Sc(H$_{2}$O)$_{2}$[BP$_{2}$O$_{8}$]$\cdot$H$_{2}$O Structure: AB4C12D2E_hP120_179_b_2c_6c_c_b-001
| Prototype | BH$_{4}$O$_{12}$P$_{2}$Sc |
| AFLOW prototype label | AB4C12D2E_hP120_179_b_2c_6c_c_b-001 |
| ICSD | 416075 |
| Pearson symbol | hP120 |
| Space group number | 179 |
| Space group symbol | $P6_522$ |
| AFLOW prototype command |
aflow --proto=AB4C12D2E_hP120_179_b_2c_6c_c_b-001
--params=$a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak x_{2}, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak y_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}$ |
- The O-VI (12c) site, actually a water molecule with undetermined hydrogen positions, is only occupied 50% of the time.
- There is also a
dehydrated
form, Sc(H$_{2}$O)$_{2}$[BP$_{2}$O$_{8}$], where these water molecules are not present. - This structure can also be found in the enantiomorphic space group $P6_{1}22$ #178.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+2 x_{1} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{1} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{1} \,\mathbf{\hat{y}}+\frac{3}{4}c \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{2}}$ | = | $- 2 x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}+\frac{5}{12} \, \mathbf{a}_{3}$ | = | $- \frac{3}{2}a x_{1} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{1} \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{3}}$ | = | $x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}+\frac{1}{12} \, \mathbf{a}_{3}$ | = | $- \sqrt{3}a x_{1} \,\mathbf{\hat{y}}+\frac{1}{12}c \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{4}}$ | = | $- x_{1} \, \mathbf{a}_{1}- 2 x_{1} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $- \frac{3}{2}a x_{1} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{1} \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{5}}$ | = | $2 x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+\frac{11}{12} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{1} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{1} \,\mathbf{\hat{y}}+\frac{11}{12}c \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{6}}$ | = | $- x_{1} \, \mathbf{a}_{1}+x_{1} \, \mathbf{a}_{2}+\frac{7}{12} \, \mathbf{a}_{3}$ | = | $\sqrt{3}a x_{1} \,\mathbf{\hat{y}}+\frac{7}{12}c \,\mathbf{\hat{z}}$ | (6b) | B I |
| $\mathbf{B_{7}}$ | = | $x_{2} \, \mathbf{a}_{1}+2 x_{2} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{2} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+\frac{3}{4}c \,\mathbf{\hat{z}}$ | (6b) | Sc I |
| $\mathbf{B_{8}}$ | = | $- 2 x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\frac{5}{12} \, \mathbf{a}_{3}$ | = | $- \frac{3}{2}a x_{2} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+\frac{5}{12}c \,\mathbf{\hat{z}}$ | (6b) | Sc I |
| $\mathbf{B_{9}}$ | = | $x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\frac{1}{12} \, \mathbf{a}_{3}$ | = | $- \sqrt{3}a x_{2} \,\mathbf{\hat{y}}+\frac{1}{12}c \,\mathbf{\hat{z}}$ | (6b) | Sc I |
| $\mathbf{B_{10}}$ | = | $- x_{2} \, \mathbf{a}_{1}- 2 x_{2} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $- \frac{3}{2}a x_{2} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (6b) | Sc I |
| $\mathbf{B_{11}}$ | = | $2 x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+\frac{11}{12} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{2} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+\frac{11}{12}c \,\mathbf{\hat{z}}$ | (6b) | Sc I |
| $\mathbf{B_{12}}$ | = | $- x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+\frac{7}{12} \, \mathbf{a}_{3}$ | = | $\sqrt{3}a x_{2} \,\mathbf{\hat{y}}+\frac{7}{12}c \,\mathbf{\hat{z}}$ | (6b) | Sc I |
| $\mathbf{B_{13}}$ | = | $x_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} + y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{14}}$ | = | $- y_{3} \, \mathbf{a}_{1}+\left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}+\left(z_{3} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - 2 y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{3} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{3} + 2\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{15}}$ | = | $- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+\left(z_{3} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{3} - y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{3} \,\mathbf{\hat{y}}+c \left(z_{3} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{16}}$ | = | $- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{3} + y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{y}}+c \left(z_{3} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{17}}$ | = | $y_{3} \, \mathbf{a}_{1}- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}+\left(z_{3} + \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{3} + 2 y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{3} \,\mathbf{\hat{y}}+c \left(z_{3} + \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{18}}$ | = | $\left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+\left(z_{3} + \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{3} - y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{3} \,\mathbf{\hat{y}}+\frac{1}{6}c \left(6 z_{3} + 5\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{19}}$ | = | $y_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}- \left(z_{3} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} + y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{3} - 2\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{20}}$ | = | $\left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - 2 y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{3} \,\mathbf{\hat{y}}- c z_{3} \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{21}}$ | = | $- x_{3} \, \mathbf{a}_{1}- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}- \left(z_{3} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{3} - y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{3} \,\mathbf{\hat{y}}- c \left(z_{3} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{22}}$ | = | $- y_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- \left(z_{3} - \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{3} + y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{3} - y_{3}\right) \,\mathbf{\hat{y}}- c \left(z_{3} - \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{23}}$ | = | $- \left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}- \left(z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{3} + 2 y_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{3} \,\mathbf{\hat{y}}- c \left(z_{3} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{24}}$ | = | $x_{3} \, \mathbf{a}_{1}+\left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}- \left(z_{3} - \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{3} - y_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{3} \,\mathbf{\hat{y}}- \frac{1}{6}c \left(6 z_{3} - 5\right) \,\mathbf{\hat{z}}$ | (12c) | H I |
| $\mathbf{B_{25}}$ | = | $x_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} + y_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{4} - y_{4}\right) \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{26}}$ | = | $- y_{4} \, \mathbf{a}_{1}+\left(x_{4} - y_{4}\right) \, \mathbf{a}_{2}+\left(z_{4} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} - 2 y_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{4} + 2\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{27}}$ | = | $- \left(x_{4} - y_{4}\right) \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}+\left(z_{4} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{4} - y_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{4} \,\mathbf{\hat{y}}+c \left(z_{4} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{28}}$ | = | $- x_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}+\left(z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{4} + y_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{4} - y_{4}\right) \,\mathbf{\hat{y}}+c \left(z_{4} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{29}}$ | = | $y_{4} \, \mathbf{a}_{1}- \left(x_{4} - y_{4}\right) \, \mathbf{a}_{2}+\left(z_{4} + \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{4} + 2 y_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}+c \left(z_{4} + \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{30}}$ | = | $\left(x_{4} - y_{4}\right) \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+\left(z_{4} + \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{4} - y_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{4} \,\mathbf{\hat{y}}+\frac{1}{6}c \left(6 z_{4} + 5\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{31}}$ | = | $y_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}- \left(z_{4} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} + y_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{4} - y_{4}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{4} - 2\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{32}}$ | = | $\left(x_{4} - y_{4}\right) \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} - 2 y_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}- c z_{4} \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{33}}$ | = | $- x_{4} \, \mathbf{a}_{1}- \left(x_{4} - y_{4}\right) \, \mathbf{a}_{2}- \left(z_{4} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{4} - y_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{4} \,\mathbf{\hat{y}}- c \left(z_{4} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{34}}$ | = | $- y_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- \left(z_{4} - \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{4} + y_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{4} - y_{4}\right) \,\mathbf{\hat{y}}- c \left(z_{4} - \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{35}}$ | = | $- \left(x_{4} - y_{4}\right) \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}- \left(z_{4} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{4} + 2 y_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{4} \,\mathbf{\hat{y}}- c \left(z_{4} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{36}}$ | = | $x_{4} \, \mathbf{a}_{1}+\left(x_{4} - y_{4}\right) \, \mathbf{a}_{2}- \left(z_{4} - \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{4} - y_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{4} \,\mathbf{\hat{y}}- \frac{1}{6}c \left(6 z_{4} - 5\right) \,\mathbf{\hat{z}}$ | (12c) | H II |
