α-BaB$_{2}$O$_{4}$ (Low Temperature) Structure: A2BC4_hR42_161_2b_b_4b-001
| Prototype | B$_{2}$BaO$_{4}$ |
| AFLOW prototype label | A2BC4_hR42_161_2b_b_4b-001 |
| ICSD | 30885 |
| Pearson symbol | hR42 |
| Space group number | 161 |
| Space group symbol | $R3c$ |
| AFLOW prototype command |
aflow --proto=A2BC4_hR42_161_2b_b_4b-001
--params=$a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak y_{1}, \allowbreak z_{1}, \allowbreak x_{2}, \allowbreak y_{2}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak y_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}$ |
- This is the low-temperature structure. Heating to temperatures between 100-400°C it transforms into $\beta$–BaB$_{2}$O$_{4}$. The principle difference between the two forms is the lack of inversion symmetry in the low-temperature structure.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{\sqrt{3}}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+y_{1} \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{1} - 2 y_{1} + z_{1}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{1} + y_{1} + z_{1}\right) \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{2}}$ | = | $z_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+y_{1} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{1} - z_{1}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{1} - y_{1} - z_{1}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{1} + y_{1} + z_{1}\right) \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{3}}$ | = | $y_{1} \, \mathbf{a}_{1}+z_{1} \, \mathbf{a}_{2}+x_{1} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{1} - y_{1}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{1} + y_{1} - 2 z_{1}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{1} + y_{1} + z_{1}\right) \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{4}}$ | = | $\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{1} - z_{1}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{1} - 2 y_{1} + z_{1}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{1} + 2 y_{1} + 2 z_{1} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{5}}$ | = | $\left(y_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{1} - z_{1}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{1} - y_{1} - z_{1}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{1} + 2 y_{1} + 2 z_{1} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{6}}$ | = | $\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{1} - y_{1}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{1} + y_{1} - 2 z_{1}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{1} + 2 y_{1} + 2 z_{1} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{7}}$ | = | $x_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{2} - 2 y_{2} + z_{2}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{2} + y_{2} + z_{2}\right) \,\mathbf{\hat{z}}$ | (6b) | B II |
| $\mathbf{B_{8}}$ | = | $z_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+y_{2} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{2} - z_{2}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{2} - y_{2} - z_{2}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{2} + y_{2} + z_{2}\right) \,\mathbf{\hat{z}}$ | (6b) | B II |
| $\mathbf{B_{9}}$ | = | $y_{2} \, \mathbf{a}_{1}+z_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{2} - y_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{2} + y_{2} - 2 z_{2}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{2} + y_{2} + z_{2}\right) \,\mathbf{\hat{z}}$ | (6b) | B II |
| $\mathbf{B_{10}}$ | = | $\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{2} - z_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{2} - 2 y_{2} + z_{2}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{2} + 2 y_{2} + 2 z_{2} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | B II |
| $\mathbf{B_{11}}$ | = | $\left(y_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{2} - z_{2}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{2} - y_{2} - z_{2}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{2} + 2 y_{2} + 2 z_{2} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | B II |
| $\mathbf{B_{12}}$ | = | $\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} - y_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{2} + y_{2} - 2 z_{2}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{2} + 2 y_{2} + 2 z_{2} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | B II |
| $\mathbf{B_{13}}$ | = | $x_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{3} - 2 y_{3} + z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{3} + y_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6b) | Ba I |
| $\mathbf{B_{14}}$ | = | $z_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+y_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{3} - z_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{3} - y_{3} - z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{3} + y_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6b) | Ba I |
| $\mathbf{B_{15}}$ | = | $y_{3} \, \mathbf{a}_{1}+z_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{3} + y_{3} - 2 z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{3} + y_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6b) | Ba I |
| $\mathbf{B_{16}}$ | = | $\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{3} - 2 y_{3} + z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{3} + 2 y_{3} + 2 z_{3} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | Ba I |
| $\mathbf{B_{17}}$ | = | $\left(y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{3} - z_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{3} - y_{3} - z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{3} + 2 y_{3} + 2 z_{3} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | Ba I |
| $\mathbf{B_{18}}$ | = | $\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{3} + y_{3} - 2 z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{3} + 2 y_{3} + 2 z_{3} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | Ba I |
| $\mathbf{B_{19}}$ | = | $x_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{4} - 2 y_{4} + z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{4} + y_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6b) | O I |
