Rhombohedral BaTiO$_{3}$ Structure: AB3C_hR5_160_a_b_a-001
| Prototype | BaO$_{3}$Ti |
| AFLOW prototype label | AB3C_hR5_160_a_b_a-001 |
| ICSD | 6102 |
| Pearson symbol | hR5 |
| Space group number | 160 |
| Space group symbol | $R3m$ |
| AFLOW prototype command |
aflow --proto=AB3C_hR5_160_a_b_a-001
--params=$a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak x_{2}, \allowbreak x_{3}, \allowbreak z_{3}$ |
- The perovskite BaTiO$_{3}$ undergoes a variety of temperature driven phase transitions. (Shirane, 1957)
- The first three structures are ferroelectric:
- Below 193K the structure is rhombohedral. (This structure)
- Between 193K and 278K the structure is orthorhombic.
- Between 278K and 393K the structure is tetragonal. This is the room-temperature form of the material.
- Above 393K the compound is a cubic perovskite ($E2_{1}$).
- Hexagonal BaTiO$_{3}$ can be stabilized by alloying the titanium sites with other transition metals. (Dickson, 1961) The pure structure has been grown at 1853K and cooled to room temperature. (Akimo, 1994)
- The data was taken at 77.4K.
- Rhombohedral BaTiO$_{3}$ is isostructural with $\gamma$–KNO$_{3}$ and KBrO$_{3}$ ($G0_{7}$), but the structural parameters are sufficiently different to warrant adding another structure to the database.
- Space group $R3m$ #160 does not specify the origin of the $z$-axis. We follow (Hewat, 1974) and place the origin so that the titanium atom is at $1/2 c \hat{z}$.
- 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 | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $c x_{1} \,\mathbf{\hat{z}}$ | (1a) | Ba I |
| $\mathbf{B_{2}}$ | = | $x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $c x_{2} \,\mathbf{\hat{z}}$ | (1a) | Ti I |
| $\mathbf{B_{3}}$ | = | $x_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (3b) | O I |
| $\mathbf{B_{4}}$ | = | $z_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (3b) | O I |
| $\mathbf{B_{5}}$ | = | $x_{3} \, \mathbf{a}_{1}+z_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{\sqrt{3}}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (3b) | O I |
References
- G. Shirane, H. Danner, and R. Pepinsky, Neutron Diffraction Study of Orthorhombic BaTiO$_{3}$, Phys. Rev. 105, 856–860 (1957), doi:10.1103/PhysRev.105.856.
- J. G. Dickson, L. Katz, and R. Ward, Compounds with the Hexagonal Barium Titanate Structure, J. Am. Chem. Soc. 83, 3026–3029 (1961), doi:10.1021/ja01475a012.
- A. W. Hewat, Structure of rhombohedral ferroelectric barium titanate, Ferroelectrics 6, 215–218 (1974), doi:10.1080/00150197408243970.
- J. Akimoto, Y. Gotoh, and Y. Oosawa, Refinement of Hexagonal BaTiO$_{3}$, Acta Crystallogr. Sect. C 50, 160–161 (1994), doi:10.1107/S0108270193008637.
Prototype Generator
aflow --proto=AB3C_hR5_160_a_b_a --params=$a,c/a,x_{1},x_{2},x_{3},z_{3}$