| $\mathbf{B_{37}}$ | = | $x_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} + y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{38}}$ | = | $- y_{5} \, \mathbf{a}_{1}+\left(x_{5} - y_{5}\right) \, \mathbf{a}_{2}+\left(z_{5} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - 2 y_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{5} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{5} + 2\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{39}}$ | = | $- \left(x_{5} - y_{5}\right) \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{5} - y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{5} \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{40}}$ | = | $- x_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} + y_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{41}}$ | = | $y_{5} \, \mathbf{a}_{1}- \left(x_{5} - y_{5}\right) \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{5} + 2 y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{5} \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{42}}$ | = | $\left(x_{5} - y_{5}\right) \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}+\left(z_{5} + \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{5} - y_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{5} \,\mathbf{\hat{y}}+\frac{1}{6}c \left(6 z_{5} + 5\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{43}}$ | = | $y_{5} \, \mathbf{a}_{1}+x_{5} \, \mathbf{a}_{2}- \left(z_{5} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} + y_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{5} - 2\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{44}}$ | = | $\left(x_{5} - y_{5}\right) \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{5} - 2 y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{5} \,\mathbf{\hat{y}}- c z_{5} \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{45}}$ | = | $- x_{5} \, \mathbf{a}_{1}- \left(x_{5} - y_{5}\right) \, \mathbf{a}_{2}- \left(z_{5} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{5} - y_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{5} \,\mathbf{\hat{y}}- c \left(z_{5} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{46}}$ | = | $- y_{5} \, \mathbf{a}_{1}- x_{5} \, \mathbf{a}_{2}- \left(z_{5} - \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{5} + y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{5} - y_{5}\right) \,\mathbf{\hat{y}}- c \left(z_{5} - \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{47}}$ | = | $- \left(x_{5} - y_{5}\right) \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}- \left(z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{5} + 2 y_{5}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{5} \,\mathbf{\hat{y}}- c \left(z_{5} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{48}}$ | = | $x_{5} \, \mathbf{a}_{1}+\left(x_{5} - y_{5}\right) \, \mathbf{a}_{2}- \left(z_{5} - \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{5} - y_{5}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{5} \,\mathbf{\hat{y}}- \frac{1}{6}c \left(6 z_{5} - 5\right) \,\mathbf{\hat{z}}$ | (12c) | O I |
| $\mathbf{B_{49}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} + y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{50}}$ | = | $- y_{6} \, \mathbf{a}_{1}+\left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}+\left(z_{6} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - 2 y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{6} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{6} + 2\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{51}}$ | = | $- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{6} \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{52}}$ | = | $- x_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} + y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{53}}$ | = | $y_{6} \, \mathbf{a}_{1}- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{6} + 2 y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{6} \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{54}}$ | = | $\left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}+\left(z_{6} + \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{6} - y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{6} \,\mathbf{\hat{y}}+\frac{1}{6}c \left(6 z_{6} + 5\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{55}}$ | = | $y_{6} \, \mathbf{a}_{1}+x_{6} \, \mathbf{a}_{2}- \left(z_{6} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} + y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{6} - 2\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{56}}$ | = | $\left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{6} - 2 y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{6} \,\mathbf{\hat{y}}- c z_{6} \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{57}}$ | = | $- x_{6} \, \mathbf{a}_{1}- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}- \left(z_{6} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{6} - y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{6} \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{58}}$ | = | $- y_{6} \, \mathbf{a}_{1}- x_{6} \, \mathbf{a}_{2}- \left(z_{6} - \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{6} + y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{6} - y_{6}\right) \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{59}}$ | = | $- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{6} + 2 y_{6}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{6} \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{60}}$ | = | $x_{6} \, \mathbf{a}_{1}+\left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}- \left(z_{6} - \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{6} - y_{6}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{6} \,\mathbf{\hat{y}}- \frac{1}{6}c \left(6 z_{6} - 5\right) \,\mathbf{\hat{z}}$ | (12c) | O II |
