Low Temperature GdBO$_{3}$ Structure: ABC3_hR30_155_de_f_de2f-001
| Prototype | BGdO$_{3}$ |
| AFLOW prototype label | ABC3_hR30_155_de_f_de2f-001 |
| ICSD | 87778 |
| Pearson symbol | hR30 |
| Space group number | 155 |
| Space group symbol | $R32$ |
| AFLOW prototype command |
aflow --proto=ABC3_hR30_155_de_f_de2f-001
--params=$a, \allowbreak c/a, \allowbreak y_{1}, \allowbreak y_{2}, \allowbreak y_{3}, \allowbreak y_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}$ |
- (Ren, 1999) found two structures for GdBO$_{3}$: this low-temperature rhombohedral structure, and a high-temperature hexagonal structure.
- There is large thermal hysteresis in this system, with the LT $\rightarrow$ HT transition taking place at 1109K and the HT $\rightarrow$ LT transition at 819K.
- Hexagonal settings of this structure 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 | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $y_{1} \, \mathbf{a}_{2}- y_{1} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a y_{1} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{1} \,\mathbf{\hat{y}}$ | (3d) | B I |
| $\mathbf{B_{2}}$ | = | $- y_{1} \, \mathbf{a}_{1}+y_{1} \, \mathbf{a}_{3}$ | = | $- a y_{1} \,\mathbf{\hat{x}}$ | (3d) | B I |
| $\mathbf{B_{3}}$ | = | $y_{1} \, \mathbf{a}_{1}- y_{1} \, \mathbf{a}_{2}$ | = | $\frac{1}{2}a y_{1} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{1} \,\mathbf{\hat{y}}$ | (3d) | B I |
| $\mathbf{B_{4}}$ | = | $y_{2} \, \mathbf{a}_{2}- y_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a y_{2} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{2} \,\mathbf{\hat{y}}$ | (3d) | O I |
| $\mathbf{B_{5}}$ | = | $- y_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{3}$ | = | $- a y_{2} \,\mathbf{\hat{x}}$ | (3d) | O I |
| $\mathbf{B_{6}}$ | = | $y_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}$ | = | $\frac{1}{2}a y_{2} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{2} \,\mathbf{\hat{y}}$ | (3d) | O I |
| $\mathbf{B_{7}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}- y_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \left(2 y_{3} + 1\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{12}a \left(6 y_{3} - 1\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3e) | B II |
| $\mathbf{B_{8}}$ | = | $- y_{3} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+y_{3} \, \mathbf{a}_{3}$ | = | $- a y_{3} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3e) | B II |
| $\mathbf{B_{9}}$ | = | $y_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \left(2 y_{3} - 1\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{12}a \left(6 y_{3} + 1\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3e) | B II |
| $\mathbf{B_{10}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}- y_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \left(2 y_{4} + 1\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{12}a \left(6 y_{4} - 1\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3e) | O II |
| $\mathbf{B_{11}}$ | = | $- y_{4} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+y_{4} \, \mathbf{a}_{3}$ | = | $- a y_{4} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3e) | O II |
| $\mathbf{B_{12}}$ | = | $y_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \left(2 y_{4} - 1\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{12}a \left(6 y_{4} + 1\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3e) | O II |
| $\mathbf{B_{13}}$ | = | $x_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} - 2 y_{5} + z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Gd I |
| $\mathbf{B_{14}}$ | = | $z_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+y_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{5} - z_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{5} - y_{5} - z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Gd I |
| $\mathbf{B_{15}}$ | = | $y_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} + y_{5} - 2 z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Gd I |
| $\mathbf{B_{16}}$ | = | $- z_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{5} - 2 y_{5} + z_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Gd I |
| $\mathbf{B_{17}}$ | = | $- y_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{5} - z_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{5} - y_{5} - z_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Gd I |
| $\mathbf{B_{18}}$ | = | $- x_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- y_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{5} + y_{5} - 2 z_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Gd I |
| $\mathbf{B_{19}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | O III |
| $\mathbf{B_{20}}$ | = | $z_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+y_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | O III |
| $\mathbf{B_{21}}$ | = | $y_{6} \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | O III |
| $\mathbf{B_{22}}$ | = | $- z_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | O III |
| $\mathbf{B_{23}}$ | = | $- y_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | O III |
| $\mathbf{B_{24}}$ | = | $- x_{6} \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}- y_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | O III |
| $\mathbf{B_{25}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - z_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} - 2 y_{7} + z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (6f) | O IV |
| $\mathbf{B_{26}}$ | = | $z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{7} - z_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{7} - y_{7} - z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (6f) | O IV |
| $\mathbf{B_{27}}$ | = | $y_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} + y_{7} - 2 z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (6f) | O IV |
| $\mathbf{B_{28}}$ | = | $- z_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - z_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{7} - 2 y_{7} + z_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (6f) | O IV |
| $\mathbf{B_{29}}$ | = | $- y_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{7} - z_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{7} - y_{7} - z_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (6f) | O IV |
| $\mathbf{B_{30}}$ | = | $- x_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{7} + y_{7} - 2 z_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (6f) | O IV |
References
- M. Ren, J. H. Lin, Y. Dong, L. Q. Yang, M. Z. Su, and L. P. You, Structure and Phase Transition of GdBO$_{3}$, Chem. Mater. 11, 1576–1580 (1999), doi:10.1021/cm990022o.
Prototype Generator
aflow --proto=ABC3_hR30_155_de_f_de2f --params=$a,c/a,y_{1},y_{2},y_{3},y_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7}$