V$_{6}$C$_{5}$ Structure: A5B6_hP33_151_3a2b_3c-001
| Prototype | C$_{5}$V$_{6}$ |
| AFLOW prototype label | A5B6_hP33_151_3a2b_3c-001 |
| ICSD | 654841 |
| Pearson symbol | hP33 |
| Space group number | 151 |
| Space group symbol | $P3_112$ |
| AFLOW prototype command |
aflow --proto=A5B6_hP33_151_3a2b_3c-001
--params=$a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak x_{2}, \allowbreak x_{3}, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}$ |
- This is the CuPt ($L1_{1}$) structure with one of every six carbon atoms removed. Placing carbon atoms on another (3a) site with $x = 1/9$ will restore the $L1_{1}$ structure.
- (Venables, 1968) put this structure in space group $P3_{1}$ #144 or its enantiomorph $P3_{2}$ #145. (Cenzual, 1991) showed that with the given coordinates the space group is actually $P3_{1}12$ #151.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}+\frac{1}{3} \, \mathbf{a}_{3}$ | = | $- \sqrt{3}a x_{1} \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}$ | (3a) | C I |
| $\mathbf{B_{2}}$ | = | $x_{1} \, \mathbf{a}_{1}+2 x_{1} \, \mathbf{a}_{2}+\frac{2}{3} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{1} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{1} \,\mathbf{\hat{y}}+\frac{2}{3}c \,\mathbf{\hat{z}}$ | (3a) | C I |
| $\mathbf{B_{3}}$ | = | $- 2 x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}$ | = | $- \frac{3}{2}a x_{1} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{1} \,\mathbf{\hat{y}}$ | (3a) | C I |
| $\mathbf{B_{4}}$ | = | $x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\frac{1}{3} \, \mathbf{a}_{3}$ | = | $- \sqrt{3}a x_{2} \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}$ | (3a) | C II |
| $\mathbf{B_{5}}$ | = | $x_{2} \, \mathbf{a}_{1}+2 x_{2} \, \mathbf{a}_{2}+\frac{2}{3} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{2} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+\frac{2}{3}c \,\mathbf{\hat{z}}$ | (3a) | C II |
| $\mathbf{B_{6}}$ | = | $- 2 x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}$ | = | $- \frac{3}{2}a x_{2} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}$ | (3a) | C II |
| $\mathbf{B_{7}}$ | = | $x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+\frac{1}{3} \, \mathbf{a}_{3}$ | = | $- \sqrt{3}a x_{3} \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}$ | (3a) | C III |
| $\mathbf{B_{8}}$ | = | $x_{3} \, \mathbf{a}_{1}+2 x_{3} \, \mathbf{a}_{2}+\frac{2}{3} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{3} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{3} \,\mathbf{\hat{y}}+\frac{2}{3}c \,\mathbf{\hat{z}}$ | (3a) | C III |
| $\mathbf{B_{9}}$ | = | $- 2 x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}$ | = | $- \frac{3}{2}a x_{3} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{3} \,\mathbf{\hat{y}}$ | (3a) | C III |
| $\mathbf{B_{10}}$ | = | $x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}+\frac{5}{6} \, \mathbf{a}_{3}$ | = | $- \sqrt{3}a x_{4} \,\mathbf{\hat{y}}+\frac{5}{6}c \,\mathbf{\hat{z}}$ | (3b) | C IV |
| $\mathbf{B_{11}}$ | = | $x_{4} \, \mathbf{a}_{1}+2 x_{4} \, \mathbf{a}_{2}+\frac{1}{6} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{4} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3b) | C IV |
| $\mathbf{B_{12}}$ | = | $- 2 x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- \frac{3}{2}a x_{4} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (3b) | C IV |
| $\mathbf{B_{13}}$ | = | $x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+\frac{5}{6} \, \mathbf{a}_{3}$ | = | $- \sqrt{3}a x_{5} \,\mathbf{\hat{y}}+\frac{5}{6}c \,\mathbf{\hat{z}}$ | (3b) | C V |
| $\mathbf{B_{14}}$ | = | $x_{5} \, \mathbf{a}_{1}+2 x_{5} \, \mathbf{a}_{2}+\frac{1}{6} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{5} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{5} \,\mathbf{\hat{y}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3b) | C V |
| $\mathbf{B_{15}}$ | = | $- 2 x_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- \frac{3}{2}a x_{5} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{5} \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (3b) | C V |
| $\mathbf{B_{16}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} + y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (6c) | V I |
| $\mathbf{B_{17}}$ | = | $- y_{6} \, \mathbf{a}_{1}+\left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - 2 y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{6} \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | V I |
