Th$_{2}$Zn$_{17}$ Structure: A2B17_hR19_166_c_cdfh-001
| Prototype | Th$_{2}$Zn$_{17}$ |
| AFLOW prototype label | A2B17_hR19_166_c_cdfh-001 |
| ICSD | 20238 |
| Pearson symbol | hR19 |
| Space group number | 166 |
| Space group symbol | $R\overline{3}m$ |
| AFLOW prototype command |
aflow --proto=A2B17_hR19_166_c_cdfh-001
--params=$a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak x_{2}, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak z_{5}$ |
Other compounds with this structure
Ba$_{2}$Mg$_{17}$, Ce$_{2}$Co$_{17}$, Ce$_{2}$Fe$_{17}$, Ce$_{2}$Zn$_{17}$, Dy$_{2}$Zn$_{17}$, Er$_{2}$Zn$_{17}$, Gd$_{2}$Co$_{17}$, Gd$_{2}$Fe$_{17}$, Gd$_{2}$Zn$_{17}$, Ho$_{2}$Zn$_{17}$, La$_{2}$Zn$_{17}$, Lu$_{2}$Zn$_{17}$, Nd$_{2}$Co$_{17}$, Nd$_{2}$Fe$_{17}$, Nd$_{2}$Zn$_{17}$, Pr$_{2}$Co$_{17}$, Pr$_{2}$Zn$_{17}$, Sm$_{2}$Co$_{17}$, Sm$_{2}$Zn$_{17}$, Tb$_{2}$Co$_{17}$, Tb$_{2}$Zn$_{17}$, Th$_{2}$Co$_{17}$, Th$_{2}$Fe$_{17}$, Tm$_{2}$Zn$_{17}$, U$_{2}$Zn$_{17}$, Yb$_{2}$Zn$_{17}$, Al$_{2}$Ce$_{2}$Co$_{15}$, Ce$_{2}$Mn$_{7}$Al$_{10}$
- 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 | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $c x_{1} \,\mathbf{\hat{z}}$ | (2c) | Th I |
| $\mathbf{B_{2}}$ | = | $- x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $- c x_{1} \,\mathbf{\hat{z}}$ | (2c) | Th I |
| $\mathbf{B_{3}}$ | = | $x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $c x_{2} \,\mathbf{\hat{z}}$ | (2c) | Zn I |
| $\mathbf{B_{4}}$ | = | $- x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $- c x_{2} \,\mathbf{\hat{z}}$ | (2c) | Zn I |
| $\mathbf{B_{5}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{12}a \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3d) | Zn II |
| $\mathbf{B_{6}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}$ | = | $\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3d) | Zn II |
| $\mathbf{B_{7}}$ | = | $\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- \frac{1}{4}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{12}a \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3d) | Zn II |
| $\mathbf{B_{8}}$ | = | $x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}$ | = | $\frac{1}{2}a x_{4} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}$ | (6f) | Zn III |
| $\mathbf{B_{9}}$ | = | $x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a x_{4} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}$ | (6f) | Zn III |
| $\mathbf{B_{10}}$ | = | $- x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}$ | (6f) | Zn III |
| $\mathbf{B_{11}}$ | = | $- x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}$ | = | $- \frac{1}{2}a x_{4} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}$ | (6f) | Zn III |
| $\mathbf{B_{12}}$ | = | $- x_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a x_{4} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}$ | (6f) | Zn III |
| $\mathbf{B_{13}}$ | = | $x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}$ | (6f) | Zn III |
| $\mathbf{B_{14}}$ | = | $x_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6h) | Zn IV |
| $\mathbf{B_{15}}$ | = | $z_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6h) | Zn IV |
| $\mathbf{B_{16}}$ | = | $x_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{\sqrt{3}}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6h) | Zn IV |
| $\mathbf{B_{17}}$ | = | $- z_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6h) | Zn IV |
| $\mathbf{B_{18}}$ | = | $- x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6h) | Zn IV |
| $\mathbf{B_{19}}$ | = | $- x_{5} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{\sqrt{3}}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6h) | Zn IV |
References
- E. S. Makarov and S. I. Vinogradov, Crystal Structure of Th$_{2}$Zn$_{17}$ and U$_{2}$Zn$_{17}$, Sov. Phys. Crystallogr. 1, 499–504 (1956). Translated from Kristallografiya 1, 634 (1956).
Found in
- W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, Volume 2, International Series of Monographs on Metal Physics and Physical Metallurgy, vol. 8 (Pergamon Press, Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig, 1967).
Prototype Generator
aflow --proto=A2B17_hR19_166_c_cdfh --params=$a,c/a,x_{1},x_{2},x_{4},x_{5},z_{5}$