Manganese-leonite 185K [K$_{2}$Mn(SO$_{4}$)$_{2}\cdot$4H$_{2}$O] Structure: A8B2CD12E2_mC200_15_8f_2f_ae_2e11f_2f-001
| Prototype | H$_{8}$K4$_{2}$MnO$_{12}$S$_{2}$ |
| AFLOW prototype label | A8B2CD12E2_mC200_15_8f_2f_ae_2e11f_2f-001 |
| Mineral name | manganese-leonite |
| ICSD | 92701 |
| Pearson symbol | mC200 |
| Space group number | 15 |
| Space group symbol | $C2/c$ |
| AFLOW prototype command |
aflow --proto=A8B2CD12E2_mC200_15_8f_2f_ae_2e11f_2f-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak \beta, \allowbreak y_{2}, \allowbreak y_{3}, \allowbreak y_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}, \allowbreak x_{13}, \allowbreak y_{13}, \allowbreak z_{13}, \allowbreak x_{14}, \allowbreak y_{14}, \allowbreak z_{14}, \allowbreak x_{15}, \allowbreak y_{15}, \allowbreak z_{15}, \allowbreak x_{16}, \allowbreak y_{16}, \allowbreak z_{16}, \allowbreak x_{17}, \allowbreak y_{17}, \allowbreak z_{17}, \allowbreak x_{18}, \allowbreak y_{18}, \allowbreak z_{18}, \allowbreak x_{19}, \allowbreak y_{19}, \allowbreak z_{19}, \allowbreak x_{20}, \allowbreak y_{20}, \allowbreak z_{20}, \allowbreak x_{21}, \allowbreak y_{21}, \allowbreak z_{21}, \allowbreak x_{22}, \allowbreak y_{22}, \allowbreak z_{22}, \allowbreak x_{23}, \allowbreak y_{23}, \allowbreak z_{23}, \allowbreak x_{24}, \allowbreak y_{24}, \allowbreak z_{24}, \allowbreak x_{25}, \allowbreak y_{25}, \allowbreak z_{25}, \allowbreak x_{26}, \allowbreak y_{26}, \allowbreak z_{26}, \allowbreak x_{27}, \allowbreak y_{27}, \allowbreak z_{27}$ |
Other compounds with this structure
K$_{2}$Mg(SO$_{4}$)$_{2}\cdot$4H$_{2}$O (leonite), K$_{2}$Fe(SO$_{4}$)$_{2}\cdot$4H$_{2}$O (mereiterite)
- Manganese-leonite is found in three forms:
- A low-temperature structure, stable below 168K.
- This intermediate-temperature structure, stable in between 168 and 205K.
- The room temperature structure, Strukturbericht $H4_{23}$, stable above 205K.
- The current structure orders all of the SO$_{4}$ radicals. The data was taken from a sample held at 185K.
- (Hertweck, 2001) give crystallographic information of the 185K phase in the $I2/a$ setting of space group #15, with the origin supposedly shifted by $(\frac14,\frac14,\frac12)$ from the -1 point on the $a$-glide plane (the symmetry operations for this setting may be found here). We were unable to use their data to construct a realistic crystal structure. Instead, we used the interpretation of their results by (Villars, 2016) to put the structure in the standard $C2/c$ setting of space group #15. Unfortunately this does not agree with the ICSD entry. We are investigating further and will update this page when we have more information.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}b \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \cos{\beta} \,\mathbf{\hat{x}}+c \sin{\beta} \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (4a) | Mn I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}c \cos{\beta} \,\mathbf{\hat{x}}+\frac{1}{2}c \sin{\beta} \,\mathbf{\hat{z}}$ | (4a) | Mn I |
| $\mathbf{B_{3}}$ | = | $- y_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}c \cos{\beta} \,\mathbf{\hat{x}}+b y_{2} \,\mathbf{\hat{y}}+\frac{1}{4}c \sin{\beta} \,\mathbf{\hat{z}}$ | (4e) | Mn II |
| $\mathbf{B_{4}}$ | = | $y_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{4}c \cos{\beta} \,\mathbf{\hat{x}}- b y_{2} \,\mathbf{\hat{y}}+\frac{3}{4}c \sin{\beta} \,\mathbf{\hat{z}}$ | (4e) | Mn II |
| $\mathbf{B_{5}}$ | = | $- y_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}c \cos{\beta} \,\mathbf{\hat{x}}+b y_{3} \,\mathbf{\hat{y}}+\frac{1}{4}c \sin{\beta} \,\mathbf{\hat{z}}$ | (4e) | O I |
| $\mathbf{B_{6}}$ | = | $y_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{4}c \cos{\beta} \,\mathbf{\hat{x}}- b y_{3} \,\mathbf{\hat{y}}+\frac{3}{4}c \sin{\beta} \,\mathbf{\hat{z}}$ | (4e) | O I |
| $\mathbf{B_{7}}$ | = | $- y_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}c \cos{\beta} \,\mathbf{\hat{x}}+b y_{4} \,\mathbf{\hat{y}}+\frac{1}{4}c \sin{\beta} \,\mathbf{\hat{z}}$ | (4e) | O II |
| $\mathbf{B_{8}}$ | = | $y_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{3}{4}c \cos{\beta} \,\mathbf{\hat{x}}- b y_{4} \,\mathbf{\hat{y}}+\frac{3}{4}c \sin{\beta} \,\mathbf{\hat{z}}$ | (4e) | O II |
| $\mathbf{B_{9}}$ | = | $\left(x_{5} - y_{5}\right) \, \mathbf{a}_{1}+\left(x_{5} + y_{5}\right) \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\left(a x_{5} + c z_{5} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{5} \,\mathbf{\hat{y}}+c z_{5} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H I |
