Cs$_{6}$W$_{11}$O$_{36}$ Structure: A6B36C11_hR106_167_3c_6f_cef-001
| Prototype | Cs$_{6}$O$_{36}$W$_{11}$ |
| AFLOW prototype label | A6B36C11_hR106_167_3c_6f_cef-001 |
| ICSD | 80133 |
| CCDC | 1640359 |
| Pearson symbol | hR106 |
| Space group number | 167 |
| Space group symbol | $R\overline{3}c$ |
| AFLOW prototype command |
aflow --proto=A6B36C11_hR106_167_3c_6f_cef-001
--params=$a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak x_{2}, \allowbreak x_{3}, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}$ |
- (Okada, 1978) placed this in the monoclinic space group $Cc$ #9. (Marsh, 1995) determined that their data fit in space group $R\overline{3}c$ #167, and we show that refinement here.
- The original orientation may be found here.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{\sqrt{3}}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $c x_{1} \,\mathbf{\hat{z}}$ | (4c) | Cs I |
| $\mathbf{B_{2}}$ | = | $- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- c \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | Cs I |
| $\mathbf{B_{3}}$ | = | $- x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}$ | = | $- c x_{1} \,\mathbf{\hat{z}}$ | (4c) | Cs I |
| $\mathbf{B_{4}}$ | = | $\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $c \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | Cs I |
| $\mathbf{B_{5}}$ | = | $x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $c x_{2} \,\mathbf{\hat{z}}$ | (4c) | Cs II |
| $\mathbf{B_{6}}$ | = | $- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- c \left(x_{2} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | Cs II |
| $\mathbf{B_{7}}$ | = | $- x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $- c x_{2} \,\mathbf{\hat{z}}$ | (4c) | Cs II |
| $\mathbf{B_{8}}$ | = | $\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $c \left(x_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | Cs II |
| $\mathbf{B_{9}}$ | = | $x_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $c x_{3} \,\mathbf{\hat{z}}$ | (4c) | Cs III |
| $\mathbf{B_{10}}$ | = | $- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- c \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | Cs III |
| $\mathbf{B_{11}}$ | = | $- x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $- c x_{3} \,\mathbf{\hat{z}}$ | (4c) | Cs III |
| $\mathbf{B_{12}}$ | = | $\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $c \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | Cs III |
| $\mathbf{B_{13}}$ | = | $x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $c x_{4} \,\mathbf{\hat{z}}$ | (4c) | W I |
| $\mathbf{B_{14}}$ | = | $- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- c \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | W I |
| $\mathbf{B_{15}}$ | = | $- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $- c x_{4} \,\mathbf{\hat{z}}$ | (4c) | W I |
| $\mathbf{B_{16}}$ | = | $\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $c \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4c) | W I |
| $\mathbf{B_{17}}$ | = | $x_{5} \, \mathbf{a}_{1}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \left(4 x_{5} - 1\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{8}a \left(4 x_{5} - 1\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (6e) | W II |
| $\mathbf{B_{18}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \left(4 x_{5} - 1\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{8}a \left(4 x_{5} - 1\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (6e) | W II |
| $\mathbf{B_{19}}$ | = | $- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (6e) | W II |
| $\mathbf{B_{20}}$ | = | $- x_{5} \, \mathbf{a}_{1}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{8}a \left(4 x_{5} + 3\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{24}a \left(12 x_{5} + 1\right) \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ | (6e) | W II |
| $\mathbf{B_{21}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{8}a \left(4 x_{5} - 1\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{24}a \left(12 x_{5} + 5\right) \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ | (6e) | W II |
| $\mathbf{B_{22}}$ | = | $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}- x_{5} \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ | (6e) | W II |
| $\mathbf{B_{23}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{24}}$ | = | $z_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+y_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{25}}$ | = | $y_{6} \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{26}}$ | = | $- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{6} + 2 y_{6} + 2 z_{6} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{27}}$ | = | $- \left(y_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{6} + 2 y_{6} + 2 z_{6} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{28}}$ | = | $- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{6} + 2 y_{6} + 2 z_{6} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{29}}$ | = | $- x_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{30}}$ | = | $- z_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- y_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{31}}$ | = | $- y_{6} \, \mathbf{a}_{1}- z_{6} \, \mathbf{a}_{2}- x_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{32}}$ | = | $\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{6} + 2 y_{6} + 2 z_{6} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{33}}$ | = | $\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{6} + 2 y_{6} + 2 z_{6} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{34}}$ | = | $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{6} + 2 y_{6} + 2 z_{6} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O I |
| $\mathbf{B_{35}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - z_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} - 2 y_{7} + z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{36}}$ | = | $z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{7} - z_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{7} - y_{7} - z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{37}}$ | = | $y_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} + y_{7} - 2 z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{38}}$ | = | $- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - z_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{7} - 2 y_{7} + z_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{7} + 2 y_{7} + 2 z_{7} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{39}}$ | = | $- \left(y_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{7} - z_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{7} - y_{7} - z_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{7} + 2 y_{7} + 2 z_{7} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{40}}$ | = | $- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{7} + y_{7} - 