α-B (R-12) Structure: A_hR12_166_2h-001
| Prototype | B |
| AFLOW prototype label | A_hR12_166_2h-001 |
| ICSD | 26487 |
| Pearson symbol | hR12 |
| Space group number | 166 |
| Space group symbol | $R\overline{3}m$ |
| AFLOW prototype command |
aflow --proto=A_hR12_166_2h-001
--params=$a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak z_{1}, \allowbreak x_{2}, \allowbreak z_{2}$ |
- This is a metastable phase of boron, also known as rhombohedral-12 boron (Donohue, 1982). It is the simplest known phase (the ground state, $\beta$–B, has 105 atoms in the primitive cell).
- Note the relationship between the icosahedra in this structure, $\beta$–B and T-50 B.
- Hexagonal settings for 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 | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{1} + z_{1}\right) \,\mathbf{\hat{z}}$ | (6h) | B I |
| $\mathbf{B_{2}}$ | = | $z_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{1} + z_{1}\right) \,\mathbf{\hat{z}}$ | (6h) | B I |
| $\mathbf{B_{3}}$ | = | $x_{1} \, \mathbf{a}_{1}+z_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $- \frac{1}{\sqrt{3}}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{1} + z_{1}\right) \,\mathbf{\hat{z}}$ | (6h) | B I |
| $\mathbf{B_{4}}$ | = | $- z_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{1} + z_{1}\right) \,\mathbf{\hat{z}}$ | (6h) | B I |
| $\mathbf{B_{5}}$ | = | $- x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}- z_{1} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{1} + z_{1}\right) \,\mathbf{\hat{z}}$ | (6h) | B I |
| $\mathbf{B_{6}}$ | = | $- x_{1} \, \mathbf{a}_{1}- z_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $\frac{1}{\sqrt{3}}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{1} + z_{1}\right) \,\mathbf{\hat{z}}$ | (6h) | B I |
| $\mathbf{B_{7}}$ | = | $x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{2} + z_{2}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{8}}$ | = | $z_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{2} + z_{2}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{9}}$ | = | $x_{2} \, \mathbf{a}_{1}+z_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $- \frac{1}{\sqrt{3}}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{2} + z_{2}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{10}}$ | = | $- z_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{2} + z_{2}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{11}}$ | = | $- x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{2} + z_{2}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{12}}$ | = | $- x_{2} \, \mathbf{a}_{1}- z_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{\sqrt{3}}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{2} + z_{2}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
References
- B. F. Decker and J. S. Kasper, The crystal structure of a simple rhombohedral form of boron, Acta Cryst. 12, 503–506 (1959), doi:10.1107/S0365110X59001529.
Found in
- J. Donohue, The Structures of the Elements (Robert E. Krieger Publishing Company, New York, 1974).
Prototype Generator
aflow --proto=A_hR12_166_2h --params=$a,c/a,x_{1},z_{1},x_{2},z_{2}$