Orthorhombic Bi$_{4}$Ti$_{3}$O$_{12}$ $m = 3$ Aurivillius Structure: A4B12C3_oF76_69_2g_cf2gl_ag-001
| Prototype | Bi$_{4}$O$_{12}$Ti$_{3}$ |
| AFLOW prototype label | A4B12C3_oF76_69_2g_cf2gl_ag-001 |
| ICSD | 24735 |
| Pearson symbol | oF76 |
| Space group number | 69 |
| Space group symbol | $Fmmm$ |
| AFLOW prototype command |
aflow --proto=A4B12C3_oF76_69_2g_cf2gl_ag-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak x_{6}, \allowbreak x_{7}, \allowbreak x_{8}, \allowbreak x_{9}$ |
- 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)
- This is the original structural determination by (Aurivillius, 1949). It should not be confused with the obsolete $m=3$ Aurivillius structure in space group $Aea2$ #41.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}b \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (4a) | Ti I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}$ | = | $\frac{1}{4}b \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (8c) | O I |
| $\mathbf{B_{3}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}b \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (8c) | O I |
| $\mathbf{B_{4}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{1}{4}b \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (8f) | O II |
| $\mathbf{B_{5}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{4}a \,\mathbf{\hat{x}}+\frac{3}{4}b \,\mathbf{\hat{y}}+\frac{3}{4}c \,\mathbf{\hat{z}}$ | (8f) | O II |
| $\mathbf{B_{6}}$ | = | $- x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}$ | (8g) | Bi I |
| $\mathbf{B_{7}}$ | = | $x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}$ | (8g) | Bi I |
| $\mathbf{B_{8}}$ | = | $- x_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}$ | (8g) | Bi II |
| $\mathbf{B_{9}}$ | = | $x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}$ | (8g) | Bi II |
| $\mathbf{B_{10}}$ | = | $- x_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}$ | (8g) | O III |
| $\mathbf{B_{11}}$ | = | $x_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}$ | (8g) | O III |
| $\mathbf{B_{12}}$ | = | $- x_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}$ | (8g) | O IV |
| $\mathbf{B_{13}}$ | = | $x_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}$ | (8g) | O IV |
| $\mathbf{B_{14}}$ | = | $- x_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}$ | (8g) | Ti II |
| $\mathbf{B_{15}}$ | = | $x_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}$ | (8g) | Ti II |
| $\mathbf{B_{16}}$ | = | $- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{9} \, \mathbf{a}_{2}+x_{9} \, \mathbf{a}_{3}$ | = | $a x_{9} \,\mathbf{\hat{x}}+\frac{1}{4}b \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16l) | O V |
| $\mathbf{B_{17}}$ | = | $x_{9} \, \mathbf{a}_{1}- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{9} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+\frac{1}{4}b \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16l) | O V |
| $\mathbf{B_{18}}$ | = | $\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}- x_{9} \, \mathbf{a}_{3}$ | = | $- a x_{9} \,\mathbf{\hat{x}}+\frac{1}{4}b \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16l) | O V |
| $\mathbf{B_{19}}$ | = | $- x_{9} \, \mathbf{a}_{1}+\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{9} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+\frac{1}{4}b \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16l) | O V |
References
- B. Aurivillius, Mixed bismuth oxides with layer lattices II. Structure of Bi$_{4}$Ti$_{3}$O$_{12}$, Arkiv för Kemi 1, 499–512 (1949).
- 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.
- Y.-Y. Guo, A. S. Gibbs, J. M. Perez-Matoc, and P. Lightfoot, Unexpected phase transition sequence in the ferroelectric Bi$_{4}$Ti$_{3}$O$_{12}$, IUCrJ 6, 438–446 (2019), doi:10.1107/S2052252519003804.
- J. F. Dorrian, R. E. Newnham, D. K. Smith, and M. I. Kay, Crystal Structure of Bi$_{4}$Ti$_{3}$O$_{12}$, Ferroelectrics 3, 17–27 (1972), doi:10.1080/00150197108237680.
Prototype Generator
aflow --proto=A4B12C3_oF76_69_2g_cf2gl_ag --params=$a,b/a,c/a,x_{4},x_{5},x_{6},x_{7},x_{8},x_{9}$