Ce$_{5}$Ga$_{7}$Rh$_{12}$ Structure: A5B7C12_hR24_166_a2c_b3c_2h-001
| Prototype | Ce$_{5}$Ga$_{7}$Rh$_{12}$ |
| AFLOW prototype label | A5B7C12_hR24_166_a2c_b3c_2h-001 |
| ICSD | 432161 |
| CCDC | 1791222 |
| Pearson symbol | hR24 |
| Space group number | 166 |
| Space group symbol | $R\overline{3}m$ |
| AFLOW prototype command |
aflow --proto=A5B7C12_hR24_166_a2c_b3c_2h-001
--params=$a, \allowbreak c/a, \allowbreak x_{3}, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak x_{6}, \allowbreak x_{7}, \allowbreak x_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak z_{9}$ |
Other compounds with this structure
Nd$_{5}$Ga$_{7}$Rh$_{12}$
- This is the ternary version of the Ce$_{5}$Co$_{19}$ structure. We have arbitrarily chosen Ce$_{5}$Ga$_{7}$Rh$_{12}$ as the prototype.
- Ce$_{4}$Ga$_{5}$Rh$_{9}$ and Ce$_{7}$Ga$_{11}$Rh$_{18}$ are slightly gallium-enriched forms of this structure.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{\sqrt{3}}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (1a) | Ce 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) | Ga I |
| $\mathbf{B_{3}}$ | = | $x_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $c x_{3} \,\mathbf{\hat{z}}$ | (2c) | Ce II |
| $\mathbf{B_{4}}$ | = | $- x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $- c x_{3} \,\mathbf{\hat{z}}$ | (2c) | Ce II |
| $\mathbf{B_{5}}$ | = | $x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $c x_{4} \,\mathbf{\hat{z}}$ | (2c) | Ce III |
| $\mathbf{B_{6}}$ | = | $- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $- c x_{4} \,\mathbf{\hat{z}}$ | (2c) | Ce III |
| $\mathbf{B_{7}}$ | = | $x_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ | = | $c x_{5} \,\mathbf{\hat{z}}$ | (2c) | Ga II |
| $\mathbf{B_{8}}$ | = | $- x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ | = | $- c x_{5} \,\mathbf{\hat{z}}$ | (2c) | Ga II |
| $\mathbf{B_{9}}$ | = | $x_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ | = | $c x_{6} \,\mathbf{\hat{z}}$ | (2c) | Ga III |
| $\mathbf{B_{10}}$ | = | $- x_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ | = | $- c x_{6} \,\mathbf{\hat{z}}$ | (2c) | Ga III |
| $\mathbf{B_{11}}$ | = | $x_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $c x_{7} \,\mathbf{\hat{z}}$ | (2c) | Ga IV |
| $\mathbf{B_{12}}$ | = | $- x_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $- c x_{7} \,\mathbf{\hat{z}}$ | (2c) | Ga IV |
| $\mathbf{B_{13}}$ | = | $x_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh I |
| $\mathbf{B_{14}}$ | = | $z_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh I |
| $\mathbf{B_{15}}$ | = | $x_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $- \frac{1}{\sqrt{3}}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh I |
| $\mathbf{B_{16}}$ | = | $- z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh I |
| $\mathbf{B_{17}}$ | = | $- x_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh I |
| $\mathbf{B_{18}}$ | = | $- x_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{\sqrt{3}}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh I |
| $\mathbf{B_{19}}$ | = | $x_{9} \, \mathbf{a}_{1}+x_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh II |
| $\mathbf{B_{20}}$ | = | $z_{9} \, \mathbf{a}_{1}+x_{9} \, \mathbf{a}_{2}+x_{9} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh II |
| $\mathbf{B_{21}}$ | = | $x_{9} \, \mathbf{a}_{1}+z_{9} \, \mathbf{a}_{2}+x_{9} \, \mathbf{a}_{3}$ | = | $- \frac{1}{\sqrt{3}}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh II |
| $\mathbf{B_{22}}$ | = | $- z_{9} \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}- x_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh II |
| $\mathbf{B_{23}}$ | = | $- x_{9} \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh II |
| $\mathbf{B_{24}}$ | = | $- x_{9} \, \mathbf{a}_{1}- z_{9} \, \mathbf{a}_{2}- x_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{\sqrt{3}}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (6h) | Rh II |
References
- S. Seidel, U. C. Rodewald, O. Janka, and R. Pöttgen, Ternary gallides RE$_{4}$Rh$_{9}$Ga$_{5}$, RE$_{5}$Rh$_{12}$Ga$_{7}$ and RE$_{7}$Rh$_{18}$Ga$_{11}$ (RE=Y, La–Nd, Sm, Gd, Tb) – intergrowth structures with MgCu$_{2}$ and CaCu$_{5}$ related slabs, Z. Krystallogr. 232, 365–374 (2017), doi:10.1515/zkri-2016-2017.
Found in
- Inorganic Crystal Structure Database. Entry 432161 (Ce5Ga7Rh12).
First cited in
- N. Anderson, M. J. Mehl, H. Eckert, S. Divilov, X. Campilongo, S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 5. Submitted to Computational Materials Science (2026).
Geometry files
Prototype Generator
aflow --proto=A5B7C12_hR24_166_a2c_b3c_2h --params=$a,c/a,x_{3},x_{4},x_{5},x_{6},x_{7},x_{8},z_{8},x_{9},z_{9}$