| $\mathbf{B_{10}}$ | = | $- \left(x_{5} + y_{5}\right) \, \mathbf{a}_{1}- \left(x_{5} - y_{5}\right) \, \mathbf{a}_{2}- \left(z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{5} + c \left(z_{5} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{5} \,\mathbf{\hat{y}}- c \left(z_{5} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H I |
| $\mathbf{B_{11}}$ | = | $- \left(x_{5} - y_{5}\right) \, \mathbf{a}_{1}- \left(x_{5} + y_{5}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $- \left(a x_{5} + c z_{5} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{5} \,\mathbf{\hat{y}}- c z_{5} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H I |
| $\mathbf{B_{12}}$ | = | $\left(x_{5} + y_{5}\right) \, \mathbf{a}_{1}+\left(x_{5} - y_{5}\right) \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{5} + c \left(z_{5} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{5} \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H I |
| $\mathbf{B_{13}}$ | = | $\left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}+\left(x_{6} + y_{6}\right) \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}+c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H II |
| $\mathbf{B_{14}}$ | = | $- \left(x_{6} + y_{6}\right) \, \mathbf{a}_{1}- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}- \left(z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{6} + c \left(z_{6} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H II |
| $\mathbf{B_{15}}$ | = | $- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{1}- \left(x_{6} + y_{6}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $- \left(a x_{6} + c z_{6} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{6} \,\mathbf{\hat{y}}- c z_{6} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H II |
| $\mathbf{B_{16}}$ | = | $\left(x_{6} + y_{6}\right) \, \mathbf{a}_{1}+\left(x_{6} - y_{6}\right) \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{6} + c \left(z_{6} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{6} \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H II |
| $\mathbf{B_{17}}$ | = | $\left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}+\left(x_{7} + y_{7}\right) \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $\left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}+c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H III |
| $\mathbf{B_{18}}$ | = | $- \left(x_{7} + y_{7}\right) \, \mathbf{a}_{1}- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}- \left(z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{7} + c \left(z_{7} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H III |
| $\mathbf{B_{19}}$ | = | $- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{1}- \left(x_{7} + y_{7}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}$ | = | $- \left(a x_{7} + c z_{7} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{7} \,\mathbf{\hat{y}}- c z_{7} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H III |
| $\mathbf{B_{20}}$ | = | $\left(x_{7} + y_{7}\right) \, \mathbf{a}_{1}+\left(x_{7} - y_{7}\right) \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{7} + c \left(z_{7} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{7} \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H III |
| $\mathbf{B_{21}}$ | = | $\left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} + y_{8}\right) \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\left(a x_{8} + c z_{8} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{8} \,\mathbf{\hat{y}}+c z_{8} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H IV |
| $\mathbf{B_{22}}$ | = | $- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}- \left(z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{8} + c \left(z_{8} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{8} \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H IV |
| $\mathbf{B_{23}}$ | = | $- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- \left(a x_{8} + c z_{8} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{8} \,\mathbf{\hat{y}}- c z_{8} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H IV |
| $\mathbf{B_{24}}$ | = | $\left(x_{8} + y_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{8} + c \left(z_{8} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{8} \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H IV |
