Bi$_{7}$(Fe,Ti)$_{6}$O$_{21}$ $m = 6$ Aurivillius Structure: A7B6C21_oF136_42_a3c_3c_ab3c3e-001
| Prototype | Bi$_{7}$Fe$_{3}$O$_{21}$ |
| AFLOW prototype label | A7B6C21_oF136_42_a3c_3c_ab3c3e-001 |
| ICSD | 155931 |
| Pearson symbol | oF136 |
| Space group number | 42 |
| Space group symbol | $Fmm2$ |
| AFLOW prototype command |
aflow --proto=A7B6C21_oF136_42_a3c_3c_ab3c3e-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak z_{1}, \allowbreak z_{2}, \allowbreak z_{3}, \allowbreak y_{4}, \allowbreak z_{4}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak y_{8}, \allowbreak z_{8}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak y_{12}, \allowbreak z_{12}, \allowbreak x_{13}, \allowbreak y_{13}, \allowbreak z_{13}, \allowbreak x_{14}, \allowbreak y_{14}, \allowbreak z_{14}, \allowbreak x_{15}, \allowbreak y_{15}, \allowbreak z_{15}$ |
- Aurivillius phases are layered tetragonal materials with composition (Me$'_{2}$O$_{2}$)$^{2+}$(Me$_{m-1}$R$_{m}$O$_{3m+1}$)$^{2-}$ (Me$_{m-1}$Me$'_{2}$R$_{m}$O$_{3(m+1)}$), where Me and Me' are metals and R is a transition metal with a charge of +4 or +5. (Subbaro, 1962).
- The iron and titanium atoms are randomly distributed across the transition metal (8c) sites. We have arbitrarily labeled them as iron.
- (Krzhizhanovskaya, 2005) gave the structure in the $F2mm$ setting of space group #42. We swapped the $y-$ and $z-$axes to transform this to the standard $Fmm2$ setting.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}b \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $z_{1} \, \mathbf{a}_{1}+z_{1} \, \mathbf{a}_{2}- z_{1} \, \mathbf{a}_{3}$ | = | $c z_{1} \,\mathbf{\hat{z}}$ | (4a) | Bi I |
| $\mathbf{B_{2}}$ | = | $z_{2} \, \mathbf{a}_{1}+z_{2} \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}$ | = | $c z_{2} \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{3}}$ | = | $z_{3} \, \mathbf{a}_{1}+z_{3} \, \mathbf{a}_{2}- \left(z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}b \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (8b) | O II |
| $\mathbf{B_{4}}$ | = | $\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}b \,\mathbf{\hat{y}}+c \left(z_{3} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (8b) | O II |
| $\mathbf{B_{5}}$ | = | $\left(y_{4} + z_{4}\right) \, \mathbf{a}_{1}- \left(y_{4} - z_{4}\right) \, \mathbf{a}_{2}+\left(y_{4} - z_{4}\right) \, \mathbf{a}_{3}$ | = | $b y_{4} \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (8c) | Bi II |
| $\mathbf{B_{6}}$ | = | $- \left(y_{4} - z_{4}\right) \, \mathbf{a}_{1}+\left(y_{4} + z_{4}\right) \, \mathbf{a}_{2}- \left(y_{4} + z_{4}\right) \, \mathbf{a}_{3}$ | = | $- b y_{4} \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (8c) | Bi II |
| $\mathbf{B_{7}}$ | = | $\left(y_{5} + z_{5}\right) \, \mathbf{a}_{1}- \left(y_{5} - z_{5}\right) \, \mathbf{a}_{2}+\left(y_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $b y_{5} \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (8c) | Bi III |
| $\mathbf{B_{8}}$ | = | $- \left(y_{5} - z_{5}\right) \, \mathbf{a}_{1}+\left(y_{5} + z_{5}\right) \, \mathbf{a}_{2}- \left(y_{5} + z_{5}\right) \, \mathbf{a}_{3}$ | = | $- b y_{5} \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (8c) | Bi III |
| $\mathbf{B_{9}}$ | = | $\left(y_{6} + z_{6}\right) \, \mathbf{a}_{1}- \left(y_{6} - z_{6}\right) \, \mathbf{a}_{2}+\left(y_{6} - z_{6}\right) \, \mathbf{a}_{3}$ | = | $b y_{6} \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (8c) | Bi IV |
| $\mathbf{B_{10}}$ | = | $- \left(y_{6} - z_{6}\right) \, \mathbf{a}_{1}+\left(y_{6} + z_{6}\right) \, \mathbf{a}_{2}- \left(y_{6} + z_{6}\right) \, \mathbf{a}_{3}$ | = | $- b y_{6} \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (8c) | Bi IV |
