α-LiZnPO$_{4}$ Structure: AB4CD_mC224_9_8a_32a_8a_8a-001
| Prototype | LiO$_{4}$PZn |
| AFLOW prototype label | AB4CD_mC224_9_8a_32a_8a_8a-001 |
| ICSD | 65764 |
| Pearson symbol | mC224 |
| Space group number | 9 |
| Space group symbol | $Cc$ |
| AFLOW prototype command |
aflow --proto=AB4CD_mC224_9_8a_32a_8a_8a-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak \beta, \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}, \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}, \allowbreak x_{13}, \allowbreak y_{13}, \allowbreak z_{13}, \allowbreak x_{14}, \allowbreak y_{14}, \allowbreak z_{14}, \allowbreak x_{15}, \allowbreak y_{15}, \allowbreak z_{15}, \allowbreak x_{16}, \allowbreak y_{16}, \allowbreak z_{16}, \allowbreak x_{17}, \allowbreak y_{17}, \allowbreak z_{17}, \allowbreak x_{18}, \allowbreak y_{18}, \allowbreak z_{18}, \allowbreak x_{19}, \allowbreak y_{19}, \allowbreak z_{19}, \allowbreak x_{20}, \allowbreak y_{20}, \allowbreak z_{20}, \allowbreak x_{21}, \allowbreak y_{21}, \allowbreak z_{21}, \allowbreak x_{22}, \allowbreak y_{22}, \allowbreak z_{22}, \allowbreak x_{23}, \allowbreak y_{23}, \allowbreak z_{23}, \allowbreak x_{24}, \allowbreak y_{24}, \allowbreak z_{24}, \allowbreak x_{25}, \allowbreak y_{25}, \allowbreak z_{25}, \allowbreak x_{26}, \allowbreak y_{26}, \allowbreak z_{26}, \allowbreak x_{27}, \allowbreak y_{27}, \allowbreak z_{27}, \allowbreak x_{28}, \allowbreak y_{28}, \allowbreak z_{28}, \allowbreak x_{29}, \allowbreak y_{29}, \allowbreak z_{29}, \allowbreak x_{30}, \allowbreak y_{30}, \allowbreak z_{30}, \allowbreak x_{31}, \allowbreak y_{31}, \allowbreak z_{31}, \allowbreak x_{32}, \allowbreak y_{32}, \allowbreak z_{32}, \allowbreak x_{33}, \allowbreak y_{33}, \allowbreak z_{33}, \allowbreak x_{34}, \allowbreak y_{34}, \allowbreak z_{34}, \allowbreak x_{35}, \allowbreak y_{35}, \allowbreak z_{35}, \allowbreak x_{36}, \allowbreak y_{36}, \allowbreak z_{36}, \allowbreak x_{37}, \allowbreak y_{37}, \allowbreak z_{37}, \allowbreak x_{38}, \allowbreak y_{38}, \allowbreak z_{38}, \allowbreak x_{39}, \allowbreak y_{39}, \allowbreak z_{39}, \allowbreak x_{40}, \allowbreak y_{40}, \allowbreak z_{40}, \allowbreak x_{41}, \allowbreak y_{41}, \allowbreak z_{41}, \allowbreak x_{42}, \allowbreak y_{42}, \allowbreak z_{42}, \allowbreak x_{43}, \allowbreak y_{43}, \allowbreak z_{43}, \allowbreak x_{44}, \allowbreak y_{44}, \allowbreak z_{44}, \allowbreak x_{45}, \allowbreak y_{45}, \allowbreak z_{45}, \allowbreak x_{46}, \allowbreak y_{46}, \allowbreak z_{46}, \allowbreak x_{47}, \allowbreak y_{47}, \allowbreak z_{47}, \allowbreak x_{48}, \allowbreak y_{48}, \allowbreak z_{48}, \allowbreak x_{49}, \allowbreak y_{49}, \allowbreak z_{49}, \allowbreak x_{50}, \allowbreak y_{50}, \allowbreak z_{50}, \allowbreak x_{51}, \allowbreak y_{51}, \allowbreak z_{51}, \allowbreak x_{52}, \allowbreak y_{52}, \allowbreak z_{52}, \allowbreak x_{53}, \allowbreak y_{53}, \allowbreak z_{53}, \allowbreak x_{54}, \allowbreak y_{54}, \allowbreak z_{54}, \allowbreak x_{55}, \allowbreak y_{55}, \allowbreak z_{55}, \allowbreak x_{56}, \allowbreak y_{56}, \allowbreak z_{56}$ |
- LiZnPO$_{4}$ has been observed in a variety of structures. We list those which have know crystal structures:
- $\alpha$–LiZnPO$_{4}$ (this structure)
- $\delta_{1}$-LiZnPO$_{4}$, and
- rhombohedral LiZnPO$_{4}$.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}b \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \cos{\beta} \,\mathbf{\hat{x}}+c \sin{\beta} \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $\left(x_{1} - y_{1}\right) \, \mathbf{a}_{1}+\left(x_{1} + y_{1}\right) \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $\left(a x_{1} + c z_{1} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{1} \,\mathbf{\hat{y}}+c z_{1} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li I |
| $\mathbf{B_{2}}$ | = | $\left(x_{1} + y_{1}\right) \, \mathbf{a}_{1}+\left(x_{1} - y_{1}\right) \, \mathbf{a}_{2}+\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{1} + c \left(z_{1} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{1} \,\mathbf{\hat{y}}+c \left(z_{1} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li I |
| $\mathbf{B_{3}}$ | = | $\left(x_{2} - y_{2}\right) \, \mathbf{a}_{1}+\left(x_{2} + y_{2}\right) \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\left(a x_{2} + c z_{2} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{2} \,\mathbf{\hat{y}}+c z_{2} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li II |
| $\mathbf{B_{4}}$ | = | $\left(x_{2} + y_{2}\right) \, \mathbf{a}_{1}+\left(x_{2} - y_{2}\right) \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{2} + c \left(z_{2} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{2} \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li II |
| $\mathbf{B_{5}}$ | = | $\left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} + y_{3}\right) \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\left(a x_{3} + c z_{3} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{3} \,\mathbf{\hat{y}}+c z_{3} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li III |