| $\mathbf{B_{25}}$ | = | $\left(x_{9} - y_{9}\right) \, \mathbf{a}_{1}+\left(x_{9} + y_{9}\right) \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\left(a x_{9} + c z_{9} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{9} \,\mathbf{\hat{y}}+c z_{9} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H V |
| $\mathbf{B_{26}}$ | = | $- \left(x_{9} + y_{9}\right) \, \mathbf{a}_{1}- \left(x_{9} - y_{9}\right) \, \mathbf{a}_{2}- \left(z_{9} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{9} + c \left(z_{9} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{9} \,\mathbf{\hat{y}}- c \left(z_{9} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H V |
| $\mathbf{B_{27}}$ | = | $- \left(x_{9} - y_{9}\right) \, \mathbf{a}_{1}- \left(x_{9} + y_{9}\right) \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $- \left(a x_{9} + c z_{9} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{9} \,\mathbf{\hat{y}}- c z_{9} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H V |
| $\mathbf{B_{28}}$ | = | $\left(x_{9} + y_{9}\right) \, \mathbf{a}_{1}+\left(x_{9} - y_{9}\right) \, \mathbf{a}_{2}+\left(z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{9} + c \left(z_{9} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{9} \,\mathbf{\hat{y}}+c \left(z_{9} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H V |
| $\mathbf{B_{29}}$ | = | $\left(x_{10} - y_{10}\right) \, \mathbf{a}_{1}+\left(x_{10} + y_{10}\right) \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $\left(a x_{10} + c z_{10} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{10} \,\mathbf{\hat{y}}+c z_{10} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VI |
| $\mathbf{B_{30}}$ | = | $- \left(x_{10} + y_{10}\right) \, \mathbf{a}_{1}- \left(x_{10} - y_{10}\right) \, \mathbf{a}_{2}- \left(z_{10} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{10} + c \left(z_{10} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{10} \,\mathbf{\hat{y}}- c \left(z_{10} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VI |
| $\mathbf{B_{31}}$ | = | $- \left(x_{10} - y_{10}\right) \, \mathbf{a}_{1}- \left(x_{10} + y_{10}\right) \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}$ | = | $- \left(a x_{10} + c z_{10} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{10} \,\mathbf{\hat{y}}- c z_{10} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VI |
| $\mathbf{B_{32}}$ | = | $\left(x_{10} + y_{10}\right) \, \mathbf{a}_{1}+\left(x_{10} - y_{10}\right) \, \mathbf{a}_{2}+\left(z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{10} + c \left(z_{10} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{10} \,\mathbf{\hat{y}}+c \left(z_{10} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VI |
| $\mathbf{B_{33}}$ | = | $\left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}+\left(x_{11} + y_{11}\right) \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\left(a x_{11} + c z_{11} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{11} \,\mathbf{\hat{y}}+c z_{11} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VII |
| $\mathbf{B_{34}}$ | = | $- \left(x_{11} + y_{11}\right) \, \mathbf{a}_{1}- \left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}- \left(z_{11} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{11} + c \left(z_{11} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{11} \,\mathbf{\hat{y}}- c \left(z_{11} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VII |
| $\mathbf{B_{35}}$ | = | $- \left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}- \left(x_{11} + y_{11}\right) \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}$ | = | $- \left(a x_{11} + c z_{11} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{11} \,\mathbf{\hat{y}}- c z_{11} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VII |
| $\mathbf{B_{36}}$ | = | $\left(x_{11} + y_{11}\right) \, \mathbf{a}_{1}+\left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}+\left(z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{11} + c \left(z_{11} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{11} \,\mathbf{\hat{y}}+c \left(z_{11} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VII |
| $\mathbf{B_{37}}$ | = | $\left(x_{12} - y_{12}\right) \, \mathbf{a}_{1}+\left(x_{12} + y_{12}\right) \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $\left(a x_{12} + c z_{12} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c z_{12} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VIII |
