Boropollucite [Cs(Si$_{2}$B)O$_{6}$] Structure: AB6C3_cI160_230_b_h_g-001
| Prototype | BCsO$_{6}$Si$_{2}$ |
| AFLOW prototype label | AB6C3_cI160_230_b_h_g-001 |
| Mineral name | boropollucite |
| ICSD | 413224 |
| CCDC | 1728269 |
| Pearson symbol | cI160 |
| Space group number | 230 |
| Space group symbol | $Ia\overline{3}d$ |
| AFLOW prototype command |
aflow --proto=AB6C3_cI160_230_b_h_g-001
--params=$a, \allowbreak y_{2}, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}$ |
- According to (Bubnova, 2004) the Cs (16b) site is 81.4% filled, and the Si (48g) site is 65.9% silicon and 36.5% boron. This gives the final stoichiometry of Cs$_{0.814}$B$_{1.095}$Si$_{1.977}$O$_{6}$
- Other entries in the ICSD show that the Cs site may also contain calcium.
- The Wyckoff labels listed in Table 3 of (Bubnova, 2004) are incorrect. They are corrected in the ICSD entry.
- The Wyckoff positions in the ICSD differ slightly from the published values. We use the values from the ICSD CIF.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}a \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}- \frac{1}{2}a \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (16b) | Cs I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{8}a \,\mathbf{\hat{x}}+\frac{3}{8}a \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (16b) | Cs I |
| $\mathbf{B_{3}}$ | = | $\frac{1}{4} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}- \frac{1}{8}a \,\mathbf{\hat{z}}$ | (16b) | Cs I |
| $\mathbf{B_{4}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}- \frac{1}{8}a \,\mathbf{\hat{y}}+\frac{3}{8}a \,\mathbf{\hat{z}}$ | (16b) | Cs I |
| $\mathbf{B_{5}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}+\frac{3}{8}a \,\mathbf{\hat{y}}+\frac{3}{8}a \,\mathbf{\hat{z}}$ | (16b) | Cs I |
| $\mathbf{B_{6}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}+\frac{5}{8}a \,\mathbf{\hat{y}}- \frac{1}{8}a \,\mathbf{\hat{z}}$ | (16b) | Cs I |
| $\mathbf{B_{7}}$ | = | $\frac{3}{4} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{5}{8}a \,\mathbf{\hat{x}}- \frac{1}{8}a \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (16b) | Cs I |
| $\mathbf{B_{8}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}$ | = | $- \frac{1}{8}a \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}+\frac{5}{8}a \,\mathbf{\hat{z}}$ | (16b) | Cs I |
| $\mathbf{B_{9}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}- \left(y_{2} - \frac{3}{8}\right) \, \mathbf{a}_{2}+\left(y_{2} + \frac{1}{8}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}+a y_{2} \,\mathbf{\hat{y}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{10}}$ | = | $- \left(2 y_{2} - \frac{3}{4}\right) \, \mathbf{a}_{1}- \left(y_{2} - \frac{1}{8}\right) \, \mathbf{a}_{2}- \left(y_{2} - \frac{3}{8}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{8}a \,\mathbf{\hat{x}}- a \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{11}}$ | = | $\left(2 y_{2} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\left(y_{2} + \frac{1}{8}\right) \, \mathbf{a}_{2}+\left(y_{2} + \frac{3}{8}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{8}a \,\mathbf{\hat{x}}+a \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{12}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+\left(y_{2} + \frac{3}{8}\right) \, \mathbf{a}_{2}- \left(y_{2} - \frac{1}{8}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{8}a \,\mathbf{\hat{x}}- a y_{2} \,\mathbf{\hat{y}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{13}}$ | = | $\left(y_{2} + \frac{1}{8}\right) \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}- \left(y_{2} - \frac{3}{8}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}+a y_{2} \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{14}}$ | = | $- \left(y_{2} - \frac{3}{8}\right) \, \mathbf{a}_{1}- \left(2 y_{2} - \frac{3}{4}\right) \, \mathbf{a}_{2}- \left(y_{2} - \frac{1}{8}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- \frac{1}{8}a \,\mathbf{\hat{y}}- a \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{15}}$ | = | $\left(y_{2} + \frac{3}{8}\right) \, \mathbf{a}_{1}+\left(2 y_{2} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\left(y_{2} + \frac{1}{8}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- \frac{1}{8}a \,\mathbf{\hat{y}}+a \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{16}}$ | = | $- \left(y_{2} - \frac{1}{8}\right) \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+\left(y_{2} + \frac{3}{8}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{1}{8}a \,\mathbf{\hat{y}}- a y_{2} \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{17}}$ | = | $- \left(y_{2} - \frac{3}{8}\right) \, \mathbf{a}_{1}+\left(y_{2} + \frac{1}{8}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $a y_{2} \,\mathbf{\hat{x}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{18}}$ | = | $- \left(y_{2} - \frac{1}{8}\right) \, \mathbf{a}_{1}- \left(y_{2} - \frac{3}{8}\right) \, \mathbf{a}_{2}- \left(2 