Tetragonal Potassium Bronze (K$_{3}$W$_{5}$O$_{15}$) Structure: A3B15C5_tP46_127_bh_cg2ij_di-001
| Prototype | K$_{3}$O$_{15}$W$_{3}$ |
| AFLOW prototype label | A3B15C5_tP46_127_bh_cg2ij_di-001 |
| Mineral name | bronze |
| ICSD | 24730 |
| Pearson symbol | tP46 |
| Space group number | 127 |
| Space group symbol | $P4/mbm$ |
| AFLOW prototype command |
aflow --proto=A3B15C5_tP46_127_bh_cg2ij_di-001
--params=$a, \allowbreak c/a, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak x_{9}, \allowbreak y_{9}$ |
- (Hyde, 1973) notes that this is a derivative of the $\alpha$–ReO$_{3}$ ($D0_{9}$) structure.
- The measured structure is actually deficient in potassium. We use the lattice constants from (Magnéli, 1949) for K$_{0.57}$WO$_{3}$.
- Magnéli's X-ray data could not determine the oxygen positions. The coordinates for the oxygen atoms were determined by the requirement that the space group be $P4/mbm$ #127. as determined by the positions of the potassium and tungsten atoms, and by
considerations of space.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}c \,\mathbf{\hat{z}}$ | (2b) | K I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (2b) | K I |
| $\mathbf{B_{3}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (2c) | O I |
| $\mathbf{B_{4}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (2c) | O I |
| $\mathbf{B_{5}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}$ | (2d) | W I |
| $\mathbf{B_{6}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}$ | (2d) | W I |
| $\mathbf{B_{7}}$ | = | $x_{4} \, \mathbf{a}_{1}+\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $a x_{4} \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (4g) | O II |
| $\mathbf{B_{8}}$ | = | $- x_{4} \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $- a x_{4} \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (4g) | O II |
| $\mathbf{B_{9}}$ | = | $- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}$ | = | $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}$ | (4g) | O II |
| $\mathbf{B_{10}}$ | = | $\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}$ | = | $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}$ | (4g) | O II |
| $\mathbf{B_{11}}$ | = | $x_{5} \, \mathbf{a}_{1}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (4h) | K II |
| $\mathbf{B_{12}}$ | = | $- x_{5} \, \mathbf{a}_{1}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (4h) | K II |
| $\mathbf{B_{13}}$ | = | $- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{5} \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (4h) | K II |
| $\mathbf{B_{14}}$ | = | $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{5} \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (4h) | K II |
| $\mathbf{B_{15}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}$ | = | $a x_{6} \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{y}}$ | (8i) | O III |
| $\mathbf{B_{16}}$ | = | $- x_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}$ | = | $- a x_{6} \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{y}}$ | (8i) | O III |
| $\mathbf{B_{17}}$ | = | $- y_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}$ | = | $- a y_{6} \,\mathbf{\hat{x}}+a x_{6} \,\mathbf{\hat{y}}$ | (8i) | O III |
| $\mathbf{B_{18}}$ | = | $y_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}$ | = | $a y_{6} \,\mathbf{\hat{x}}- a x_{6} \,\mathbf{\hat{y}}$ | (8i) | O III |
| $\mathbf{B_{19}}$ | = | $- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(y_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | O III |
| $\mathbf{B_{20}}$ | = | $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(y_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | O III |
| $\mathbf{B_{21}}$ | = | $\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $a \left(y_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | O III |
| $\mathbf{B_{22}}$ | = | $- \left(y_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $- a \left(y_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | O III |
| $\mathbf{B_{23}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}$ | = | $a x_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}$ | (8i) | O IV |
| $\mathbf{B_{24}}$ | = | $- x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}$ | (8i) | O IV |
| $\mathbf{B_{25}}$ | = | $- y_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}$ | = | $- a y_{7} \,\mathbf{\hat{x}}+a x_{7} \,\mathbf{\hat{y}}$ | (8i) | O IV |
| $\mathbf{B_{26}}$ | = | $y_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}$ | = | $a y_{7} \,\mathbf{\hat{x}}- a x_{7} \,\mathbf{\hat{y}}$ | (8i) | O IV |
| $\mathbf{B_{27}}$ | = | $- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(y_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | O IV |
| $\mathbf{B_{28}}$ | = | $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(y_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | O IV |
| $\mathbf{B_{29}}$ | = | $\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $a \left(y_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | O IV |
| $\mathbf{B_{30}}$ | = | $- \left(y_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $- a \left(y_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | O IV |
| $\mathbf{B_{31}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}$ | (8i) | W II |
| $\mathbf{B_{32}}$ | = | $- x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}$ | (8i) | W II |
| $\mathbf{B_{33}}$ | = | $- y_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}$ | = | $- a y_{8} \,\mathbf{\hat{x}}+a x_{8} \,\mathbf{\hat{y}}$ | (8i) | W II |
| $\mathbf{B_{34}}$ | = | $y_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}$ | = | $a y_{8} \,\mathbf{\hat{x}}- a x_{8} \,\mathbf{\hat{y}}$ | (8i) | W II |
| $\mathbf{B_{35}}$ | = | $- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(y_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | W II |
| $\mathbf{B_{36}}$ | = | $\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(y_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | W II |
| $\mathbf{B_{37}}$ | = | $\left(y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $a \left(y_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | W II |
| $\mathbf{B_{38}}$ | = | $- \left(y_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}$ | = | $- a \left(y_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}$ | (8i) | W II |
| $\mathbf{B_{39}}$ | = | $x_{9} \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $a x_{9} \,\mathbf{\hat{x}}+a y_{9} \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (8j) | O V |
| $\mathbf{B_{40}}$ | = | $- x_{9} \, \mathbf{a}_{1}- y_{9} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- a x_{9} \,\mathbf{\hat{x}}- a y_{9} \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (8j) | O V |
| $\mathbf{B_{41}}$ | = | $- y_{9} \, \mathbf{a}_{1}+x_{9} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- a y_{9} \,\mathbf{\hat{x}}+a x_{9} \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (8j) | O V |
| $\mathbf{B_{42}}$ | = | $y_{9} \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $a y_{9} \,\mathbf{\hat{x}}- a x_{9} \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (8j) | O V |
| $\mathbf{B_{43}}$ | = | $- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{9} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- a \left(x_{9} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(y_{9} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (8j) | O V |
| $\mathbf{B_{44}}$ | = | $\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{9} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $a \left(x_{9} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(y_{9} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (8j) | O V |
| $\mathbf{B_{45}}$ | = | $\left(y_{9} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $a \left(y_{9} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{9} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (8j) | O V |
| $\mathbf{B_{46}}$ | = | $- \left(y_{9} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- a \left(y_{9} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{9} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (8j) | O V |
References
- A. Magnéli, The crystal structure of tetragonal potassium bronze, Arkiv Kemi 1, 213–221 (1949).
- B. G. Hyde and M. O'Keefe, Relations between the DO$_{9}$(ReO$_{3}$) structure type and some bronze and tunnel structures, Acta Crystallogr. Sect. A 29, 243–248 (1973), doi:10.1107/S056773947300063X.
Found in
- A. Magnéli and S. Nord, Bronze-Type Structure of KNb$_{2}$O$_{5}$F and KTa$_{2}$O$_{5}$, Acta Chem. Scand. p. 1510 (1965), doi:10.3891/acta.chem.scand.19-1510a.
Prototype Generator
aflow --proto=A3B15C5_tP46_127_bh_cg2ij_di --params=$a,c/a,x_{4},x_{5},x_{6},y_{6},x_{7},y_{7},x_{8},y_{8},x_{9},y_{9}$