| $\mathbf{B_{18}}$ | = | $- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}+\left(z_{6} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{6} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{6} + 2\right) \,\mathbf{\hat{z}}$ | (6c) | V I |
| $\mathbf{B_{19}}$ | = | $- y_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- \left(z_{6} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} + y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{6} - 2\right) \,\mathbf{\hat{z}}$ | (6c) | V I |
| $\mathbf{B_{20}}$ | = | $- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}- \left(z_{6} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{6} + 2 y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{6} \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | V I |
| $\mathbf{B_{21}}$ | = | $x_{6} \, \mathbf{a}_{1}+\left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{6} \,\mathbf{\hat{y}}- c z_{6} \,\mathbf{\hat{z}}$ | (6c) | V I |
| $\mathbf{B_{22}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} + y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (6c) | V II |
| $\mathbf{B_{23}}$ | = | $- y_{7} \, \mathbf{a}_{1}+\left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - 2 y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{7} \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | V II |
| $\mathbf{B_{24}}$ | = | $- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+\left(z_{7} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{7} - y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{7} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{7} + 2\right) \,\mathbf{\hat{z}}$ | (6c) | V II |
| $\mathbf{B_{25}}$ | = | $- y_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- \left(z_{7} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} + y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{7} - 2\right) \,\mathbf{\hat{z}}$ | (6c) | V II |
| $\mathbf{B_{26}}$ | = | $- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}- \left(z_{7} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{7} + 2 y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{7} \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | V II |
| $\mathbf{B_{27}}$ | = | $x_{7} \, \mathbf{a}_{1}+\left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{7} - y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{7} \,\mathbf{\hat{y}}- c z_{7} \,\mathbf{\hat{z}}$ | (6c) | V II |
| $\mathbf{B_{28}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} + y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (6c) | V III |
| $\mathbf{B_{29}}$ | = | $- y_{8} \, \mathbf{a}_{1}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - 2 y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{8} \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | V III |
| $\mathbf{B_{30}}$ | = | $- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+\left(z_{8} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{8} - y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{8} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{8} + 2\right) \,\mathbf{\hat{z}}$ | (6c) | V III |
| $\mathbf{B_{31}}$ | = | $- y_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- \left(z_{8} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} + y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{8} - 2\right) \,\mathbf{\hat{z}}$ | (6c) | V III |
| $\mathbf{B_{32}}$ | = | $- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}- \left(z_{8} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{8} + 2 y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{8} \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (6c) | V III |
| $\mathbf{B_{33}}$ | = | $x_{8} \, \mathbf{a}_{1}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{8} - y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{8} \,\mathbf{\hat{y}}- c z_{8} \,\mathbf{\hat{z}}$ | (6c) | V III |
References
- J. D. Venables, D. Kahn, and R. G. Lye, Structure of the ordered compound V$_{6}$C$_{5}$, Phil. Mag. 18, 177–192 (1968), doi:10.1080/14786436808227320.
Found in
- K. Cenzual, L. M. Gelato, M. Penzo, and E. Parthé, Inorganic structure types with revised space groups. I, Acta Crystallogr. Sect. B 47, 433–439 (1991), doi:10.1107/S0108768191000903.
Prototype Generator
aflow --proto=A5B6_hP33_151_3a2b_3c --params=$a,c/a,x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8}$