| $\mathbf{B_{38}}$ | = | $- \left(x_{12} + y_{12}\right) \, \mathbf{a}_{1}- \left(x_{12} - y_{12}\right) \, \mathbf{a}_{2}- \left(z_{12} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{12} + c \left(z_{12} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}- c \left(z_{12} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VIII |
| $\mathbf{B_{39}}$ | = | $- \left(x_{12} - y_{12}\right) \, \mathbf{a}_{1}- \left(x_{12} + y_{12}\right) \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}$ | = | $- \left(a x_{12} + c z_{12} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}- c z_{12} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VIII |
| $\mathbf{B_{40}}$ | = | $\left(x_{12} + y_{12}\right) \, \mathbf{a}_{1}+\left(x_{12} - y_{12}\right) \, \mathbf{a}_{2}+\left(z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{12} + c \left(z_{12} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}+c \left(z_{12} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | H VIII |
| $\mathbf{B_{41}}$ | = | $\left(x_{13} - y_{13}\right) \, \mathbf{a}_{1}+\left(x_{13} + y_{13}\right) \, \mathbf{a}_{2}+z_{13} \, \mathbf{a}_{3}$ | = | $\left(a x_{13} + c z_{13} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}+c z_{13} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | K I |
| $\mathbf{B_{42}}$ | = | $- \left(x_{13} + y_{13}\right) \, \mathbf{a}_{1}- \left(x_{13} - y_{13}\right) \, \mathbf{a}_{2}- \left(z_{13} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{13} + c \left(z_{13} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}- c \left(z_{13} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | K I |
| $\mathbf{B_{43}}$ | = | $- \left(x_{13} - y_{13}\right) \, \mathbf{a}_{1}- \left(x_{13} + y_{13}\right) \, \mathbf{a}_{2}- z_{13} \, \mathbf{a}_{3}$ | = | $- \left(a x_{13} + c z_{13} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{13} \,\mathbf{\hat{y}}- c z_{13} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | K I |
| $\mathbf{B_{44}}$ | = | $\left(x_{13} + y_{13}\right) \, \mathbf{a}_{1}+\left(x_{13} - y_{13}\right) \, \mathbf{a}_{2}+\left(z_{13} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{13} + c \left(z_{13} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{13} \,\mathbf{\hat{y}}+c \left(z_{13} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | K I |
| $\mathbf{B_{45}}$ | = | $\left(x_{14} - y_{14}\right) \, \mathbf{a}_{1}+\left(x_{14} + y_{14}\right) \, \mathbf{a}_{2}+z_{14} \, \mathbf{a}_{3}$ | = | $\left(a x_{14} + c z_{14} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}+c z_{14} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | K II |
| $\mathbf{B_{46}}$ | = | $- \left(x_{14} + y_{14}\right) \, \mathbf{a}_{1}- \left(x_{14} - y_{14}\right) \, \mathbf{a}_{2}- \left(z_{14} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{14} + c \left(z_{14} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}- c \left(z_{14} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | K II |
| $\mathbf{B_{47}}$ | = | $- \left(x_{14} - y_{14}\right) \, \mathbf{a}_{1}- \left(x_{14} + y_{14}\right) \, \mathbf{a}_{2}- z_{14} \, \mathbf{a}_{3}$ | = | $- \left(a x_{14} + c z_{14} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{14} \,\mathbf{\hat{y}}- c z_{14} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | K II |
| $\mathbf{B_{48}}$ | = | $\left(x_{14} + y_{14}\right) \, \mathbf{a}_{1}+\left(x_{14} - y_{14}\right) \, \mathbf{a}_{2}+\left(z_{14} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{14} + c \left(z_{14} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{14} \,\mathbf{\hat{y}}+c \left(z_{14} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | K II |
| $\mathbf{B_{49}}$ | = | $\left(x_{15} - y_{15}\right) \, \mathbf{a}_{1}+\left(x_{15} + y_{15}\right) \, \mathbf{a}_{2}+z_{15} \, \mathbf{a}_{3}$ | = | $\left(a x_{15} + c z_{15} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{15} \,\mathbf{\hat{y}}+c z_{15} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O III |
| $\mathbf{B_{50}}$ | = | $- \left(x_{15} + y_{15}\right) \, \mathbf{a}_{1}- \left(x_{15} - y_{15}\right) \, \mathbf{a}_{2}- \left(z_{15} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{15} + c \left(z_{15} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{15} \,\mathbf{\hat{y}}- c \left(z_{15} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O III |
| $\mathbf{B_{51}}$ | = | $- \left(x_{15} - y_{15}\right) \, \mathbf{a}_{1}- \left(x_{15} + y_{15}\right) \, \mathbf{a}_{2}- z_{15} \, \mathbf{a}_{3}$ | = | $- \left(a x_{15} + c z_{15} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{15} \,\mathbf{\hat{y}}- c z_{15} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O III |