| $\mathbf{B_{61}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} + y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{62}}$ | = | $- y_{7} \, \mathbf{a}_{1}+\left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}+\left(z_{7} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - 2 y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{7} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{7} + 2\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{63}}$ | = | $- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{7} - y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{7} \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{64}}$ | = | $- x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} + y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{65}}$ | = | $y_{7} \, \mathbf{a}_{1}- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{7} + 2 y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{7} \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{66}}$ | = | $\left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}+\left(z_{7} + \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{7} - y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{7} \,\mathbf{\hat{y}}+\frac{1}{6}c \left(6 z_{7} + 5\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{67}}$ | = | $y_{7} \, \mathbf{a}_{1}+x_{7} \, \mathbf{a}_{2}- \left(z_{7} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} + y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{7} - 2\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{68}}$ | = | $\left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{7} - 2 y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{7} \,\mathbf{\hat{y}}- c z_{7} \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{69}}$ | = | $- x_{7} \, \mathbf{a}_{1}- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}- \left(z_{7} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{7} - y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{7} \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{70}}$ | = | $- y_{7} \, \mathbf{a}_{1}- x_{7} \, \mathbf{a}_{2}- \left(z_{7} - \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{7} + y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{7} - y_{7}\right) \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{71}}$ | = | $- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{7} + 2 y_{7}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{7} \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{72}}$ | = | $x_{7} \, \mathbf{a}_{1}+\left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}- \left(z_{7} - \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{7} - y_{7}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{7} \,\mathbf{\hat{y}}- \frac{1}{6}c \left(6 z_{7} - 5\right) \,\mathbf{\hat{z}}$ | (12c) | O III |
| $\mathbf{B_{73}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} + y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{74}}$ | = | $- y_{8} \, \mathbf{a}_{1}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}+\left(z_{8} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - 2 y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{8} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{8} + 2\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{75}}$ | = | $- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{8} - y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{8} \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{76}}$ | = | $- x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} + y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{77}}$ | = | $y_{8} \, \mathbf{a}_{1}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{8} + 2 y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{8} \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{78}}$ | = | $\left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}+\left(z_{8} + \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{8} - y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{8} \,\mathbf{\hat{y}}+\frac{1}{6}c \left(6 z_{8} + 5\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{79}}$ | = | $y_{8} \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- \left(z_{8} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} + y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{8} - 2\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{80}}$ | = | $\left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - 2 y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{8} \,\mathbf{\hat{y}}- c z_{8} \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{81}}$ | = | $- x_{8} \, \mathbf{a}_{1}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}- \left(z_{8} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{8} - y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{8} \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{82}}$ | = | $- y_{8} \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}- \left(z_{8} - \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} + y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{83}}$ | = | $- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{8} + 2 y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{8} \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{84}}$ | = | $x_{8} \, \mathbf{a}_{1}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}- \left(z_{8} - \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{8} - y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{8} \,\mathbf{\hat{y}}- \frac{1}{6}c \left(6 z_{8} - 5\right) \,\mathbf{\hat{z}}$ | (12c) | O IV |