| $\mathbf{B_{6}}$ | = | $\left(x_{3} + y_{3}\right) \, \mathbf{a}_{1}+\left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}+\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{3} + c \left(z_{3} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{3} \,\mathbf{\hat{y}}+c \left(z_{3} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li III |
| $\mathbf{B_{7}}$ | = | $\left(x_{4} - y_{4}\right) \, \mathbf{a}_{1}+\left(x_{4} + y_{4}\right) \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\left(a x_{4} + c z_{4} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{4} \,\mathbf{\hat{y}}+c z_{4} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li IV |
| $\mathbf{B_{8}}$ | = | $\left(x_{4} + y_{4}\right) \, \mathbf{a}_{1}+\left(x_{4} - y_{4}\right) \, \mathbf{a}_{2}+\left(z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{4} + c \left(z_{4} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{4} \,\mathbf{\hat{y}}+c \left(z_{4} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li IV |
| $\mathbf{B_{9}}$ | = | $\left(x_{5} - y_{5}\right) \, \mathbf{a}_{1}+\left(x_{5} + y_{5}\right) \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\left(a x_{5} + c z_{5} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{5} \,\mathbf{\hat{y}}+c z_{5} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li V |
| $\mathbf{B_{10}}$ | = | $\left(x_{5} + y_{5}\right) \, \mathbf{a}_{1}+\left(x_{5} - y_{5}\right) \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{5} + c \left(z_{5} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{5} \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li V |
| $\mathbf{B_{11}}$ | = | $\left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}+\left(x_{6} + y_{6}\right) \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}+c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li VI |
| $\mathbf{B_{12}}$ | = | $\left(x_{6} + y_{6}\right) \, \mathbf{a}_{1}+\left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{6} + c \left(z_{6} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{6} \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li VI |
| $\mathbf{B_{13}}$ | = | $\left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}+\left(x_{7} + y_{7}\right) \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}+c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li VII |
| $\mathbf{B_{14}}$ | = | $\left(x_{7} + y_{7}\right) \, \mathbf{a}_{1}+\left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{7} + c \left(z_{7} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{7} \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li VII |
| $\mathbf{B_{15}}$ | = | $\left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} + y_{8}\right) \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\left(a x_{8} + c z_{8} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{8} \,\mathbf{\hat{y}}+c z_{8} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li VIII |
| $\mathbf{B_{16}}$ | = | $\left(x_{8} + y_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{8} + c \left(z_{8} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{8} \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Li VIII |
| $\mathbf{B_{17}}$ | = | $\left(x_{9} - y_{9}\right) \, \mathbf{a}_{1}+\left(x_{9} + y_{9}\right) \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\left(a x_{9} + c z_{9} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{9} \,\mathbf{\hat{y}}+c z_{9} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{18}}$ | = | $\left(x_{9} + y_{9}\right) \, \mathbf{a}_{1}+\left(x_{9} - y_{9}\right) \, \mathbf{a}_{2}+\left(z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{9} + c \left(z_{9} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{9} \,\mathbf{\hat{y}}+c \left(z_{9} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{19}}$ | = | $\left(x_{10} - y_{10}\right) \, \mathbf{a}_{1}+\left(x_{10} + y_{10}\right) \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $\left(a x_{10} + c z_{10} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{10} \,\mathbf{\hat{y}}+c z_{10} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{20}}$ | = | $\left(x_{10} + y_{10}\right) \, \mathbf{a}_{1}+\left(x_{10} - y_{10}\right) \, \mathbf{a}_{2}+\left(z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{10} + c \left(z_{10} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{10} \,\mathbf{\hat{y}}+c \left(z_{10} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{21}}$ | = | $\left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}+\left(x_{11} + y_{11}\right) \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\left(a x_{11} + c z_{11} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{11} \,\mathbf{\hat{y}}+c z_{11} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{22}}$ | = | $\left(x_{11} + y_{11}\right) \, \mathbf{a}_{1}+\left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}+\left(z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{11} + c \left(z_{11} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{11} \,\mathbf{\hat{y}}+c \left(z_{11} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{23}}$ | = | $\left(x_{12} - y_{12}\right) \, \mathbf{a}_{1}+\left(x_{12} + y_{12}\right) \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $\left(a