| $\mathbf{B_{52}}$ | = | $\left(x_{15} + y_{15}\right) \, \mathbf{a}_{1}+\left(x_{15} - y_{15}\right) \, \mathbf{a}_{2}+\left(z_{15} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{15} + c \left(z_{15} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{15} \,\mathbf{\hat{y}}+c \left(z_{15} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O III |
| $\mathbf{B_{53}}$ | = | $\left(x_{16} - y_{16}\right) \, \mathbf{a}_{1}+\left(x_{16} + y_{16}\right) \, \mathbf{a}_{2}+z_{16} \, \mathbf{a}_{3}$ | = | $\left(a x_{16} + c z_{16} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{16} \,\mathbf{\hat{y}}+c z_{16} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O IV |
| $\mathbf{B_{54}}$ | = | $- \left(x_{16} + y_{16}\right) \, \mathbf{a}_{1}- \left(x_{16} - y_{16}\right) \, \mathbf{a}_{2}- \left(z_{16} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{16} + c \left(z_{16} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{16} \,\mathbf{\hat{y}}- c \left(z_{16} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O IV |
| $\mathbf{B_{55}}$ | = | $- \left(x_{16} - y_{16}\right) \, \mathbf{a}_{1}- \left(x_{16} + y_{16}\right) \, \mathbf{a}_{2}- z_{16} \, \mathbf{a}_{3}$ | = | $- \left(a x_{16} + c z_{16} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{16} \,\mathbf{\hat{y}}- c z_{16} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O IV |
| $\mathbf{B_{56}}$ | = | $\left(x_{16} + y_{16}\right) \, \mathbf{a}_{1}+\left(x_{16} - y_{16}\right) \, \mathbf{a}_{2}+\left(z_{16} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{16} + c \left(z_{16} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{16} \,\mathbf{\hat{y}}+c \left(z_{16} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O IV |
| $\mathbf{B_{57}}$ | = | $\left(x_{17} - y_{17}\right) \, \mathbf{a}_{1}+\left(x_{17} + y_{17}\right) \, \mathbf{a}_{2}+z_{17} \, \mathbf{a}_{3}$ | = | $\left(a x_{17} + c z_{17} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{17} \,\mathbf{\hat{y}}+c z_{17} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O V |
| $\mathbf{B_{58}}$ | = | $- \left(x_{17} + y_{17}\right) \, \mathbf{a}_{1}- \left(x_{17} - y_{17}\right) \, \mathbf{a}_{2}- \left(z_{17} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{17} + c \left(z_{17} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{17} \,\mathbf{\hat{y}}- c \left(z_{17} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O V |
| $\mathbf{B_{59}}$ | = | $- \left(x_{17} - y_{17}\right) \, \mathbf{a}_{1}- \left(x_{17} + y_{17}\right) \, \mathbf{a}_{2}- z_{17} \, \mathbf{a}_{3}$ | = | $- \left(a x_{17} + c z_{17} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{17} \,\mathbf{\hat{y}}- c z_{17} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O V |
| $\mathbf{B_{60}}$ | = | $\left(x_{17} + y_{17}\right) \, \mathbf{a}_{1}+\left(x_{17} - y_{17}\right) \, \mathbf{a}_{2}+\left(z_{17} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{17} + c \left(z_{17} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{17} \,\mathbf{\hat{y}}+c \left(z_{17} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O V |
| $\mathbf{B_{61}}$ | = | $\left(x_{18} - y_{18}\right) \, \mathbf{a}_{1}+\left(x_{18} + y_{18}\right) \, \mathbf{a}_{2}+z_{18} \, \mathbf{a}_{3}$ | = | $\left(a x_{18} + c z_{18} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{18} \,\mathbf{\hat{y}}+c z_{18} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VI |
| $\mathbf{B_{62}}$ | = | $- \left(x_{18} + y_{18}\right) \, \mathbf{a}_{1}- \left(x_{18} - y_{18}\right) \, \mathbf{a}_{2}- \left(z_{18} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{18} + c \left(z_{18} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{18} \,\mathbf{\hat{y}}- c \left(z_{18} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VI |
| $\mathbf{B_{63}}$ | = | $- \left(x_{18} - y_{18}\right) \, \mathbf{a}_{1}- \left(x_{18} + y_{18}\right) \, \mathbf{a}_{2}- z_{18} \, \mathbf{a}_{3}$ | = | $- \left(a x_{18} + c z_{18} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{18} \,\mathbf{\hat{y}}- c z_{18} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VI |
| $\mathbf{B_{64}}$ | = | $\left(x_{18} + y_{18}\right) \, \mathbf{a}_{1}+\left(x_{18} - y_{18}\right) \, \mathbf{a}_{2}+\left(z_{18} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{18} + c \left(z_{18} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{18} \,\mathbf{\hat{y}}+c \left(z_{18} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VI |