2 z_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{7} + 2 y_{7} + 2 z_{7} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{41}}$ | = | $- x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} - z_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{7} - 2 y_{7} + z_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{42}}$ | = | $- z_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- y_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{7} - z_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{7} - y_{7} - z_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{43}}$ | = | $- y_{7} \, \mathbf{a}_{1}- z_{7} \, \mathbf{a}_{2}- x_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{7} + y_{7} - 2 z_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{44}}$ | = | $\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} - z_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} - 2 y_{7} + z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{7} + 2 y_{7} + 2 z_{7} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{45}}$ | = | $\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{7} - z_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{7} - y_{7} - z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{7} + 2 y_{7} + 2 z_{7} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{46}}$ | = | $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} + y_{7} - 2 z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{7} + 2 y_{7} + 2 z_{7} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O II |
| $\mathbf{B_{47}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{8} - 2 y_{8} + z_{8}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{8} + y_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{48}}$ | = | $z_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+y_{8} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{8} - z_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{8} - y_{8} - z_{8}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{8} + y_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{49}}$ | = | $y_{8} \, \mathbf{a}_{1}+z_{8} \, \mathbf{a}_{2}+x_{8} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{8} + y_{8} - 2 z_{8}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{8} + y_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{50}}$ | = | $- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{8} - 2 y_{8} + z_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{8} + 2 y_{8} + 2 z_{8} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{51}}$ | = | $- \left(y_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{8} - z_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{8} - y_{8} - z_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{8} + 2 y_{8} + 2 z_{8} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{52}}$ | = | $- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{8} + y_{8} - 2 z_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{8} + 2 y_{8} + 2 z_{8} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{53}}$ | = | $- x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{8} - 2 y_{8} + z_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{8} + y_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{54}}$ | = | $- z_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- y_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{8} - z_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{8} - y_{8} - z_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{8} + y_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{55}}$ | = | $- y_{8} \, \mathbf{a}_{1}- z_{8} \, \mathbf{a}_{2}- x_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{8} + y_{8} - 2 z_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{8} + y_{8} + z_{8}\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{56}}$ | = | $\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} - z_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{8} - 2 y_{8} + z_{8}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{8} + 2 y_{8} + 2 z_{8} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{57}}$ | = | $\left(y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{8} - z_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{8} - y_{8} - z_{8}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{8} + 2 y_{8} + 2 z_{8} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{58}}$ | = | $\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{8} + y_{8} - 2 z_{8}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{8} + 2 y_{8} + 2 z_{8} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O III |
| $\mathbf{B_{59}}$ | = | $x_{9} \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{9} - 2 y_{9} + z_{9}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{9} + y_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{60}}$ | = | $z_{9} \, \mathbf{a}_{1}+x_{9} \, \mathbf{a}_{2}+y_{9} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{9} - z_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{9} - y_{9} - z_{9}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{9} + y_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{61}}$ | = | $y_{9} \, \mathbf{a}_{1}+z_{9} \, \mathbf{a}_{2}+x_{9} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{9} + y_{9} - 2 z_{9}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{9} + y_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{62}}$ | = | $- \left(z_{9} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{9} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{9} - 2 y_{9} + z_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{9} + 2 y_{9} + 2 z_{9} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{63}}$ | = | $- \left(y_{9} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{9} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{9} - z_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{9} - y_{9} - z_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{9} + 2 y_{9} + 2 z_{9} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{64}}$ | = | $- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{9} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{9} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{9} + y_{9} - 2 z_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{9} + 2 y_{9} + 2 z_{9} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{65}}$ | = | $- x_{9} \, \mathbf{a}_{1}- y_{9} \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{9} - 2 y_{9} + z_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{9} + y_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{66}}$ | = | $- z_{9} \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}- y_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{9} - z_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{9} - y_{9} - z_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{9} + y_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{67}}$ | = | $- y_{9} \, \mathbf{a}_{1}- z_{9} \, \mathbf{a}_{2}- x_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{9} + y_{9} - 2 z_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{9} + y_{9} + z_{9}\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{68}}$ | = | $\left(z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{9} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{9} - z_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{9} - 2 y_{9} + z_{9}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{9} + 2 y_{9} + 2 z_{9} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{69}}$ | = | $\left(y_{9} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{9} - z_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{9} - y_{9} - z_{9}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{9} + 2 y_{9} + 2 z_{9} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{70}}$ | = | $\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{9} + y_{9} - 2 z_{9}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{9} + 2 y_{9} + 2 z_{9} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O IV |
| $\mathbf{B_{71}}$ | = | $x_{10} \, \mathbf{a}_{1}+y_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} - z_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{10} - 2 y_{10} + z_{10}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{10} + y_{10} + z_{10}\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{72}}$ | = | $z_{10} \, \mathbf{a}_{1}+x_{10} \, \mathbf{a}_{2}+y_{10} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{10} - z_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{10} - y_{10} - z_{10}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{10} + y_{10} + z_{10}\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{73}}$ | = | $y_{10} \, \mathbf{a}_{1}+z_{10} \, \mathbf{a}_{2}+x_{10} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{10} + y_{10} - 2 z_{10}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{10} + y_{10} + z_{10}\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{74}}$ | = | $- \left(z_{10} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{10} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{10} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} - z_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{10} - 2 y_{10} + z_{10}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{10} + 2 y_{10} + 2 z_{10} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{75}}$ | = | $- \left(y_{10} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{10} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{10} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{10} - z_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{10} - y_{10} - z_{10}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{10} + 2 y_{10} + 2 z_{10} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{76}}$ | = | $- \left(x_{10} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{10} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{10} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{10} + y_{10} - 2 z_{10}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{10} + 2 y_{10} + 2 z_{10} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{77}}$ | = | $- x_{10} \, \mathbf{a}_{1}- y_{10} \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{10} - z_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{10} - 2 y_{10} + z_{10}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{10} + y_{10} + z_{10}\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{78}}$ | = | $- z_{10} \, \mathbf{a}_{1}- x_{10} \, \mathbf{a}_{2}- y_{10} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{10} - z_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{10} - y_{10} - z_{10}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{10} + y_{10} + z_{10}\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{79}}$ | = | $- y_{10} \, \mathbf{a}_{1}- z_{10} \, \mathbf{a}_{2}- x_{10} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{10} + y_{10} - 2 z_{10}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{10} + y_{10} + z_{10}\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{80}}$ | = | $\left(z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{10} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{10} - z_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{10} - 2 y_{10} + z_{10}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{10} + 2 y_{10} + 2 z_{10} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{81}}$ | = | $\left(y_{10} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{10} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{10} - z_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{10} - y_{10} - z_{10}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{10} + 2 y_{10} + 2 z_{10} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{82}}$ | = | $\left(x_{10} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{10} + y_{10} - 2 z_{10}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{10} + 2 y_{10} + 2 z_{10} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O V |
| $\mathbf{B_{83}}$ | = | $x_{11} \, \mathbf{a}_{1}+y_{11} \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} - z_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{11} - 2 y_{11} + z_{11}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{11} + y_{11} + z_{11}\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{84}}$ | = | $z_{11} \, \mathbf{a}_{1}+x_{11} \, \mathbf{a}_{2}+y_{11} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{11} - z_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{11} - y_{11} - z_{11}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{11} + y_{11} + z_{11}\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{85}}$ | = | $y_{11} \, \mathbf{a}_{1}+z_{11} \, \mathbf{a}_{2}+x_{11} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{11} + y_{11} - 2 z_{11}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{11} + y_{11} + z_{11}\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{86}}$ | = | $- \left(z_{11} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{11} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{11} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} - z_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{11} - 2 y_{11} + z_{11}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{11} + 2 y_{11} + 2 z_{11} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{87}}$ | = | $- \left(y_{11} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{11} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{11} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{11} - z_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{11} - y_{11} - z_{11}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{11} + 2 y_{11} + 2 z_{11} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{88}}$ | = | $- \left(x_{11} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{11} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{11} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{11} + y_{11} - 2 z_{11}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{11} + 2 y_{11} + 2 z_{11} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{89}}$ | = | $- x_{11} \, \mathbf{a}_{1}- y_{11} \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{11} - z_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{11} - 2 y_{11} + z_{11}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{11} + y_{11} + z_{11}\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{90}}$ | = | $- z_{11} \, \mathbf{a}_{1}- x_{11} \, \mathbf{a}_{2}- y_{11} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{11} - z_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{11} - y_{11} - z_{11}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{11} + y_{11} + z_{11}\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{91}}$ | = | $- y_{11} \, \mathbf{a}_{1}- z_{11} \, \mathbf{a}_{2}- x_{11} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{11} + y_{11} - 2 z_{11}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{11} + y_{11} + z_{11}\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{92}}$ | = | $\left(z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{11} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{11} - z_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{11} - 2 y_{11} + z_{11}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{11} + 2 y_{11} + 2 z_{11} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{93}}$ | = | $\left(y_{11} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{11} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{11} - z_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{11} - y_{11} - z_{11}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{11} + 2 y_{11} + 2 z_{11} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{94}}$ | = | $\left(x_{11} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{11} + y_{11} - 2 z_{11}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{11} + 2 y_{11} + 2 z_{11} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | O VI |
| $\mathbf{B_{95}}$ | = | $x_{12} \, \mathbf{a}_{1}+y_{12} \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{12} - z_{12}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{12} - 2 y_{12} + z_{12}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{12} + y_{12} + z_{12}\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{96}}$ | = | $z_{12} \, \mathbf{a}_{1}+x_{12} \, \mathbf{a}_{2}+y_{12} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{12} - z_{12}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{12} - y_{12} - z_{12}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{12} + y_{12} + z_{12}\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{97}}$ | = | $y_{12} \, \mathbf{a}_{1}+z_{12} \, \mathbf{a}_{2}+x_{12} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{12} - y_{12}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{12} + y_{12} - 2 z_{12}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{12} + y_{12} + z_{12}\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{98}}$ | = | $- \left(z_{12} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{12} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{12} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{12} - z_{12}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{12} - 2 y_{12} + z_{12}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{12} + 2 y_{12} + 2 z_{12} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{99}}$ | = | $- \left(y_{12} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{12} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(z_{12} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{12} - z_{12}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{12} - y_{12} - z_{12}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{12} + 2 y_{12} + 2 z_{12} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{100}}$ | = | $- \left(x_{12} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(z_{12} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{12} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{12} - y_{12}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{12} + y_{12} - 2 z_{12}\right) \,\mathbf{\hat{y}}- \frac{1}{6}c \left(2 x_{12} + 2 y_{12} + 2 z_{12} - 3\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{101}}$ | = | $- x_{12} \, \mathbf{a}_{1}- y_{12} \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{12} - z_{12}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{12} - 2 y_{12} + z_{12}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{12} + y_{12} + z_{12}\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{102}}$ | = | $- z_{12} \, \mathbf{a}_{1}- x_{12} \, \mathbf{a}_{2}- y_{12} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{12} - z_{12}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(2 x_{12} - y_{12} - z_{12}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{12} + y_{12} + z_{12}\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{103}}$ | = | $- y_{12} \, \mathbf{a}_{1}- z_{12} \, \mathbf{a}_{2}- x_{12} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{12} - y_{12}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{12} + y_{12} - 2 z_{12}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(x_{12} + y_{12} + z_{12}\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{104}}$ | = | $\left(z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{12} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{12} - z_{12}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{12} - 2 y_{12} + z_{12}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{12} + 2 y_{12} + 2 z_{12} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{105}}$ | = | $\left(y_{12} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{12} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{12} - z_{12}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{12} - y_{12} - z_{12}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{12} + 2 y_{12} + 2 z_{12} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
| $\mathbf{B_{106}}$ | = | $\left(x_{12} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{12} - y_{12}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{12} + y_{12} - 2 z_{12}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{12} + 2 y_{12} + 2 z_{12} + 3\right) \,\mathbf{\hat{z}}$ | (12f) | W III |
References
- R. E. Marsh and I. Bernal, More Space-Group Changes, Acta Crystallogr. Sect. B 51, 300–307 (1995), doi:10.1107/S0108768194011857.
- K. Okada, F. Marumo, and S. Iwai, The crystal structure of Cs$_{6}$W$_{11}$O$_{36}$, Acta Crystallogr. Sect. B 34, 50–54 (1978), doi:10.1107/S0567740878014934.
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=A6B36C11_hR106_167_3c_6f_cef --params=$a,c/a,x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12}$