x_{12} + c z_{12} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c z_{12} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{24}}$ | = | $\left(x_{12} + y_{12}\right) \, \mathbf{a}_{1}+\left(x_{12} - y_{12}\right) \, \mathbf{a}_{2}+\left(z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{12} + c \left(z_{12} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}+c \left(z_{12} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{25}}$ | = | $\left(x_{13} - y_{13}\right) \, \mathbf{a}_{1}+\left(x_{13} + y_{13}\right) \, \mathbf{a}_{2}+z_{13} \, \mathbf{a}_{3}$ | = | $\left(a x_{13} + c z_{13} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}+c z_{13} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{26}}$ | = | $\left(x_{13} + y_{13}\right) \, \mathbf{a}_{1}+\left(x_{13} - y_{13}\right) \, \mathbf{a}_{2}+\left(z_{13} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{13} + c \left(z_{13} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{13} \,\mathbf{\hat{y}}+c \left(z_{13} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{27}}$ | = | $\left(x_{14} - y_{14}\right) \, \mathbf{a}_{1}+\left(x_{14} + y_{14}\right) \, \mathbf{a}_{2}+z_{14} \, \mathbf{a}_{3}$ | = | $\left(a x_{14} + c z_{14} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}+c z_{14} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{28}}$ | = | $\left(x_{14} + y_{14}\right) \, \mathbf{a}_{1}+\left(x_{14} - y_{14}\right) \, \mathbf{a}_{2}+\left(z_{14} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{14} + c \left(z_{14} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{14} \,\mathbf{\hat{y}}+c \left(z_{14} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{29}}$ | = | $\left(x_{15} - y_{15}\right) \, \mathbf{a}_{1}+\left(x_{15} + y_{15}\right) \, \mathbf{a}_{2}+z_{15} \, \mathbf{a}_{3}$ | = | $\left(a x_{15} + c z_{15} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{15} \,\mathbf{\hat{y}}+c z_{15} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{30}}$ | = | $\left(x_{15} + y_{15}\right) \, \mathbf{a}_{1}+\left(x_{15} - y_{15}\right) \, \mathbf{a}_{2}+\left(z_{15} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{15} + c \left(z_{15} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{15} \,\mathbf{\hat{y}}+c \left(z_{15} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{31}}$ | = | $\left(x_{16} - y_{16}\right) \, \mathbf{a}_{1}+\left(x_{16} + y_{16}\right) \, \mathbf{a}_{2}+z_{16} \, \mathbf{a}_{3}$ | = | $\left(a x_{16} + c z_{16} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{16} \,\mathbf{\hat{y}}+c z_{16} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{32}}$ | = | $\left(x_{16} + y_{16}\right) \, \mathbf{a}_{1}+\left(x_{16} - y_{16}\right) \, \mathbf{a}_{2}+\left(z_{16} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{16} + c \left(z_{16} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{16} \,\mathbf{\hat{y}}+c \left(z_{16} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{33}}$ | = | $\left(x_{17} - y_{17}\right) \, \mathbf{a}_{1}+\left(x_{17} + y_{17}\right) \, \mathbf{a}_{2}+z_{17} \, \mathbf{a}_{3}$ | = | $\left(a x_{17} + c z_{17} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{17} \,\mathbf{\hat{y}}+c z_{17} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{34}}$ | = | $\left(x_{17} + y_{17}\right) \, \mathbf{a}_{1}+\left(x_{17} - y_{17}\right) \, \mathbf{a}_{2}+\left(z_{17} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{17} + c \left(z_{17} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{17} \,\mathbf{\hat{y}}+c \left(z_{17} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{35}}$ | = | $\left(x_{18} - y_{18}\right) \, \mathbf{a}_{1}+\left(x_{18} + y_{18}\right) \, \mathbf{a}_{2}+z_{18} \, \mathbf{a}_{3}$ | = | $\left(a x_{18} + c z_{18} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{18} \,\mathbf{\hat{y}}+c z_{18} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{36}}$ | = | $\left(x_{18} + y_{18}\right) \, \mathbf{a}_{1}+\left(x_{18} - y_{18}\right) \, \mathbf{a}_{2}+\left(z_{18} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{18} + c \left(z_{18} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{18} \,\mathbf{\hat{y}}+c \left(z_{18} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{37}}$ | = | $\left(x_{19} - y_{19}\right) \, \mathbf{a}_{1}+\left(x_{19} + y_{19}\right) \, \mathbf{a}_{2}+z_{19} \, \mathbf{a}_{3}$ | = | $\left(a x_{19} + c z_{19} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{19} \,\mathbf{\hat{y}}+c z_{19} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{38}}$ | = | $\left(x_{19} + y_{19}\right) \, \mathbf{a}_{1}+\left(x_{19} - y_{19}\right) \, \mathbf{a}_{2}+\left(z_{19} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{19} + c \left(z_{19} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{19} \,\mathbf{\hat{y}}+c \left(z_{19} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{39}}$ | = | $\left(x_{20} - y_{20}\right) \, \mathbf{a}_{1}+\left(x_{20} + y_{20}\right) \, \mathbf{a}_{2}+z_{20} \, \mathbf{a}_{3}$ | = | $\left(a x_{20} + c z_{20} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{20} \,\mathbf{\hat{y}}+c