| $\mathbf{B_{11}}$ | = | $\left(y_{7} + z_{7}\right) \, \mathbf{a}_{1}- \left(y_{7} - z_{7}\right) \, \mathbf{a}_{2}+\left(y_{7} - z_{7}\right) \, \mathbf{a}_{3}$ | = | $b y_{7} \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (8c) | Fe I |
| $\mathbf{B_{12}}$ | = | $- \left(y_{7} - z_{7}\right) \, \mathbf{a}_{1}+\left(y_{7} + z_{7}\right) \, \mathbf{a}_{2}- \left(y_{7} + z_{7}\right) \, \mathbf{a}_{3}$ | = | $- b y_{7} \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (8c) | Fe I |
| $\mathbf{B_{13}}$ | = | $\left(y_{8} + z_{8}\right) \, \mathbf{a}_{1}- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{2}+\left(y_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $b y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (8c) | Fe II |
| $\mathbf{B_{14}}$ | = | $- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{1}+\left(y_{8} + z_{8}\right) \, \mathbf{a}_{2}- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $- b y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (8c) | Fe II |
| $\mathbf{B_{15}}$ | = | $\left(y_{9} + z_{9}\right) \, \mathbf{a}_{1}- \left(y_{9} - z_{9}\right) \, \mathbf{a}_{2}+\left(y_{9} - z_{9}\right) \, \mathbf{a}_{3}$ | = | $b y_{9} \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (8c) | Fe III |
| $\mathbf{B_{16}}$ | = | $- \left(y_{9} - z_{9}\right) \, \mathbf{a}_{1}+\left(y_{9} + z_{9}\right) \, \mathbf{a}_{2}- \left(y_{9} + z_{9}\right) \, \mathbf{a}_{3}$ | = | $- b y_{9} \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (8c) | Fe III |
| $\mathbf{B_{17}}$ | = | $\left(y_{10} + z_{10}\right) \, \mathbf{a}_{1}- \left(y_{10} - z_{10}\right) \, \mathbf{a}_{2}+\left(y_{10} - z_{10}\right) \, \mathbf{a}_{3}$ | = | $b y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (8c) | O III |
| $\mathbf{B_{18}}$ | = | $- \left(y_{10} - z_{10}\right) \, \mathbf{a}_{1}+\left(y_{10} + z_{10}\right) \, \mathbf{a}_{2}- \left(y_{10} + z_{10}\right) \, \mathbf{a}_{3}$ | = | $- b y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (8c) | O III |
| $\mathbf{B_{19}}$ | = | $\left(y_{11} + z_{11}\right) \, \mathbf{a}_{1}- \left(y_{11} - z_{11}\right) \, \mathbf{a}_{2}+\left(y_{11} - z_{11}\right) \, \mathbf{a}_{3}$ | = | $b y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (8c) | O IV |
| $\mathbf{B_{20}}$ | = | $- \left(y_{11} - z_{11}\right) \, \mathbf{a}_{1}+\left(y_{11} + z_{11}\right) \, \mathbf{a}_{2}- \left(y_{11} + z_{11}\right) \, \mathbf{a}_{3}$ | = | $- b y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (8c) | O IV |
| $\mathbf{B_{21}}$ | = | $\left(y_{12} + z_{12}\right) \, \mathbf{a}_{1}- \left(y_{12} - z_{12}\right) \, \mathbf{a}_{2}+\left(y_{12} - z_{12}\right) \, \mathbf{a}_{3}$ | = | $b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8c) | O V |
| $\mathbf{B_{22}}$ | = | $- \left(y_{12} - z_{12}\right) \, \mathbf{a}_{1}+\left(y_{12} + z_{12}\right) \, \mathbf{a}_{2}- \left(y_{12} + z_{12}\right) \, \mathbf{a}_{3}$ | = | $- b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8c) | O V |
| $\mathbf{B_{23}}$ | = | $\left(- x_{13} + y_{13} + z_{13}\right) \, \mathbf{a}_{1}+\left(x_{13} - y_{13} + z_{13}\right) \, \mathbf{a}_{2}+\left(x_{13} + y_{13} - z_{13}\right) \, \mathbf{a}_{3}$ | = | $a x_{13} \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (16e) | O VI |
| $\mathbf{B_{24}}$ | = | $\left(x_{13} - y_{13} + z_{13}\right) \, \mathbf{a}_{1}+\left(- x_{13} + y_{13} + z_{13}\right) \, \mathbf{a}_{2}- \left(x_{13} + y_{13} + z_{13}\right) \, \mathbf{a}_{3}$ | = | $- a x_{13} \,\mathbf{\hat{x}}- b y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (16e) | O VI |
