}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{8}a \left(4 x_{2} - 1\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{24}a \left(12 x_{2} + 5\right) \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ | (6e) | O I |
| $\mathbf{B_{10}}$ | = | $\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $a \left(x_{2} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ | (6e) | O I |
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.
Prototype Generator
aflow --proto=A2B3_hR10_167_c_e --params=$a,c/a,x_{1},x_{2}$