z_{20} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{40}}$ | = | $\left(x_{20} + y_{20}\right) \, \mathbf{a}_{1}+\left(x_{20} - y_{20}\right) \, \mathbf{a}_{2}+\left(z_{20} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{20} + c \left(z_{20} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{20} \,\mathbf{\hat{y}}+c \left(z_{20} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{41}}$ | = | $\left(x_{21} - y_{21}\right) \, \mathbf{a}_{1}+\left(x_{21} + y_{21}\right) \, \mathbf{a}_{2}+z_{21} \, \mathbf{a}_{3}$ | = | $\left(a x_{21} + c z_{21} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{21} \,\mathbf{\hat{y}}+c z_{21} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{42}}$ | = | $\left(x_{21} + y_{21}\right) \, \mathbf{a}_{1}+\left(x_{21} - y_{21}\right) \, \mathbf{a}_{2}+\left(z_{21} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{21} + c \left(z_{21} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{21} \,\mathbf{\hat{y}}+c \left(z_{21} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{43}}$ | = | $\left(x_{22} - y_{22}\right) \, \mathbf{a}_{1}+\left(x_{22} + y_{22}\right) \, \mathbf{a}_{2}+z_{22} \, \mathbf{a}_{3}$ | = | $\left(a x_{22} + c z_{22} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{22} \,\mathbf{\hat{y}}+c z_{22} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{44}}$ | = | $\left(x_{22} + y_{22}\right) \, \mathbf{a}_{1}+\left(x_{22} - y_{22}\right) \, \mathbf{a}_{2}+\left(z_{22} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{22} + c \left(z_{22} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{22} \,\mathbf{\hat{y}}+c \left(z_{22} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{45}}$ | = | $\left(x_{23} - y_{23}\right) \, \mathbf{a}_{1}+\left(x_{23} + y_{23}\right) \, \mathbf{a}_{2}+z_{23} \, \mathbf{a}_{3}$ | = | $\left(a x_{23} + c z_{23} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{23} \,\mathbf{\hat{y}}+c z_{23} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{46}}$ | = | $\left(x_{23} + y_{23}\right) \, \mathbf{a}_{1}+\left(x_{23} - y_{23}\right) \, \mathbf{a}_{2}+\left(z_{23} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{23} + c \left(z_{23} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{23} \,\mathbf{\hat{y}}+c \left(z_{23} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{47}}$ | = | $\left(x_{24} - y_{24}\right) \, \mathbf{a}_{1}+\left(x_{24} + y_{24}\right) \, \mathbf{a}_{2}+z_{24} \, \mathbf{a}_{3}$ | = | $\left(a x_{24} + c z_{24} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{24} \,\mathbf{\hat{y}}+c z_{24} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{48}}$ | = | $\left(x_{24} + y_{24}\right) \, \mathbf{a}_{1}+\left(x_{24} - y_{24}\right) \, \mathbf{a}_{2}+\left(z_{24} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{24} + c \left(z_{24} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{24} \,\mathbf{\hat{y}}+c \left(z_{24} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{49}}$ | = | $\left(x_{25} - y_{25}\right) \, \mathbf{a}_{1}+\left(x_{25} + y_{25}\right) \, \mathbf{a}_{2}+z_{25} \, \mathbf{a}_{3}$ | = | $\left(a x_{25} + c z_{25} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{25} \,\mathbf{\hat{y}}+c z_{25} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{50}}$ | = | $\left(x_{25} + y_{25}\right) \, \mathbf{a}_{1}+\left(x_{25} - y_{25}\right) \, \mathbf{a}_{2}+\left(z_{25} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{25} + c \left(z_{25} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{25} \,\mathbf{\hat{y}}+c \left(z_{25} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{51}}$ | = | $\left(x_{26} - y_{26}\right) \, \mathbf{a}_{1}+\left(x_{26} + y_{26}\right) \, \mathbf{a}_{2}+z_{26} \, \mathbf{a}_{3}$ | = | $\left(a x_{26} + c z_{26} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{26} \,\mathbf{\hat{y}}+c z_{26} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{52}}$ | = | $\left(x_{26} + y_{26}\right) \, \mathbf{a}_{1}+\left(x_{26} - y_{26}\right) \, \mathbf{a}_{2}+\left(z_{26} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{26} + c \left(z_{26} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{26} \,\mathbf{\hat{y}}+c \left(z_{26} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{53}}$ | = | $\left(x_{27} - y_{27}\right) \, \mathbf{a}_{1}+\left(x_{27} + y_{27}\right) \, \mathbf{a}_{2}+z_{27} \, \mathbf{a}_{3}$ | = | $\left(a x_{27} + c z_{27} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{27} \,\mathbf{\hat{y}}+c z_{27} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{54}}$ | = | $\left(x_{27} + y_{27}\right) \, \mathbf{a}_{1}+\left(x_{27} - y_{27}\right) \, \mathbf{a}_{2}+\left(z_{27} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{27} + c \left(z_{27} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{27} \,\mathbf{\hat{y}}+c \left(z_{27} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{55}}$ | = | $\left(x_{28} - y_{28}\right) \, \mathbf{a}_{1}+\left(x_{28} + y_{28}\right) \, \mathbf{a}_{2}+z_{28} \, \mathbf{a}_{3}$ | = | $\left(a x_{28} + c z_{28} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{28} \,\mathbf{\hat{y}}+c z_{28} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{56}}$ | = | $\left(x_{28} + y_{28}\right) \, \mathbf{a}_{1}+\left(x_{28} - y_{28}\right) \, \mathbf{a}_{2}+\left(z_{28} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{28} + c \left(z_{28} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{28} \,\mathbf{\hat{y}}+c \left(z_{28} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{57}}$ | = | $\left(x_{29} - y_{29}\right) \, \mathbf{a}_{1}+\left(x_{29} + y_{29}\right) \, \mathbf{a}_{2}+z_{29} \, \mathbf{a}_{3}$ | = | $\left(a x_{29} + c z_{29} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{29} \,\mathbf{\hat{y}}+c z_{29} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXI |
| $\mathbf{B_{58}}$ | = | $\left(x_{29} + y_{29}\right) \, \mathbf{a}_{1}+\left(x_{29} - y_{29}\right) \, \mathbf{a}_{2}+\left(z_{29} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{29} + c \left(z_{29} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{29} \,\mathbf{\hat{y}}+c \left(z_{29} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXI |
| $\mathbf{B_{59}}$ | = | $\left(x_{30} - y_{30}\right) \, \mathbf{a}_{1}+\left(x_{30} + y_{30}\right) \, \mathbf{a}_{2}+z_{30} \, \mathbf{a}_{3}$ | = | $\left(a x_{30} + c z_{30} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{30} \,\mathbf{\hat{y}}+c z_{30} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXII |
| $\mathbf{B_{60}}$ | = | $\left(x_{30} + y_{30}\right) \, \mathbf{a}_{1}+\left(x_{30} - y_{30}\right) \, \mathbf{a}_{2}+\left(z_{30} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{30} + c \left(z_{30} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{30} \,\mathbf{\hat{y}}+c \left(z_{30} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXII |
| $\mathbf{B_{61}}$ | = | $\left(x_{31} - y_{31}\right) \, \mathbf{a}_{1}+\left(x_{31} + y_{31}\right) \, \mathbf{a}_{2}+z_{31} \, \mathbf{a}_{3}$ | = | $\left(a x_{31} + c z_{31} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{31} \,\mathbf{\hat{y}}+c z_{31} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXIII |
| $\mathbf{B_{62}}$ | = | $\left(x_{31} + y_{31}\right) \, \mathbf{a}_{1}+\left(x_{31} - y_{31}\right) \, \mathbf{a}_{2}+\left(z_{31} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{31} + c \left(z_{31} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{31} \,\mathbf{\hat{y}}+c \left(z_{31} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXIII |
| $\mathbf{B_{63}}$ | = | $\left(x_{32} - y_{32}\right) \, \mathbf{a}_{1}+\left(x_{32} + y_{32}\right) \, \mathbf{a}_{2}+z_{32} \, \mathbf{a}_{3}$ | = | $\left(a x_{32} + c z_{32} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{32} \,\mathbf{\hat{y}}+c z_{32} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXIV |
| $\mathbf{B_{64}}$ | = | $\left(x_{32} + y_{32}\right) \, \mathbf{a}_{1}+\left(x_{32} - y_{32}\right) \, \mathbf{a}_{2}+\left(z_{32} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{32} + c \left(z_{32} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{32} \,\mathbf{\hat{y}}+c \left(z_{32} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXIV |
| $\mathbf{B_{65}}$ | = | $\left(x_{33} - y_{33}\right) \, \mathbf{a}_{1}+\left(x_{33} + y_{33}\right) \, \mathbf{a}_{2}+z_{33} \, \mathbf{a}_{3}$ | = | $\left(a x_{33} + c z_{33} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{33} \,\mathbf{\hat{y}}+c z_{33} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXV |
| $\mathbf{B_{66}}$ | = | $\left(x_{33} + y_{33}\right) \, \mathbf{a}_{1}+\left(x_{33} - y_{33}\right) \, \mathbf{a}_{2}+\left(z_{33} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{33} + c \left(z_{33} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{33} \,\mathbf{\hat{y}}+c \left(z_{33} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXV |
| $\mathbf{B_{67}}$ | = | $\left(x_{34} - y_{34}\right) \, \mathbf{a}_{1}+\left(x_{34} + y_{34}\right) \, \mathbf{a}_{2}+z_{34} \, \mathbf{a}_{3}$ | = | $\left(a x_{34} + c z_{34} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{34} \,\mathbf{\hat{y}}+c z_{34} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXVI |
| $\mathbf{B_{68}}$ | = | $\left(x_{34} + y_{34}\right) \, \mathbf{a}_{1}+\left(x_{34} - y_{34}\right) \, \mathbf{a}_{2}+\left(z_{34} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{34} + c \left(z_{34} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{34} \,\mathbf{\hat{y}}+c \left(z_{34} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXVI |
| $\mathbf{B_{69}}$ | = | $\left(x_{35} - y_{35}\right) \, \mathbf{a}_{1}+\left(x_{35} + y_{35}\right) \, \mathbf{a}_{2}+z_{35} \, \mathbf{a}_{3}$ | = | $\left(a x_{35} + c z_{35} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{35} \,\mathbf{\hat{y}}+c z_{35} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXVII |
| $\mathbf{B_{70}}$ | = | $\left(x_{35} + y_{35}\right) \, \mathbf{a}_{1}+\left(x_{35} - y_{35}\right) \, \mathbf{a}_{2}+\left(z_{35} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{35} + c \left(z_{35} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{35} \,\mathbf{\hat{y}}+c \left(z_{35} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXVII |
| $\mathbf{B_{71}}$ | = | $\left(x_{36} - y_{36}\right) \, \mathbf{a}_{1}+\left(x_{36} + y_{36}\right) \, \mathbf{a}_{2}+z_{36} \, \mathbf{a}_{3}$ | = | $\left(a x_{36} + c z_{36} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{36} \,\mathbf{\hat{y}}+c z_{36} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXVIII |
| $\mathbf{B_{72}}$ | = | $\left(x_{36} + y_{36}\right) \, \mathbf{a}_{1}+\left(x_{36} - y_{36}\right) \, \mathbf{a}_{2}+\left(z_{36} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{36} + c \left(z_{36} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{36} \,\mathbf{\hat{y}}+c \left(z_{36} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXVIII |
| $\mathbf{B_{73}}$ | = | $\left(x_{37} - y_{37}\right) \, \mathbf{a}_{1}+\left(x_{37} + y_{37}\right) \, \mathbf{a}_{2}+z_{37} \, \mathbf{a}_{3}$ | = | $\left(a x_{37} + c z_{37} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{37} \,\mathbf{\hat{y}}+c z_{37} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXIX |
| $\mathbf{B_{74}}$ | = | $\left(x_{37} + y_{37}\right) \, \mathbf{a}_{1}+\left(x_{37} - y_{37}\right) \, \mathbf{a}_{2}+\left(z_{37} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{37} + c \left(z_{37} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{37} \,\mathbf{\hat{y}}+c \left(z_{37} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXIX |
| $\mathbf{B_{75}}$ | = | $\left(x_{38} - y_{38}\right) \, \mathbf{a}_{1}+\left(x_{38} + y_{38}\right) \, \mathbf{a}_{2}+z_{38} \, \mathbf{a}_{3}$ | = | $\left(a x_{38} + c z_{38} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{38} \,\mathbf{\hat{y}}+c z_{38} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXX |
| $\mathbf{B_{76}}$ | = | $\left(x_{38} + y_{38}\right) \, \mathbf{a}_{1}+\left(x_{38} - y_{38}\right) \, \mathbf{a}_{2}+\left(z_{38} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{38} + c \left(z_{38} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{38} \,\mathbf{\hat{y}}+c \left(z_{38} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXX |
| $\mathbf{B_{77}}$ | = | $\left(x_{39} - y_{39}\right) \, \mathbf{a}_{1}+\left(x_{39} + y_{39}\right) \, \mathbf{a}_{2}+z_{39} \, \mathbf{a}_{3}$ | = | $\left(a x_{39} + c z_{39} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{39} \,\mathbf{\hat{y}}+c z_{39} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXXI |
| $\mathbf{B_{78}}$ | = | $\left(x_{39} + y_{39}\right) \, \mathbf{a}_{1}+\left(x_{39} - y_{39}\right) \, \mathbf{a}_{2}+\left(z_{39} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{39} + c \left(z_{39} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{39} \,\mathbf{\hat{y}}+c \left(z_{39} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXXI |
| $\mathbf{B_{79}}$ | = | $\left(x_{40} - y_{40}\right) \, \mathbf{a}_{1}+\left(x_{40} + y_{40}\right) \, \mathbf{a}_{2}+z_{40} \, \mathbf{a}_{3}$ | = | $\left(a x_{40} + c z_{40} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{40} \,\mathbf{\hat{y}}+c z_{40} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXXII |
| $\mathbf{B_{80}}$ | = | $\left(x_{40} + y_{40}\right) \, \mathbf{a}_{1}+\left(x_{40} - y_{40}\right) \, \mathbf{a}_{2}+\left(z_{40} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{40} + c \left(z_{40} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{40} \,\mathbf{\hat{y}}+c \left(z_{40} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | O XXXII |
| $\mathbf{B_{81}}$ | = | $\left(x_{41} - y_{41}\right) \, \mathbf{a}_{1}+\left(x_{41} + y_{41}\right) \, \mathbf{a}_{2}+z_{41} \, \mathbf{a}_{3}$ | = | $\left(a x_{41} + c z_{41} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{41} \,\mathbf{\hat{y}}+c z_{41} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P I |
| $\mathbf{B_{82}}$ | = | $\left(x_{41} + y_{41}\right) \, \mathbf{a}_{1}+\left(x_{41} - y_{41}\right) \, \mathbf{a}_{2}+\left(z_{41} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{41} + c \left(z_{41} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{41} \,\mathbf{\hat{y}}+c \left(z_{41} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P I |
| $\mathbf{B_{83}}$ | = | $\left(x_{42} - y_{42}\right) \, \mathbf{a}_{1}+\left(x_{42} + y_{42}\right) \, \mathbf{a}_{2}+z_{42} \, \mathbf{a}_{3}$ | = | $\left(a x_{42} + c z_{42} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{42} \,\mathbf{\hat{y}}+c z_{42} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P II |
| $\mathbf{B_{84}}$ | = | $\left(x_{42} + y_{42}\right) \, \mathbf{a}_{1}+\left(x_{42} - y_{42}\right) \, \mathbf{a}_{2}+\left(z_{42} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{42} + c \left(z_{42} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{42} \,\mathbf{\hat{y}}+c \left(z_{42} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P II |
| $\mathbf{B_{85}}$ | = | $\left(x_{43} - y_{43}\right) \, \mathbf{a}_{1}+\left(x_{43} + y_{43}\right) \, \mathbf{a}_{2}+z_{43} \, \mathbf{a}_{3}$ | = | $\left(a x_{43} + c z_{43} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{43} \,\mathbf{\hat{y}}+c z_{43} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P III |
| $\mathbf{B_{86}}$ | = | $\left(x_{43} + y_{43}\right) \, \mathbf{a}_{1}+\left(x_{43} - y_{43}\right) \, \mathbf{a}_{2}+\left(z_{43} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{43} + c \left(z_{43} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{43} \,\mathbf{\hat{y}}+c \left(z_{43} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P III |