| $\mathbf{B_{65}}$ | = | $\left(x_{19} - y_{19}\right) \, \mathbf{a}_{1}+\left(x_{19} + y_{19}\right) \, \mathbf{a}_{2}+z_{19} \, \mathbf{a}_{3}$ | = | $\left(a x_{19} + c z_{19} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{19} \,\mathbf{\hat{y}}+c z_{19} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VII |
| $\mathbf{B_{66}}$ | = | $- \left(x_{19} + y_{19}\right) \, \mathbf{a}_{1}- \left(x_{19} - y_{19}\right) \, \mathbf{a}_{2}- \left(z_{19} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{19} + c \left(z_{19} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{19} \,\mathbf{\hat{y}}- c \left(z_{19} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VII |
| $\mathbf{B_{67}}$ | = | $- \left(x_{19} - y_{19}\right) \, \mathbf{a}_{1}- \left(x_{19} + y_{19}\right) \, \mathbf{a}_{2}- z_{19} \, \mathbf{a}_{3}$ | = | $- \left(a x_{19} + c z_{19} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{19} \,\mathbf{\hat{y}}- c z_{19} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VII |
| $\mathbf{B_{68}}$ | = | $\left(x_{19} + y_{19}\right) \, \mathbf{a}_{1}+\left(x_{19} - y_{19}\right) \, \mathbf{a}_{2}+\left(z_{19} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{19} + c \left(z_{19} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{19} \,\mathbf{\hat{y}}+c \left(z_{19} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VII |
| $\mathbf{B_{69}}$ | = | $\left(x_{20} - y_{20}\right) \, \mathbf{a}_{1}+\left(x_{20} + y_{20}\right) \, \mathbf{a}_{2}+z_{20} \, \mathbf{a}_{3}$ | = | $\left(a x_{20} + c z_{20} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{20} \,\mathbf{\hat{y}}+c z_{20} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VIII |
| $\mathbf{B_{70}}$ | = | $- \left(x_{20} + y_{20}\right) \, \mathbf{a}_{1}- \left(x_{20} - y_{20}\right) \, \mathbf{a}_{2}- \left(z_{20} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{20} + c \left(z_{20} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{20} \,\mathbf{\hat{y}}- c \left(z_{20} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VIII |
| $\mathbf{B_{71}}$ | = | $- \left(x_{20} - y_{20}\right) \, \mathbf{a}_{1}- \left(x_{20} + y_{20}\right) \, \mathbf{a}_{2}- z_{20} \, \mathbf{a}_{3}$ | = | $- \left(a x_{20} + c z_{20} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{20} \,\mathbf{\hat{y}}- c z_{20} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VIII |
| $\mathbf{B_{72}}$ | = | $\left(x_{20} + y_{20}\right) \, \mathbf{a}_{1}+\left(x_{20} - y_{20}\right) \, \mathbf{a}_{2}+\left(z_{20} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{20} + c \left(z_{20} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{20} \,\mathbf{\hat{y}}+c \left(z_{20} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O VIII |
| $\mathbf{B_{73}}$ | = | $\left(x_{21} - y_{21}\right) \, \mathbf{a}_{1}+\left(x_{21} + y_{21}\right) \, \mathbf{a}_{2}+z_{21} \, \mathbf{a}_{3}$ | = | $\left(a x_{21} + c z_{21} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{21} \,\mathbf{\hat{y}}+c z_{21} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O IX |
| $\mathbf{B_{74}}$ | = | $- \left(x_{21} + y_{21}\right) \, \mathbf{a}_{1}- \left(x_{21} - y_{21}\right) \, \mathbf{a}_{2}- \left(z_{21} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{21} + c \left(z_{21} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{21} \,\mathbf{\hat{y}}- c \left(z_{21} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O IX |
| $\mathbf{B_{75}}$ | = | $- \left(x_{21} - y_{21}\right) \, \mathbf{a}_{1}- \left(x_{21} + y_{21}\right) \, \mathbf{a}_{2}- z_{21} \, \mathbf{a}_{3}$ | = | $- \left(a x_{21} + c z_{21} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{21} \,\mathbf{\hat{y}}- c z_{21} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O IX |
| $\mathbf{B_{76}}$ | = | $\left(x_{21} + y_{21}\right) \, \mathbf{a}_{1}+\left(x_{21} - y_{21}\right) \, \mathbf{a}_{2}+\left(z_{21} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{21} + c \left(z_{21} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{21} \,\mathbf{\hat{y}}+c \left(z_{21} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O IX |
| $\mathbf{B_{77}}$ | = | $\left(x_{22} - y_{22}\right) \, \mathbf{a}_{1}+\left(x_{22} + y_{22}\right) \, \mathbf{a}_{2}+z_{22} \, \mathbf{a}_{3}$ | = | $\left(a x_{22} + c z_{22} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{22} \,\mathbf{\hat{y}}+c z_{22} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O X |
