Trigonal B$_{2}$O$_{3}$ Structure: A2B3_hP15_152_c_ac-001
| Prototype | B$_{2}$O$_{3}$ |
| AFLOW prototype label | A2B3_hP15_152_c_ac-001 |
| ICSD | 51575 |
| Pearson symbol | hP15 |
| Space group number | 152 |
| Space group symbol | $P3_121$ |
| AFLOW prototype command |
aflow --proto=A2B3_hP15_152_c_ac-001
--params=$a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak x_{2}, \allowbreak y_{2}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}$ |
- This is the ground state structure of B$_{2}$O$_{3}$. At approximately 2 GPa it transforms to a high-pressure orthorhombic phase.
- Originally this phase was thought to be in space group $P3_{1}$ #144, but reanalysis by (Effenberger, 2001) showed that the higher symmetry group matches that data at least as well. For more history on the determination of the structure of B$_{2}$O$_{3}$, see (Prewitt, 1968).
- Space group $P3_{2}21$ #154 is the enantiomorph of $P3_{1}21$ #152, so this structure could exist in that space group as well.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a x_{1} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{1} \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}$ | (3a) | O I |
| $\mathbf{B_{2}}$ | = | $x_{1} \, \mathbf{a}_{2}+\frac{2}{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a x_{1} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{1} \,\mathbf{\hat{y}}+\frac{2}{3}c \,\mathbf{\hat{z}}$ | (3a) | O I |
| $\mathbf{B_{3}}$ | = | $- x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}$ | = | $- a x_{1} \,\mathbf{\hat{x}}$ | (3a) | O I |
| $\mathbf{B_{4}}$ | = | $x_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} + y_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{2} - y_{2}\right) \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (6c) | B I |
| $\mathbf{B_{5}}$ | = | $- y_{2} \, \mathbf{a}_{1}+\left(x_{2} - y_{2}\right) \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} - 2 y_{2}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | B I |
| $\mathbf{B_{6}}$ | = | $- \left(x_{2} - y_{2}\right) \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\left(z_{2} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{2} - y_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{2} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{2} + 2\right) \,\mathbf{\hat{z}}$ | (6c) | B I |
| $\mathbf{B_{7}}$ | = | $y_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} + y_{2}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{2} - y_{2}\right) \,\mathbf{\hat{y}}- c z_{2} \,\mathbf{\hat{z}}$ | (6c) | B I |
| $\mathbf{B_{8}}$ | = | $\left(x_{2} - y_{2}\right) \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}- \left(z_{2} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} - 2 y_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{2} - 2\right) \,\mathbf{\hat{z}}$ | (6c) | B I |
| $\mathbf{B_{9}}$ | = | $- x_{2} \, \mathbf{a}_{1}- \left(x_{2} - y_{2}\right) \, \mathbf{a}_{2}- \left(z_{2} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{2} - y_{2}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{2} \,\mathbf{\hat{y}}- c \left(z_{2} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | B I |
| $\mathbf{B_{10}}$ | = | $x_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} + y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (6c) | O II |
| $\mathbf{B_{11}}$ | = | $- y_{3} \, \mathbf{a}_{1}+\left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}+\left(z_{3} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - 2 y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{3} \,\mathbf{\hat{y}}+c \left(z_{3} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | O II |
| $\mathbf{B_{12}}$ | = | $- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+\left(z_{3} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{3} - y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{3} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{3} + 2\right) \,\mathbf{\hat{z}}$ | (6c) | O II |
| $\mathbf{B_{13}}$ | = | $y_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} + y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{y}}- c z_{3} \,\mathbf{\hat{z}}$ | (6c) | O II |
| $\mathbf{B_{14}}$ | = | $\left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}- \left(z_{3} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - 2 y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{3} \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{3} - 2\right) \,\mathbf{\hat{z}}$ | (6c) | O II |
| $\mathbf{B_{15}}$ | = | $- x_{3} \, \mathbf{a}_{1}- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}- \left(z_{3} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{3} - y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{3} \,\mathbf{\hat{y}}- c \left(z_{3} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | O II |
References
- H. Effenberger, C. L. Lengauer, and E. Parthé, Trigonal B$_{2}$O$_{3}$ with Higher Space-Group Symmetry: Results of a Reevaluation, Monatsh. Chem. 132, 1515–1517 (2001), doi:10.1007/s007060170008.
- C. T. Prewitt and R. D. Shannon, Crystal Structure of a High-Pressure Form of B$_{2}$O$_{3}$, Acta Crystallogr. Sect. B 24, 869–874 (1968), doi:10.1107/S0567740868003304.
Found in
- A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, G. D. Ceder, and K. A. Persson, {Commentary: The Materials Project: A materials genome approach to accelerating materials innovation}, APL Materials 1, 011002 (2013), doi:10.1063/1.4812323.
Prototype Generator
aflow --proto=A2B3_hP15_152_c_ac --params=$a,c/a,x_{1},x_{2},y_{2},z_{2},x_{3},y_{3},z_{3}$