| $\mathbf{B_{25}}$ | = | $- \left(x_{13} + y_{13} - z_{13}\right) \, \mathbf{a}_{1}+\left(x_{13} + y_{13} + z_{13}\right) \, \mathbf{a}_{2}+\left(x_{13} - y_{13} - z_{13}\right) \, \mathbf{a}_{3}$ | = | $a x_{13} \,\mathbf{\hat{x}}- b y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (16e) | O VI |
| $\mathbf{B_{26}}$ | = | $\left(x_{13} + y_{13} + z_{13}\right) \, \mathbf{a}_{1}- \left(x_{13} + y_{13} - z_{13}\right) \, \mathbf{a}_{2}- \left(x_{13} - y_{13} + z_{13}\right) \, \mathbf{a}_{3}$ | = | $- a x_{13} \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (16e) | O VI |
| $\mathbf{B_{27}}$ | = | $\left(- x_{14} + y_{14} + z_{14}\right) \, \mathbf{a}_{1}+\left(x_{14} - y_{14} + z_{14}\right) \, \mathbf{a}_{2}+\left(x_{14} + y_{14} - z_{14}\right) \, \mathbf{a}_{3}$ | = | $a x_{14} \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (16e) | O VII |
| $\mathbf{B_{28}}$ | = | $\left(x_{14} - y_{14} + z_{14}\right) \, \mathbf{a}_{1}+\left(- x_{14} + y_{14} + z_{14}\right) \, \mathbf{a}_{2}- \left(x_{14} + y_{14} + z_{14}\right) \, \mathbf{a}_{3}$ | = | $- a x_{14} \,\mathbf{\hat{x}}- b y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (16e) | O VII |
| $\mathbf{B_{29}}$ | = | $- \left(x_{14} + y_{14} - z_{14}\right) \, \mathbf{a}_{1}+\left(x_{14} + y_{14} + z_{14}\right) \, \mathbf{a}_{2}+\left(x_{14} - y_{14} - z_{14}\right) \, \mathbf{a}_{3}$ | = | $a x_{14} \,\mathbf{\hat{x}}- b y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (16e) | O VII |
| $\mathbf{B_{30}}$ | = | $\left(x_{14} + y_{14} + z_{14}\right) \, \mathbf{a}_{1}- \left(x_{14} + y_{14} - z_{14}\right) \, \mathbf{a}_{2}- \left(x_{14} - y_{14} + z_{14}\right) \, \mathbf{a}_{3}$ | = | $- a x_{14} \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (16e) | O VII |
| $\mathbf{B_{31}}$ | = | $\left(- x_{15} + y_{15} + z_{15}\right) \, \mathbf{a}_{1}+\left(x_{15} - y_{15} + z_{15}\right) \, \mathbf{a}_{2}+\left(x_{15} + y_{15} - z_{15}\right) \, \mathbf{a}_{3}$ | = | $a x_{15} \,\mathbf{\hat{x}}+b y_{15} \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (16e) | O VIII |
| $\mathbf{B_{32}}$ | = | $\left(x_{15} - y_{15} + z_{15}\right) \, \mathbf{a}_{1}+\left(- x_{15} + y_{15} + z_{15}\right) \, \mathbf{a}_{2}- \left(x_{15} + y_{15} + z_{15}\right) \, \mathbf{a}_{3}$ | = | $- a x_{15} \,\mathbf{\hat{x}}- b y_{15} \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (16e) | O VIII |
| $\mathbf{B_{33}}$ | = | $- \left(x_{15} + y_{15} - z_{15}\right) \, \mathbf{a}_{1}+\left(x_{15} + y_{15} + z_{15}\right) \, \mathbf{a}_{2}+\left(x_{15} - y_{15} - z_{15}\right) \, \mathbf{a}_{3}$ | = | $a x_{15} \,\mathbf{\hat{x}}- b y_{15} \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (16e) | O VIII |
| $\mathbf{B_{34}}$ | = | $\left(x_{15} + y_{15} + z_{15}\right) \, \mathbf{a}_{1}- \left(x_{15} + y_{15} - z_{15}\right) \, \mathbf{a}_{2}- \left(x_{15} - y_{15} + z_{15}\right) \, \mathbf{a}_{3}$ | = | $- a x_{15} \,\mathbf{\hat{x}}+b y_{15} \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (16e) | O VIII |
References
- M. Krzhizhanovskaya, S. Filatov, V. Gusarov, P. Paufler, R. Bubnova, M. Morozov, and D. C. Meyer, Aurivillius Phases in the Bi$_{4}$Ti$_{3}$O$_{12}$/BiFeO$_{3}$ System: Thermal Behaviour and Crystal Structure, Z. Anorganische und Allgemeine Chemie 631, 1603–1608 (2005), doi:10.1002/zaac.200500130.
- E. C. Subbarao, A family of ferroelectric bismuth compounds, J. Phys.: Conf. Ser. 23, 665–676 (1962), doi:10.1016/0022-3697(62)90526-7.
Prototype Generator
aflow --proto=A7B6C21_oF136_42_a3c_3c_ab3c3e --params=$a,b/a,c/a,z_{1},z_{2},z_{3},y_{4},z_{4},y_{5},z_{5},y_{6},z_{6},y_{7},z_{7},y_{8},z_{8},y_{9},z_{9},y_{10},z_{10},y_{11},z_{11},y_{12},z_{12},x_{13},y_{13},z_{13},x_{14},y_{14},z_{14},x_{15},y_{15},z_{15}$