| $\mathbf{B_{78}}$ | = | $- \left(x_{22} + y_{22}\right) \, \mathbf{a}_{1}- \left(x_{22} - y_{22}\right) \, \mathbf{a}_{2}- \left(z_{22} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{22} + c \left(z_{22} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{22} \,\mathbf{\hat{y}}- c \left(z_{22} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O X |
| $\mathbf{B_{79}}$ | = | $- \left(x_{22} - y_{22}\right) \, \mathbf{a}_{1}- \left(x_{22} + y_{22}\right) \, \mathbf{a}_{2}- z_{22} \, \mathbf{a}_{3}$ | = | $- \left(a x_{22} + c z_{22} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{22} \,\mathbf{\hat{y}}- c z_{22} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O X |
| $\mathbf{B_{80}}$ | = | $\left(x_{22} + y_{22}\right) \, \mathbf{a}_{1}+\left(x_{22} - y_{22}\right) \, \mathbf{a}_{2}+\left(z_{22} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{22} + c \left(z_{22} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{22} \,\mathbf{\hat{y}}+c \left(z_{22} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O X |
| $\mathbf{B_{81}}$ | = | $\left(x_{23} - y_{23}\right) \, \mathbf{a}_{1}+\left(x_{23} + y_{23}\right) \, \mathbf{a}_{2}+z_{23} \, \mathbf{a}_{3}$ | = | $\left(a x_{23} + c z_{23} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{23} \,\mathbf{\hat{y}}+c z_{23} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XI |
| $\mathbf{B_{82}}$ | = | $- \left(x_{23} + y_{23}\right) \, \mathbf{a}_{1}- \left(x_{23} - y_{23}\right) \, \mathbf{a}_{2}- \left(z_{23} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{23} + c \left(z_{23} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{23} \,\mathbf{\hat{y}}- c \left(z_{23} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XI |
| $\mathbf{B_{83}}$ | = | $- \left(x_{23} - y_{23}\right) \, \mathbf{a}_{1}- \left(x_{23} + y_{23}\right) \, \mathbf{a}_{2}- z_{23} \, \mathbf{a}_{3}$ | = | $- \left(a x_{23} + c z_{23} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{23} \,\mathbf{\hat{y}}- c z_{23} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XI |
| $\mathbf{B_{84}}$ | = | $\left(x_{23} + y_{23}\right) \, \mathbf{a}_{1}+\left(x_{23} - y_{23}\right) \, \mathbf{a}_{2}+\left(z_{23} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{23} + c \left(z_{23} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{23} \,\mathbf{\hat{y}}+c \left(z_{23} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XI |
| $\mathbf{B_{85}}$ | = | $\left(x_{24} - y_{24}\right) \, \mathbf{a}_{1}+\left(x_{24} + y_{24}\right) \, \mathbf{a}_{2}+z_{24} \, \mathbf{a}_{3}$ | = | $\left(a x_{24} + c z_{24} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{24} \,\mathbf{\hat{y}}+c z_{24} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XII |
| $\mathbf{B_{86}}$ | = | $- \left(x_{24} + y_{24}\right) \, \mathbf{a}_{1}- \left(x_{24} - y_{24}\right) \, \mathbf{a}_{2}- \left(z_{24} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{24} + c \left(z_{24} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{24} \,\mathbf{\hat{y}}- c \left(z_{24} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XII |
| $\mathbf{B_{87}}$ | = | $- \left(x_{24} - y_{24}\right) \, \mathbf{a}_{1}- \left(x_{24} + y_{24}\right) \, \mathbf{a}_{2}- z_{24} \, \mathbf{a}_{3}$ | = | $- \left(a x_{24} + c z_{24} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{24} \,\mathbf{\hat{y}}- c z_{24} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XII |
| $\mathbf{B_{88}}$ | = | $\left(x_{24} + y_{24}\right) \, \mathbf{a}_{1}+\left(x_{24} - y_{24}\right) \, \mathbf{a}_{2}+\left(z_{24} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{24} + c \left(z_{24} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{24} \,\mathbf{\hat{y}}+c \left(z_{24} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XII |
| $\mathbf{B_{89}}$ | = | $\left(x_{25} - y_{25}\right) \, \mathbf{a}_{1}+\left(x_{25} + y_{25}\right) \, \mathbf{a}_{2}+z_{25} \, \mathbf{a}_{3}$ | = | $\left(a x_{25} + c z_{25} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{25} \,\mathbf{\hat{y}}+c z_{25} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XIII |
| $\mathbf{B_{90}}$ | = | $- \left(x_{25} + y_{25}\right) \, \mathbf{a}_{1}- \left(x_{25} - y_{25}\right) \, \mathbf{a}_{2}- \left(z_{25} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{25} + c \left(z_{25} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{25} \,\mathbf{\hat{y}}- c \left(z_{25} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XIII |