| $\mathbf{B_{85}}$ | = | $x_{9} \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} + y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{86}}$ | = | $- y_{9} \, \mathbf{a}_{1}+\left(x_{9} - y_{9}\right) \, \mathbf{a}_{2}+\left(z_{9} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} - 2 y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{9} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{9} + 2\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{87}}$ | = | $- \left(x_{9} - y_{9}\right) \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}+\left(z_{9} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{9} - y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{9} \,\mathbf{\hat{y}}+c \left(z_{9} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{88}}$ | = | $- x_{9} \, \mathbf{a}_{1}- y_{9} \, \mathbf{a}_{2}+\left(z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{9} + y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{y}}+c \left(z_{9} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{89}}$ | = | $y_{9} \, \mathbf{a}_{1}- \left(x_{9} - y_{9}\right) \, \mathbf{a}_{2}+\left(z_{9} + \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{9} + 2 y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{9} \,\mathbf{\hat{y}}+c \left(z_{9} + \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{90}}$ | = | $\left(x_{9} - y_{9}\right) \, \mathbf{a}_{1}+x_{9} \, \mathbf{a}_{2}+\left(z_{9} + \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{9} - y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{9} \,\mathbf{\hat{y}}+\frac{1}{6}c \left(6 z_{9} + 5\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{91}}$ | = | $y_{9} \, \mathbf{a}_{1}+x_{9} \, \mathbf{a}_{2}- \left(z_{9} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} + y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{9} - 2\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{92}}$ | = | $\left(x_{9} - y_{9}\right) \, \mathbf{a}_{1}- y_{9} \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} - 2 y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{9} \,\mathbf{\hat{y}}- c z_{9} \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{93}}$ | = | $- x_{9} \, \mathbf{a}_{1}- \left(x_{9} - y_{9}\right) \, \mathbf{a}_{2}- \left(z_{9} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{9} - y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{9} \,\mathbf{\hat{y}}- c \left(z_{9} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{94}}$ | = | $- y_{9} \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}- \left(z_{9} - \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{9} + y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{y}}- c \left(z_{9} - \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{95}}$ | = | $- \left(x_{9} - y_{9}\right) \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}- \left(z_{9} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{9} + 2 y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{9} \,\mathbf{\hat{y}}- c \left(z_{9} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{96}}$ | = | $x_{9} \, \mathbf{a}_{1}+\left(x_{9} - y_{9}\right) \, \mathbf{a}_{2}- \left(z_{9} - \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{9} - y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{9} \,\mathbf{\hat{y}}- \frac{1}{6}c \left(6 z_{9} - 5\right) \,\mathbf{\hat{z}}$ | (12c) | O V |
| $\mathbf{B_{97}}$ | = | $x_{10} \, \mathbf{a}_{1}+y_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} + y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{98}}$ | = | $- y_{10} \, \mathbf{a}_{1}+\left(x_{10} - y_{10}\right) \, \mathbf{a}_{2}+\left(z_{10} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} - 2 y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{10} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{10} + 2\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{99}}$ | = | $- \left(x_{10} - y_{10}\right) \, \mathbf{a}_{1}- x_{10} \, \mathbf{a}_{2}+\left(z_{10} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{10} - y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{10} \,\mathbf{\hat{y}}+c \left(z_{10} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{100}}$ | = | $- x_{10} \, \mathbf{a}_{1}- y_{10} \, \mathbf{a}_{2}+\left(z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{10} + y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{y}}+c \left(z_{10} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{101}}$ | = | $y_{10} \, \mathbf{a}_{1}- \left(x_{10} - y_{10}\right) \, \mathbf{a}_{2}+\left(z_{10} + \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{10} + 2 y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{10} \,\mathbf{\hat{y}}+c \left(z_{10} + \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{102}}$ | = | $\left(x_{10} - y_{10}\right) \, \mathbf{a}_{1}+x_{10} \, \mathbf{a}_{2}+\left(z_{10} + \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{10} - y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{10} \,\mathbf{\hat{y}}+\frac{1}{6}c \left(6 z_{10} + 5\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{103}}$ | = | $y_{10} \, \mathbf{a}_{1}+x_{10} \, \mathbf{a}_{2}- \left(z_{10} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} + y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{10} - 2\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{104}}$ | = | $\left(x_{10} - y_{10}\right) \, \mathbf{a}_{1}- y_{10} \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} - 