| $\mathbf{B_{87}}$ | = | $\left(x_{44} - y_{44}\right) \, \mathbf{a}_{1}+\left(x_{44} + y_{44}\right) \, \mathbf{a}_{2}+z_{44} \, \mathbf{a}_{3}$ | = | $\left(a x_{44} + c z_{44} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{44} \,\mathbf{\hat{y}}+c z_{44} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P IV |
| $\mathbf{B_{88}}$ | = | $\left(x_{44} + y_{44}\right) \, \mathbf{a}_{1}+\left(x_{44} - y_{44}\right) \, \mathbf{a}_{2}+\left(z_{44} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{44} + c \left(z_{44} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{44} \,\mathbf{\hat{y}}+c \left(z_{44} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P IV |
| $\mathbf{B_{89}}$ | = | $\left(x_{45} - y_{45}\right) \, \mathbf{a}_{1}+\left(x_{45} + y_{45}\right) \, \mathbf{a}_{2}+z_{45} \, \mathbf{a}_{3}$ | = | $\left(a x_{45} + c z_{45} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{45} \,\mathbf{\hat{y}}+c z_{45} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P V |
| $\mathbf{B_{90}}$ | = | $\left(x_{45} + y_{45}\right) \, \mathbf{a}_{1}+\left(x_{45} - y_{45}\right) \, \mathbf{a}_{2}+\left(z_{45} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{45} + c \left(z_{45} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{45} \,\mathbf{\hat{y}}+c \left(z_{45} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P V |
| $\mathbf{B_{91}}$ | = | $\left(x_{46} - y_{46}\right) \, \mathbf{a}_{1}+\left(x_{46} + y_{46}\right) \, \mathbf{a}_{2}+z_{46} \, \mathbf{a}_{3}$ | = | $\left(a x_{46} + c z_{46} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{46} \,\mathbf{\hat{y}}+c z_{46} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P VI |
| $\mathbf{B_{92}}$ | = | $\left(x_{46} + y_{46}\right) \, \mathbf{a}_{1}+\left(x_{46} - y_{46}\right) \, \mathbf{a}_{2}+\left(z_{46} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{46} + c \left(z_{46} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{46} \,\mathbf{\hat{y}}+c \left(z_{46} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P VI |
| $\mathbf{B_{93}}$ | = | $\left(x_{47} - y_{47}\right) \, \mathbf{a}_{1}+\left(x_{47} + y_{47}\right) \, \mathbf{a}_{2}+z_{47} \, \mathbf{a}_{3}$ | = | $\left(a x_{47} + c z_{47} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{47} \,\mathbf{\hat{y}}+c z_{47} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P VII |
| $\mathbf{B_{94}}$ | = | $\left(x_{47} + y_{47}\right) \, \mathbf{a}_{1}+\left(x_{47} - y_{47}\right) \, \mathbf{a}_{2}+\left(z_{47} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{47} + c \left(z_{47} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{47} \,\mathbf{\hat{y}}+c \left(z_{47} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P VII |
| $\mathbf{B_{95}}$ | = | $\left(x_{48} - y_{48}\right) \, \mathbf{a}_{1}+\left(x_{48} + y_{48}\right) \, \mathbf{a}_{2}+z_{48} \, \mathbf{a}_{3}$ | = | $\left(a x_{48} + c z_{48} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{48} \,\mathbf{\hat{y}}+c z_{48} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P VIII |
| $\mathbf{B_{96}}$ | = | $\left(x_{48} + y_{48}\right) \, \mathbf{a}_{1}+\left(x_{48} - y_{48}\right) \, \mathbf{a}_{2}+\left(z_{48} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{48} + c \left(z_{48} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{48} \,\mathbf{\hat{y}}+c \left(z_{48} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | P VIII |
| $\mathbf{B_{97}}$ | = | $\left(x_{49} - y_{49}\right) \, \mathbf{a}_{1}+\left(x_{49} + y_{49}\right) \, \mathbf{a}_{2}+z_{49} \, \mathbf{a}_{3}$ | = | $\left(a x_{49} + c z_{49} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{49} \,\mathbf{\hat{y}}+c z_{49} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn I |
| $\mathbf{B_{98}}$ | = | $\left(x_{49} + y_{49}\right) \, \mathbf{a}_{1}+\left(x_{49} - y_{49}\right) \, \mathbf{a}_{2}+\left(z_{49} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{49} + c \left(z_{49} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{49} \,\mathbf{\hat{y}}+c \left(z_{49} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn I |
| $\mathbf{B_{99}}$ | = | $\left(x_{50} - y_{50}\right) \, \mathbf{a}_{1}+\left(x_{50} + y_{50}\right) \, \mathbf{a}_{2}+z_{50} \, \mathbf{a}_{3}$ | = | $\left(a x_{50} + c z_{50} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{50} \,\mathbf{\hat{y}}+c z_{50} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn II |
| $\mathbf{B_{100}}$ | = | $\left(x_{50} + y_{50}\right) \, \mathbf{a}_{1}+\left(x_{50} - y_{50}\right) \, \mathbf{a}_{2}+\left(z_{50} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{50} + c \left(z_{50} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{50} \,\mathbf{\hat{y}}+c \left(z_{50} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn II |
| $\mathbf{B_{101}}$ | = | $\left(x_{51} - y_{51}\right) \, \mathbf{a}_{1}+\left(x_{51} + y_{51}\right) \, \mathbf{a}_{2}+z_{51} \, \mathbf{a}_{3}$ | = | $\left(a x_{51} + c z_{51} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{51} \,\mathbf{\hat{y}}+c z_{51} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn III |