y_{2} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- \frac{1}{8}a \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{19}}$ | = | $\left(y_{2} + \frac{1}{8}\right) \, \mathbf{a}_{1}+\left(y_{2} + \frac{3}{8}\right) \, \mathbf{a}_{2}+\left(2 y_{2} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- \frac{1}{8}a \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{20}}$ | = | $\left(y_{2} + \frac{3}{8}\right) \, \mathbf{a}_{1}- \left(y_{2} - \frac{1}{8}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $- a y_{2} \,\mathbf{\hat{x}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}a \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{21}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}+\left(y_{2} + \frac{5}{8}\right) \, \mathbf{a}_{2}- \left(y_{2} - \frac{7}{8}\right) \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}- a \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{22}}$ | = | $\left(2 y_{2} + \frac{1}{4}\right) \, \mathbf{a}_{1}+\left(y_{2} + \frac{7}{8}\right) \, \mathbf{a}_{2}+\left(y_{2} + \frac{5}{8}\right) \, \mathbf{a}_{3}$ | = | $\frac{5}{8}a \,\mathbf{\hat{x}}+a y_{2} \,\mathbf{\hat{y}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{23}}$ | = | $- \left(2 y_{2} - \frac{1}{4}\right) \, \mathbf{a}_{1}- \left(y_{2} - \frac{7}{8}\right) \, \mathbf{a}_{2}- \left(y_{2} - \frac{5}{8}\right) \, \mathbf{a}_{3}$ | = | $\frac{5}{8}a \,\mathbf{\hat{x}}- a y_{2} \,\mathbf{\hat{y}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{24}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}- \left(y_{2} - \frac{5}{8}\right) \, \mathbf{a}_{2}+\left(y_{2} + \frac{7}{8}\right) \, \mathbf{a}_{3}$ | = | $\frac{3}{8}a \,\mathbf{\hat{x}}+a \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{25}}$ | = | $- \left(y_{2} - \frac{7}{8}\right) \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\left(y_{2} + \frac{5}{8}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{3}{8}a \,\mathbf{\hat{y}}- a \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{26}}$ | = | $\left(y_{2} + \frac{5}{8}\right) \, \mathbf{a}_{1}+\left(2 y_{2} + \frac{1}{4}\right) \, \mathbf{a}_{2}+\left(y_{2} + \frac{7}{8}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{5}{8}a \,\mathbf{\hat{y}}+a y_{2} \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{27}}$ | = | $- \left(y_{2} - \frac{5}{8}\right) \, \mathbf{a}_{1}- \left(2 y_{2} - \frac{1}{4}\right) \, \mathbf{a}_{2}- \left(y_{2} - \frac{7}{8}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{5}{8}a \,\mathbf{\hat{y}}- a y_{2} \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{28}}$ | = | $\left(y_{2} + \frac{7}{8}\right) \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}- \left(y_{2} - \frac{5}{8}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+\frac{3}{8}a \,\mathbf{\hat{y}}+a \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{29}}$ | = | $\left(y_{2} + \frac{5}{8}\right) \, \mathbf{a}_{1}- \left(y_{2} - \frac{7}{8}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $- a \left(y_{2} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}a \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{30}}$ | = | $\left(y_{2} + \frac{7}{8}\right) \, \mathbf{a}_{1}+\left(y_{2} + \frac{5}{8}\right) \, \mathbf{a}_{2}+\left(2 y_{2} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $a y_{2} \,\mathbf{\hat{x}}+a \left(y_{2} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{5}{8}a \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{31}}$ | = | $- \left(y_{2} - \frac{7}{8}\right) \, \mathbf{a}_{1}- \left(y_{2} - \frac{5}{8}\right) \, \mathbf{a}_{2}- \left(2 y_{2} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- a y_{2} \,\mathbf{\hat{x}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{5}{8}a \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{32}}$ | = | $- \left(y_{2} - \frac{5}{8}\right) \, \mathbf{a}_{1}+\left(y_{2} + \frac{7}{8}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $a \left(y_{2} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(y_{2} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}a \,\mathbf{\hat{z}}$ | (48g) | Si I |
| $\mathbf{B_{33}}$ | = | $\left(y_{3} + z_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}+\left(x_{3} + y_{3}\right) \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+a y_{3} \,\mathbf{\hat{y}}+a z_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{34}}$ | = | $\left(- y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{3} - z_{3}\right) \, \mathbf{a}_{2}- \left(x_{3} + y_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}- a \left(y_{3} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a z_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{35}}$ | = | $\left(y_{3} - z_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a y_{3} \,\mathbf{\hat{y}}- a z_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{36}}$ | = | $- \left(y_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} - y_{3}\right) \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}- a y_{3} \,\mathbf{\hat{y}}- a \left(z_{3} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{37}}$ | = | $\left(x_{3} + y_{3}\right) \, \mathbf{a}_{1}+\left(y_{3} + z_{3}\right) \, \mathbf{a}_{2}+\left(x_{3} + z_{3}\right) \, \mathbf{a}_{3}$ | = | $a z_{3} \,\mathbf{\hat{x}}+a x_{3} \,\mathbf{\hat{y}}+a y_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{38}}$ | = | $- \left(x_{3} + y_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(- y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{3} - z_{3}\right) \, \mathbf{a}_{3}$ | = | $a z_{3} \,\mathbf{\hat{x}}- a x_{3} \,\mathbf{\hat{y}}- a \left(y_{3} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{39}}$ | = | $\left(- x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{3} - z_{3}\right) \, \mathbf{a}_{2}- \left(x_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a z_{3} \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a y_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{40}}$ | = | $\left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}- \left(y_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(z_{3} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a x_{3} \,\mathbf{\hat{y}}- a y_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{41}}$ | = | $\left(x_{3} + z_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} + y_{3}\right) \, \mathbf{a}_{2}+\left(y_{3} + z_{3}\right) \, \mathbf{a}_{3}$ | = | $a y_{3} \,\mathbf{\hat{x}}+a z_{3} \,\mathbf{\hat{y}}+a x_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{42}}$ | = | $- \left(x_{3} - z_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + y_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{3} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a z_{3} \,\mathbf{\hat{y}}- a x_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{43}}$ | = | $- \left(x_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(- x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{3} - z_{3}\right) \, \mathbf{a}_{3}$ | = | $a y_{3} \,\mathbf{\hat{x}}- a z_{3} \,\mathbf{\hat{y}}- a \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{44}}$ | = | $\left(x_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}- \left(y_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a y_{3} \,\mathbf{\hat{x}}- a \left(z_{3} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+a x_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{45}}$ | = | $\left(x_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{3} - z_{3}\right) \, \mathbf{a}_{2}+\left(x_{3} + y_{3}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{46}}$ | = | $- \left(x_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{3} + y_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{47}}$ | = | $- \left(x_{3} - z_{3}\right) \, \mathbf{a}_{1}+\left(y_{3} + z_{3}\right) \, \mathbf{a}_{2}+\left(- x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{48}}$ | = | $\left(x_{3} + z_{3}\right) \, \mathbf{a}_{1}+\left(- y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} - y_{3}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{49}}$ | = | $\left(- y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}+\left(x_{3} + z_{3}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{50}}$ | = | $\left(y_{3} + z_{3}\right) \, \mathbf{a}_{1}+\left(- x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{3} - z_{3}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{51}}$ | = | $- \left(y_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{3} + y_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{52}}$ | = | $\left(y_{3} - z_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} + y_{3}\right) \, \mathbf{a}_{2}+\left(x_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{53}}$ | = | $\left(- x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{3} - z_{3}\right) \, \mathbf{a}_{2}+\left(y_{3} + z_{3}\right) \, \mathbf{a}_{3}$ | = | $a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{54}}$ | = | $\left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}+\left(- y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{55}}$ | = | $\left(x_{3} + y_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{3} - z_{3}\right) \, \mathbf{a}_{3}$ | = | $- a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{56}}$ | = | $- \left(x_{3} + y_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{57}}$ | = | $- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}- a y_{3} \,\mathbf{\hat{y}}- a z_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{58}}$ | = | $\left(y_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} - z_{3}\right) \, \mathbf{a}_{2}+\left(x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+a \left(y_{3} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a z_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{59}}$ | = | $- \left(y_{3} - z_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} - y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a y_{3} \,\mathbf{\hat{y}}+a z_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{60}}$ | = | $\left(y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(- x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}+a y_{3} \,\mathbf{\hat{y}}+a \left(z_{3} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{61}}$ | = | $- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{1}- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{2}- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{3}$ | = | $- a z_{3} \,\mathbf{\hat{x}}- a x_{3} \,\mathbf{\hat{y}}- a y_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{62}}$ | = | $\left(x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} - z_{3}\right) \, \mathbf{a}_{3}$ | = | $- a z_{3} \,\mathbf{\hat{x}}+a x_{3} \,\mathbf{\hat{y}}+a \left(y_{3} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{63}}$ | = | $\left(x_{3} - y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{3} - z_{3}\right) \, \mathbf{a}_{2}+\left(x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a z_{3} \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a y_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{64}}$ | = | $- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}+\left(y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(z_{3} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a x_{3} \,\mathbf{\hat{y}}+a y_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{65}}$ | = | $- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{2}- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{3}$ | = | $- a y_{3} \,\mathbf{\hat{x}}- a z_{3} \,\mathbf{\hat{y}}- a x_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{66}}$ | = | $\left(x_{3} - z_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{3} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a z_{3} \,\mathbf{\hat{y}}+a x_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{67}}$ | = | $\left(x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} - y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{3} - z_{3}\right) \, \mathbf{a}_{3}$ | = | $- a y_{3} \,\mathbf{\hat{x}}+a z_{3} \,\mathbf{\hat{y}}+a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{68}}$ | = | $\left(- x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}+\left(y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a y_{3} \,\mathbf{\hat{x}}+a \left(z_{3} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- a x_{3} \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{69}}$ | = | $\left(- x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{3} - z_{3}\right) \, \mathbf{a}_{2}- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{70}}$ | = | $\left(x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{71}}$ | = | $\left(x_{3} - z_{3}\right) \, \mathbf{a}_{1}- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{2}+\left(x_{3} - y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{72}}$ | = | $- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{1}+\left(y_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{73}}$ | = | $\left(y_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{74}}$ | = | $- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} - y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} - z_{3}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{75}}$ | = | $\left(y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{76}}$ | = | $- \left(y_{3} - z_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{2}+\left(- x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{77}}$ | = | $\left(x_{3} - y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} - z_{3}\right) \, \mathbf{a}_{2}- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{3}$ | = | $- a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(y_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{78}}$ | = | $- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}+\left(y_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(z_{3} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{79}}$ | = | $- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{1}+\left(- x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(y_{3} - z_{3}\right) \, \mathbf{a}_{3}$ | = | $a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
| $\mathbf{B_{80}}$ | = | $\left(x_{3} + y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(z_{3} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(y_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (96h) | O I |
References
- R. S. Bubnova, N. K. Stepanov, A. A. Levin, S. K. Filatov, P. Paufler, and D. C. Meyer, Crystal structure and thermal behaviour of boropollucite CsBSi$_{2}$O$_{6}$, Solid State Sci. 6, 629–637 (2004), doi:10.1016/j.solidstatesciences.2004.03.015.
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=AB6C3_cI160_230_b_h_g --params=$a,y_{2},x_{3},y_{3},z_{3}$