PdAl Structure: AB_hR26_148_a2f_b2f-001
| Prototype | AlPd |
| AFLOW prototype label | AB_hR26_148_a2f_b2f-001 |
| ICSD | 58112 |
| Pearson symbol | hR26 |
| Space group number | 148 |
| Space group symbol | $R\overline{3}$ |
| AFLOW prototype command |
aflow --proto=AB_hR26_148_a2f_b2f-001
--params=$a, \allowbreak c/a, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak y_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}$ |
- 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 | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (1a) | Al I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}c \,\mathbf{\hat{z}}$ | (1b) | Pd I |
| $\mathbf{B_{3}}$ | = | $x_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{3} - 2 y_{3} + z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{3} + y_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6f) | Al II |
| $\mathbf{B_{4}}$ | = | $z_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+y_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{3} - z_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{3} - y_{3} - z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{3} + y_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6f) | Al II |
| $\mathbf{B_{5}}$ | = | $y_{3} \, \mathbf{a}_{1}+z_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{3} + y_{3} - 2 z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{3} + y_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6f) | Al II |
| $\mathbf{B_{6}}$ | = | $- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{3} - 2 y_{3} + z_{3}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{3} + y_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6f) | Al II |
| $\mathbf{B_{7}}$ | = | $- z_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- y_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{3} - z_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{3} - y_{3} - z_{3}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{3} + y_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6f) | Al II |
| $\mathbf{B_{8}}$ | = | $- y_{3} \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{3} + y_{3} - 2 z_{3}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{3} + y_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6f) | Al II |
| $\mathbf{B_{9}}$ | = | $x_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{4} - 2 y_{4} + z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{4} + y_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6f) | Al III |
| $\mathbf{B_{10}}$ | = | $z_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+y_{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{4} - z_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{4} - y_{4} - z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{4} + y_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6f) | Al III |
| $\mathbf{B_{11}}$ | = | $y_{4} \, \mathbf{a}_{1}+z_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{4} - y_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{4} + y_{4} - 2 z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{4} + y_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6f) | Al III |
| $\mathbf{B_{12}}$ | = | $- x_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{4} - 2 y_{4} + z_{4}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{4} + y_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6f) | Al III |
| $\mathbf{B_{13}}$ | = | $- z_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- y_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{4} - z_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{4} - y_{4} - z_{4}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{4} + y_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6f) | Al III |
| $\mathbf{B_{14}}$ | = | $- y_{4} \, \mathbf{a}_{1}- z_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} - y_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{4} + y_{4} - 2 z_{4}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{4} + y_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6f) | Al III |
| $\mathbf{B_{15}}$ | = | $x_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} - 2 y_{5} + z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd II |
| $\mathbf{B_{16}}$ | = | $z_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+y_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{5} - z_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{5} - y_{5} - z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd II |
| $\mathbf{B_{17}}$ | = | $y_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} + y_{5} - 2 z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd II |
| $\mathbf{B_{18}}$ | = | $- x_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{5} - 2 y_{5} + z_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd II |
| $\mathbf{B_{19}}$ | = | $- z_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- y_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{5} - z_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{5} - y_{5} - z_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd II |
| $\mathbf{B_{20}}$ | = | $- y_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{5} + y_{5} - 2 z_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd II |
| $\mathbf{B_{21}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd III |
| $\mathbf{B_{22}}$ | = | $z_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+y_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd III |
| $\mathbf{B_{23}}$ | = | $y_{6} \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd III |
| $\mathbf{B_{24}}$ | = | $- x_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd III |
| $\mathbf{B_{25}}$ | = | $- z_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- y_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd III |
| $\mathbf{B_{26}}$ | = | $- y_{6} \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6f) | Pd III |
References
- T. Matković and K. Schubert, Kristallstruktur vo PdAl.r, J. Less-Common Met. 55, 45–52 (1977), doi:10.1016/0022-5088(77)90258-2.
Found in
- P. Villars, K. Cenzual, J. Daams, R. Gladyshevskii, O. Shcherban, V. Dubenskyy, V. Kuprysyuk, and I. Savesyuk, Landolt–Börnstein – Group III Condensed Matter (Springer–Verlag GmbH, Heidelberg, 2010), chap. Structure Types. Part 8: Space Groups (156)P3m1 – (148)R-3 · PdAl rt, doi:10.1007/978-3-540-70892-6_476. Part 8.
Prototype Generator
aflow --proto=AB_hR26_148_a2f_b2f --params=$a,c/a,x_{3},y_{3},z_{3},x_{4},y_{4},z_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6}$