| $\mathbf{B_{102}}$ | = | $\left(x_{51} + y_{51}\right) \, \mathbf{a}_{1}+\left(x_{51} - y_{51}\right) \, \mathbf{a}_{2}+\left(z_{51} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{51} + c \left(z_{51} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{51} \,\mathbf{\hat{y}}+c \left(z_{51} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn III |
| $\mathbf{B_{103}}$ | = | $\left(x_{52} - y_{52}\right) \, \mathbf{a}_{1}+\left(x_{52} + y_{52}\right) \, \mathbf{a}_{2}+z_{52} \, \mathbf{a}_{3}$ | = | $\left(a x_{52} + c z_{52} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{52} \,\mathbf{\hat{y}}+c z_{52} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn IV |
| $\mathbf{B_{104}}$ | = | $\left(x_{52} + y_{52}\right) \, \mathbf{a}_{1}+\left(x_{52} - y_{52}\right) \, \mathbf{a}_{2}+\left(z_{52} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{52} + c \left(z_{52} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{52} \,\mathbf{\hat{y}}+c \left(z_{52} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn IV |
| $\mathbf{B_{105}}$ | = | $\left(x_{53} - y_{53}\right) \, \mathbf{a}_{1}+\left(x_{53} + y_{53}\right) \, \mathbf{a}_{2}+z_{53} \, \mathbf{a}_{3}$ | = | $\left(a x_{53} + c z_{53} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{53} \,\mathbf{\hat{y}}+c z_{53} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn V |
| $\mathbf{B_{106}}$ | = | $\left(x_{53} + y_{53}\right) \, \mathbf{a}_{1}+\left(x_{53} - y_{53}\right) \, \mathbf{a}_{2}+\left(z_{53} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{53} + c \left(z_{53} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{53} \,\mathbf{\hat{y}}+c \left(z_{53} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn V |
| $\mathbf{B_{107}}$ | = | $\left(x_{54} - y_{54}\right) \, \mathbf{a}_{1}+\left(x_{54} + y_{54}\right) \, \mathbf{a}_{2}+z_{54} \, \mathbf{a}_{3}$ | = | $\left(a x_{54} + c z_{54} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{54} \,\mathbf{\hat{y}}+c z_{54} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn VI |
| $\mathbf{B_{108}}$ | = | $\left(x_{54} + y_{54}\right) \, \mathbf{a}_{1}+\left(x_{54} - y_{54}\right) \, \mathbf{a}_{2}+\left(z_{54} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{54} + c \left(z_{54} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{54} \,\mathbf{\hat{y}}+c \left(z_{54} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn VI |
| $\mathbf{B_{109}}$ | = | $\left(x_{55} - y_{55}\right) \, \mathbf{a}_{1}+\left(x_{55} + y_{55}\right) \, \mathbf{a}_{2}+z_{55} \, \mathbf{a}_{3}$ | = | $\left(a x_{55} + c z_{55} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{55} \,\mathbf{\hat{y}}+c z_{55} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn VII |
| $\mathbf{B_{110}}$ | = | $\left(x_{55} + y_{55}\right) \, \mathbf{a}_{1}+\left(x_{55} - y_{55}\right) \, \mathbf{a}_{2}+\left(z_{55} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{55} + c \left(z_{55} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{55} \,\mathbf{\hat{y}}+c \left(z_{55} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn VII |
| $\mathbf{B_{111}}$ | = | $\left(x_{56} - y_{56}\right) \, \mathbf{a}_{1}+\left(x_{56} + y_{56}\right) \, \mathbf{a}_{2}+z_{56} \, \mathbf{a}_{3}$ | = | $\left(a x_{56} + c z_{56} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{56} \,\mathbf{\hat{y}}+c z_{56} \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn VIII |
| $\mathbf{B_{112}}$ | = | $\left(x_{56} + y_{56}\right) \, \mathbf{a}_{1}+\left(x_{56} - y_{56}\right) \, \mathbf{a}_{2}+\left(z_{56} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{56} + c \left(z_{56} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{56} \,\mathbf{\hat{y}}+c \left(z_{56} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Zn VIII |
References
- L. Elammari and B. Elouadi, Structure of α-LiZnPO$_{4}$, Acta Crystallogr. Sect. C 45, 1864–1867 (1989), doi:10.1107/S0108270189003094.
Prototype Generator
aflow --proto=AB4CD_mC224_9_8a_32a_8a_8a --params=$a,b/a,c/a,\beta,x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},x_{3},y_{3},z_{3},x_{4},y_{4},z_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12},x_{13},y_{13},z_{13},x_{14},y_{14},z_{14},x_{15},y_{15},z_{15},x_{16},y_{16},z_{16},x_{17},y_{17},z_{17},x_{18},y_{18},z_{18},x_{19},y_{19},z_{19},x_{20},y_{20},z_{20},x_{21},y_{21},z_{21},x_{22},y_{22},z_{22},x_{23},y_{23},z_{23},x_{24},y_{24},z_{24},x_{25},y_{25},z_{25},x_{26},y_{26},z_{26},x_{27},y_{27},z_{27},x_{28},y_{28},z_{28},x_{29},y_{29},z_{29},x_{30},y_{30},z_{30},x_{31},y_{31},z_{31},x_{32},y_{32},z_{32},x_{33},y_{33},z_{33},x_{34},y_{34},z_{34},x_{35},y_{35},z_{35},x_{36},y_{36},z_{36},x_{37},y_{37},z_{37},x_{38},y_{38},z_{38},x_{39},y_{39},z_{39},x_{40},y_{40},z_{40},x_{41},y_{41},z_{41},x_{42},y_{42},z_{42},x_{43},y_{43},z_{43},x_{44},y_{44},z_{44},x_{45},y_{45},z_{45},x_{46},y_{46},z_{46},x_{47},y_{47},z_{47},x_{48},y_{48},z_{48},x_{49},y_{49},z_{49},x_{50},y_{50},z_{50},x_{51},y_{51},z_{51},x_{52},y_{52},z_{52},x_{53},y_{53},z_{53},x_{54},y_{54},z_{54},x_{55},y_{55},z_{55},x_{56},y_{56},z_{56}$