| $\mathbf{B_{20}}$ | = | $z_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+y_{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{4} - z_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{4} - y_{4} - z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{4} + y_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6b) | O I |
| $\mathbf{B_{21}}$ | = | $y_{4} \, \mathbf{a}_{1}+z_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{4} - y_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{4} + y_{4} - 2 z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{4} + y_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6b) | O I |
| $\mathbf{B_{22}}$ | = | $\left(z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{4} - 2 y_{4} + z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{4} + 2 y_{4} + 2 z_{4} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O I |
| $\mathbf{B_{23}}$ | = | $\left(y_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{4} - z_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{4} - y_{4} - z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{4} + 2 y_{4} + 2 z_{4} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O I |
| $\mathbf{B_{24}}$ | = | $\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} - y_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{4} + y_{4} - 2 z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{4} + 2 y_{4} + 2 z_{4} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O I |
| $\mathbf{B_{25}}$ | = | $x_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} - 2 y_{5} + z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6b) | O II |
| $\mathbf{B_{26}}$ | = | $z_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+y_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{5} - z_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{5} - y_{5} - z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6b) | O II |
| $\mathbf{B_{27}}$ | = | $y_{5} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{2}+x_{5} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} + y_{5} - 2 z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{5} + y_{5} + z_{5}\right) \,\mathbf{\hat{z}}$ | (6b) | O II |
| $\mathbf{B_{28}}$ | = | $\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} - z_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} - 2 y_{5} + z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{5} + 2 y_{5} + 2 z_{5} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O II |
| $\mathbf{B_{29}}$ | = | $\left(y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{5} - z_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{5} - y_{5} - z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{5} + 2 y_{5} + 2 z_{5} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O II |
| $\mathbf{B_{30}}$ | = | $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{5} + y_{5} - 2 z_{5}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{5} + 2 y_{5} + 2 z_{5} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O II |
| $\mathbf{B_{31}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6b) | O III |
| $\mathbf{B_{32}}$ | = | $z_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+y_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6b) | O III |
| $\mathbf{B_{33}}$ | = | $y_{6} \, \mathbf{a}_{1}+z_{6} \, \mathbf{a}_{2}+x_{6} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{6} + y_{6} + z_{6}\right) \,\mathbf{\hat{z}}$ | (6b) | O III |
| $\mathbf{B_{34}}$ | = | $\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} - z_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} - 2 y_{6} + z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{6} + 2 y_{6} + 2 z_{6} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O III |
| $\mathbf{B_{35}}$ | = | $\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{6} - z_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{6} - y_{6} - z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{6} + 2 y_{6} + 2 z_{6} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O III |
| $\mathbf{B_{36}}$ | = | $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{6} + y_{6} - 2 z_{6}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{6} + 2 y_{6} + 2 z_{6} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O III |
| $\mathbf{B_{37}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - z_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} - 2 y_{7} + z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (6b) | O IV |
| $\mathbf{B_{38}}$ | = | $z_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+y_{7} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(y_{7} - z_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{7} - y_{7} - z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (6b) | O IV |
| $\mathbf{B_{39}}$ | = | $y_{7} \, \mathbf{a}_{1}+z_{7} \, \mathbf{a}_{2}+x_{7} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} + y_{7} - 2 z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(x_{7} + y_{7} + z_{7}\right) \,\mathbf{\hat{z}}$ | (6b) | O IV |
| $\mathbf{B_{40}}$ | = | $\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} - z_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} - 2 y_{7} + z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{7} + 2 y_{7} + 2 z_{7} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O IV |
| $\mathbf{B_{41}}$ | = | $\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(y_{7} - z_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(2 x_{7} - y_{7} - z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{7} + 2 y_{7} + 2 z_{7} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O IV |
| $\mathbf{B_{42}}$ | = | $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{7} + y_{7} - 2 z_{7}\right) \,\mathbf{\hat{y}}+\frac{1}{6}c \left(2 x_{7} + 2 y_{7} + 2 z_{7} + 3\right) \,\mathbf{\hat{z}}$ | (6b) | O IV |
References
- R. Fröhlich, Crystal Structure of the low-temperature form of BaB$_{2}$O$_{4}$, Z. Kristallogr. 168, 109–112 (1984), doi:10.1524/zkri.1984.168.14.109.
Prototype Generator
aflow --proto=A2BC4_hR42_161_2b_b_4b --params=$a,c/a,x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},x_{3},y_{3},z_{3},x_{4},y_{4},z_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7}$