Mg$_{3}$Ru$_{2}$ Structure: A3B2_cP20_213_d_c-001
| Prototype | Mg$_{3}$Ru$_{2}$ |
| AFLOW prototype label | A3B2_cP20_213_d_c-001 |
| ICSD | 260022 |
| Pearson symbol | cP20 |
| Space group number | 213 |
| Space group symbol | $P4_132$ |
| AFLOW prototype command |
aflow --proto=A3B2_cP20_213_d_c-001
--params=$a, \allowbreak x_{1}, \allowbreak y_{2}$ |
- This is the binary form of the $\beta$–Mn ($A13$) structure.
- This structure may also be found in the enantiomorphic space group $P4_{3}32$ #212.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&a \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $a x_{1} \,\mathbf{\hat{x}}+a x_{1} \,\mathbf{\hat{y}}+a x_{1} \,\mathbf{\hat{z}}$ | (8c) | Ru I |
| $\mathbf{B_{2}}$ | = | $- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{1} \,\mathbf{\hat{y}}+a \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (8c) | Ru I |
| $\mathbf{B_{3}}$ | = | $- x_{1} \, \mathbf{a}_{1}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{1} \,\mathbf{\hat{x}}+a \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (8c) | Ru I |
| $\mathbf{B_{4}}$ | = | $\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $a \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{1} \,\mathbf{\hat{z}}$ | (8c) | Ru I |
| $\mathbf{B_{5}}$ | = | $\left(x_{1} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\left(x_{1} + \frac{1}{4}\right) \, \mathbf{a}_{2}- \left(x_{1} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{1} + \frac{3}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{1} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(x_{1} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (8c) | Ru I |
| $\mathbf{B_{6}}$ | = | $- \left(x_{1} - \frac{3}{4}\right) \, \mathbf{a}_{1}- \left(x_{1} - \frac{3}{4}\right) \, \mathbf{a}_{2}- \left(x_{1} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{1} - \frac{3}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{1} - \frac{3}{4}\right) \,\mathbf{\hat{y}}- a \left(x_{1} - \frac{3}{4}\right) \,\mathbf{\hat{z}}$ | (8c) | Ru I |
| $\mathbf{B_{7}}$ | = | $\left(x_{1} + \frac{1}{4}\right) \, \mathbf{a}_{1}- \left(x_{1} - \frac{1}{4}\right) \, \mathbf{a}_{2}+\left(x_{1} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{1} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{1} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(x_{1} + \frac{3}{4}\right) \,\mathbf{\hat{z}}$ | (8c) | Ru I |
| $\mathbf{B_{8}}$ | = | $- \left(x_{1} - \frac{1}{4}\right) \, \mathbf{a}_{1}+\left(x_{1} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\left(x_{1} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{1} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{1} + \frac{3}{4}\right) \,\mathbf{\hat{y}}+a \left(x_{1} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (8c) | Ru I |
| $\mathbf{B_{9}}$ | = | $\frac{1}{8} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+\left(y_{2} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}+a y_{2} \,\mathbf{\hat{y}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{10}}$ | = | $\frac{3}{8} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+\left(y_{2} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}- a y_{2} \,\mathbf{\hat{y}}+a \left(y_{2} + \frac{3}{4}\right) \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{11}}$ | = | $\frac{7}{8} \, \mathbf{a}_{1}+\left(y_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{2} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{7}{8}a \,\mathbf{\hat{x}}+a \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{12}}$ | = | $\frac{5}{8} \, \mathbf{a}_{1}- \left(y_{2} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{2} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{5}{8}a \,\mathbf{\hat{x}}- a \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(y_{2} - \frac{3}{4}\right) \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{13}}$ | = | $\left(y_{2} + \frac{1}{4}\right) \, \mathbf{a}_{1}+\frac{1}{8} \, \mathbf{a}_{2}+y_{2} \, \mathbf{a}_{3}$ | = | $a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}+a y_{2} \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{14}}$ | = | $\left(y_{2} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\frac{3}{8} \, \mathbf{a}_{2}- y_{2} \, \mathbf{a}_{3}$ | = | $a \left(y_{2} + \frac{3}{4}\right) \,\mathbf{\hat{x}}+\frac{3}{8}a \,\mathbf{\hat{y}}- a y_{2} \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{15}}$ | = | $- \left(y_{2} - \frac{1}{4}\right) \, \mathbf{a}_{1}+\frac{7}{8} \, \mathbf{a}_{2}+\left(y_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{7}{8}a \,\mathbf{\hat{y}}+a \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{16}}$ | = | $- \left(y_{2} - \frac{3}{4}\right) \, \mathbf{a}_{1}+\frac{5}{8} \, \mathbf{a}_{2}- \left(y_{2} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{2} - \frac{3}{4}\right) \,\mathbf{\hat{x}}+\frac{5}{8}a \,\mathbf{\hat{y}}- a \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{17}}$ | = | $y_{2} \, \mathbf{a}_{1}+\left(y_{2} + \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{1}{8} \, \mathbf{a}_{3}$ | = | $a y_{2} \,\mathbf{\hat{x}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{18}}$ | = | $- y_{2} \, \mathbf{a}_{1}+\left(y_{2} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\frac{3}{8} \, \mathbf{a}_{3}$ | = | $- a y_{2} \,\mathbf{\hat{x}}+a \left(y_{2} + \frac{3}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}a \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{19}}$ | = | $\left(y_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{2} - \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{7}{8} \, \mathbf{a}_{3}$ | = | $a \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{7}{8}a \,\mathbf{\hat{z}}$ | (12d) | Mg I |
| $\mathbf{B_{20}}$ | = | $- \left(y_{2} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{2} - \frac{3}{4}\right) \, \mathbf{a}_{2}+\frac{5}{8} \, \mathbf{a}_{3}$ | = | $- a \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(y_{2} - \frac{3}{4}\right) \,\mathbf{\hat{y}}+\frac{5}{8}a \,\mathbf{\hat{z}}$ | (12d) | Mg I |
References
- R. Pöttgen, V. Hlukhyy, A. Baranov, and Y. Grin, Crystal Structure and Chemical Bonding of Mg$_{3}$Ru$_{2}$, Inorg. Chem. 47, 6051–6055 (2008), doi:10.1021/ic800387a.
Prototype Generator
aflow --proto=A3B2_cP20_213_d_c --params=$a,x_{1},y_{2}$