| $\mathbf{B_{91}}$ | = | $- \left(x_{25} - y_{25}\right) \, \mathbf{a}_{1}- \left(x_{25} + y_{25}\right) \, \mathbf{a}_{2}- z_{25} \, \mathbf{a}_{3}$ | = | $- \left(a x_{25} + c z_{25} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{25} \,\mathbf{\hat{y}}- c z_{25} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XIII |
| $\mathbf{B_{92}}$ | = | $\left(x_{25} + y_{25}\right) \, \mathbf{a}_{1}+\left(x_{25} - y_{25}\right) \, \mathbf{a}_{2}+\left(z_{25} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{25} + c \left(z_{25} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{25} \,\mathbf{\hat{y}}+c \left(z_{25} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | O XIII |
| $\mathbf{B_{93}}$ | = | $\left(x_{26} - y_{26}\right) \, \mathbf{a}_{1}+\left(x_{26} + y_{26}\right) \, \mathbf{a}_{2}+z_{26} \, \mathbf{a}_{3}$ | = | $\left(a x_{26} + c z_{26} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{26} \,\mathbf{\hat{y}}+c z_{26} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | S I |
| $\mathbf{B_{94}}$ | = | $- \left(x_{26} + y_{26}\right) \, \mathbf{a}_{1}- \left(x_{26} - y_{26}\right) \, \mathbf{a}_{2}- \left(z_{26} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{26} + c \left(z_{26} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{26} \,\mathbf{\hat{y}}- c \left(z_{26} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | S I |
| $\mathbf{B_{95}}$ | = | $- \left(x_{26} - y_{26}\right) \, \mathbf{a}_{1}- \left(x_{26} + y_{26}\right) \, \mathbf{a}_{2}- z_{26} \, \mathbf{a}_{3}$ | = | $- \left(a x_{26} + c z_{26} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{26} \,\mathbf{\hat{y}}- c z_{26} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | S I |
| $\mathbf{B_{96}}$ | = | $\left(x_{26} + y_{26}\right) \, \mathbf{a}_{1}+\left(x_{26} - y_{26}\right) \, \mathbf{a}_{2}+\left(z_{26} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{26} + c \left(z_{26} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{26} \,\mathbf{\hat{y}}+c \left(z_{26} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | S I |
| $\mathbf{B_{97}}$ | = | $\left(x_{27} - y_{27}\right) \, \mathbf{a}_{1}+\left(x_{27} + y_{27}\right) \, \mathbf{a}_{2}+z_{27} \, \mathbf{a}_{3}$ | = | $\left(a x_{27} + c z_{27} \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{27} \,\mathbf{\hat{y}}+c z_{27} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | S II |
| $\mathbf{B_{98}}$ | = | $- \left(x_{27} + y_{27}\right) \, \mathbf{a}_{1}- \left(x_{27} - y_{27}\right) \, \mathbf{a}_{2}- \left(z_{27} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \left(a x_{27} + c \left(z_{27} - \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}+b y_{27} \,\mathbf{\hat{y}}- c \left(z_{27} - \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | S II |
| $\mathbf{B_{99}}$ | = | $- \left(x_{27} - y_{27}\right) \, \mathbf{a}_{1}- \left(x_{27} + y_{27}\right) \, \mathbf{a}_{2}- z_{27} \, \mathbf{a}_{3}$ | = | $- \left(a x_{27} + c z_{27} \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{27} \,\mathbf{\hat{y}}- c z_{27} \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | S II |
| $\mathbf{B_{100}}$ | = | $\left(x_{27} + y_{27}\right) \, \mathbf{a}_{1}+\left(x_{27} - y_{27}\right) \, \mathbf{a}_{2}+\left(z_{27} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\left(a x_{27} + c \left(z_{27} + \frac{1}{2}\right) \cos{\beta}\right) \,\mathbf{\hat{x}}- b y_{27} \,\mathbf{\hat{y}}+c \left(z_{27} + \frac{1}{2}\right) \sin{\beta} \,\mathbf{\hat{z}}$ | (8f) | S II |
References
- B. Hertweck, G. Giester, and E. Libowitzky, The crystal structures of the low-temperature phases of leonite-type compounds, K$_{2}$Me(SO$_{4}$)$_{2}$$\cdot$4H$_{2}$O (Me$^{2+}$ = Mg, Mn, Fe), Am. Mineral. 86, 1282–1292 (2001), doi:10.2138/am-2001-1016.
- K2Mn(SO4)2{\textperiodcentered}4H2O (K2Mn[SO4]2[H2O]4 mon1, T = 185 K) Crystal Structure: Datasheet from
PAULING FILE Multinaries Edition – 2012
in SpringerMaterials (https://materials.springer.com/isp/crystallographic/docs/sd{_}1811721)}. Copyright 2016 Springer-Verlag Berlin Heidelberg {&} Material Phases Data System (MPDS), Switzerland {& National Institute for Materials Science (NIMS), Japan.
Prototype Generator
aflow --proto=A8B2CD12E2_mC200_15_8f_2f_ae_2e11f_2f --params=$a,b/a,c/a,\beta,y_{2},y_{3},y_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12},x_{13},y_{13},z_{13},x_{14},y_{14},z_{14},x_{15},y_{15},z_{15},x_{16},y_{16},z_{16},x_{17},y_{17},z_{17},x_{18},y_{18},z_{18},x_{19},y_{19},z_{19},x_{20},y_{20},z_{20},x_{21},y_{21},z_{21},x_{22},y_{22},z_{22},x_{23},y_{23},z_{23},x_{24},y_{24},z_{24},x_{25},y_{25},z_{25},x_{26},y_{26},z_{26},x_{27},y_{27},z_{27}$