2 y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{10} \,\mathbf{\hat{y}}- c z_{10} \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{105}}$ | = | $- x_{10} \, \mathbf{a}_{1}- \left(x_{10} - y_{10}\right) \, \mathbf{a}_{2}- \left(z_{10} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{10} - y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{10} \,\mathbf{\hat{y}}- c \left(z_{10} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{106}}$ | = | $- y_{10} \, \mathbf{a}_{1}- x_{10} \, \mathbf{a}_{2}- \left(z_{10} - \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{10} + y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{y}}- c \left(z_{10} - \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{107}}$ | = | $- \left(x_{10} - y_{10}\right) \, \mathbf{a}_{1}+y_{10} \, \mathbf{a}_{2}- \left(z_{10} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{10} + 2 y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{10} \,\mathbf{\hat{y}}- c \left(z_{10} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{108}}$ | = | $x_{10} \, \mathbf{a}_{1}+\left(x_{10} - y_{10}\right) \, \mathbf{a}_{2}- \left(z_{10} - \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{10} - y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{10} \,\mathbf{\hat{y}}- \frac{1}{6}c \left(6 z_{10} - 5\right) \,\mathbf{\hat{z}}$ | (12c) | O VI |
| $\mathbf{B_{109}}$ | = | $x_{11} \, \mathbf{a}_{1}+y_{11} \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} + y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{110}}$ | = | $- y_{11} \, \mathbf{a}_{1}+\left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}+\left(z_{11} + \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} - 2 y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{11} \,\mathbf{\hat{y}}+\frac{1}{3}c \left(3 z_{11} + 2\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{111}}$ | = | $- \left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}- x_{11} \, \mathbf{a}_{2}+\left(z_{11} + \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{11} - y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{11} \,\mathbf{\hat{y}}+c \left(z_{11} + \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{112}}$ | = | $- x_{11} \, \mathbf{a}_{1}- y_{11} \, \mathbf{a}_{2}+\left(z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{11} + y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{y}}+c \left(z_{11} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{113}}$ | = | $y_{11} \, \mathbf{a}_{1}- \left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}+\left(z_{11} + \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{11} + 2 y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{11} \,\mathbf{\hat{y}}+c \left(z_{11} + \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{114}}$ | = | $\left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}+x_{11} \, \mathbf{a}_{2}+\left(z_{11} + \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{11} - y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{11} \,\mathbf{\hat{y}}+\frac{1}{6}c \left(6 z_{11} + 5\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{115}}$ | = | $y_{11} \, \mathbf{a}_{1}+x_{11} \, \mathbf{a}_{2}- \left(z_{11} - \frac{2}{3}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} + y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(3 z_{11} - 2\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{116}}$ | = | $\left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}- y_{11} \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} - 2 y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{11} \,\mathbf{\hat{y}}- c z_{11} \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{117}}$ | = | $- x_{11} \, \mathbf{a}_{1}- \left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}- \left(z_{11} - \frac{1}{3}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{11} - y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{11} \,\mathbf{\hat{y}}- c \left(z_{11} - \frac{1}{3}\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{118}}$ | = | $- y_{11} \, \mathbf{a}_{1}- x_{11} \, \mathbf{a}_{2}- \left(z_{11} - \frac{1}{6}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{11} + y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{y}}- c \left(z_{11} - \frac{1}{6}\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{119}}$ | = | $- \left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}+y_{11} \, \mathbf{a}_{2}- \left(z_{11} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{11} + 2 y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{11} \,\mathbf{\hat{y}}- c \left(z_{11} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
| $\mathbf{B_{120}}$ | = | $x_{11} \, \mathbf{a}_{1}+\left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}- \left(z_{11} - \frac{5}{6}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{11} - y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{11} \,\mathbf{\hat{y}}- \frac{1}{6}c \left(6 z_{11} - 5\right) \,\mathbf{\hat{z}}$ | (12c) | P I |
References
- B. Ewald, Y. Prots, C. Kudla, D. Grüner, R. Cardoso-Gil, and R. Kniep, Crystal Structure and Thermochemical Properties of a First Scandium Borophosphate, Sc(H$_{2}$O)$_{2}$[BP$_{2}$O$_{8}$]$\cdot$H$_{2}$O, Chem. Mater. 18, 673–679 (2006), doi:10.1021/cm051577o.
Prototype Generator
aflow --proto=AB4C12D2E_hP120_179_b_2c_6c_c_b --params=$a,c/a,x_{1},x_{2},x_{3},y_{3},z_{3},x_{4},y_{4},z_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11}$