# Encyclopedia of Crystallographic Prototypes

M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 1, Comp. Mat. Sci. 136, S1-S828 (2017). (doi=10.1016/j.commatsci.2017.01.017)
D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comp. Mat. Sci. 161, S1-S1011 (2019). (doi=10.1016/j.commatsci.2018.10.043)

## Manganese–leonite 185 K [K2Mn(SO4)2·4H2O] Structure: A8B2CD12E2_mC200_15_8f_2f_ae_2e11f_2f

 Prototype : H8K2MnO12S2 AFLOW prototype label : A8B2CD12E2_mC200_15_8f_2f_ae_2e11f_2f Strukturbericht designation : None 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 --params=$a$,$b/a$,$c/a$,$\beta$,$y_{2}$,$y_{3}$,$y_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$,$x_{10}$,$y_{10}$,$z_{10}$,$x_{11}$,$y_{11}$,$z_{11}$,$x_{12}$,$y_{12}$,$z_{12}$,$x_{13}$,$y_{13}$,$z_{13}$,$x_{14}$,$y_{14}$,$z_{14}$,$x_{15}$,$y_{15}$,$z_{15}$,$x_{16}$,$y_{16}$,$z_{16}$,$x_{17}$,$y_{17}$,$z_{17}$,$x_{18}$,$y_{18}$,$z_{18}$,$x_{19}$,$y_{19}$,$z_{19}$,$x_{20}$,$y_{20}$,$z_{20}$,$x_{21}$,$y_{21}$,$z_{21}$,$x_{22}$,$y_{22}$,$z_{22}$,$x_{23}$,$y_{23}$,$z_{23}$,$x_{24}$,$y_{24}$,$z_{24}$,$x_{25}$,$y_{25}$,$z_{25}$,$x_{26}$,$y_{26}$,$z_{26}$,$x_{27}$,$y_{27}$,$z_{27}$

• This is the intermediate-low temperature structure of leonite. For Mn–leonite, discussed here, it is stable between 168 K and 205 K, and we show the structure at 185 K. The room temperature structure is Strukturbericht H423, space group C2/m #13. The current structure doubles the size of the unit cell and orders all of the SO4 radicals. The low temperature structure has space group P2_1/c #14. (Hertweck, 2001) give crystallographic information of the 185 K phase in the I2/a setting of space group #15, with the origin shifted by $(1/4, 1/4, 1/2)$ from the -1 point on the $a$-glide plane (\the symmetry operations for this setting may be found here). We have restored the origin to its standard position and then used FINDSYM to change to the standard C2/c setting. This resulted in a significant change in the primitive vectors used to describe the conventional cell.

### Base-centered Monoclinic primitive vectors:

$\begin{array}{ccc} \mathbf{a}_1 & = & \frac12 \, a \, \mathbf{\hat{x}} - \frac12 \, b \, \mathbf{\hat{y}} \\ \mathbf{a}_2 & = & \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, b \, \mathbf{\hat{y}} \\ \mathbf{a}_3 & = & c \cos\beta \, \mathbf{\hat{x}} + c \sin\beta \, \mathbf{\hat{z}} \\ \end{array}$

Basis vectors:

$\begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & 0 \, \mathbf{a}_{1} + 0 \, \mathbf{a}_{2} + 0 \, \mathbf{a}_{3} & = & 0 \, \mathbf{\hat{x}} + 0 \, \mathbf{\hat{y}} + 0 \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{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}} & \left(4a\right) & \mbox{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}} + y_{2}b \, \mathbf{\hat{y}} + \frac{1}{4}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{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}}-y_{2}b \, \mathbf{\hat{y}} + \frac{3}{4}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{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}} + y_{3}b \, \mathbf{\hat{y}} + \frac{1}{4}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{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}}-y_{3}b \, \mathbf{\hat{y}} + \frac{3}{4}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{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}} + y_{4}b \, \mathbf{\hat{y}} + \frac{1}{4}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{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}}-y_{4}b \, \mathbf{\hat{y}} + \frac{3}{4}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{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(x_{5}a+z_{5}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + z_{5}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H I} \\ \mathbf{B}_{10} & = & \left(-x_{5}-y_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{5}a - z_{5}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{5}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{5}a-z_{5}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}}-z_{5}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H I} \\ \mathbf{B}_{12} & = & \left(x_{5}+y_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}-y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{5}a + z_{5}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{6}a+z_{6}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}} + z_{6}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H II} \\ \mathbf{B}_{14} & = & \left(-x_{6}-y_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{6}a - z_{6}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{6}a-z_{6}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}}-z_{6}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H II} \\ \mathbf{B}_{16} & = & \left(x_{6}+y_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-y_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{6}a + z_{6}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{7}a+z_{7}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{7}b \, \mathbf{\hat{y}} + z_{7}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H III} \\ \mathbf{B}_{18} & = & \left(-x_{7}-y_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}+y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{7}a - z_{7}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{7}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{7}a-z_{7}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{7}b \, \mathbf{\hat{y}}-z_{7}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H III} \\ \mathbf{B}_{20} & = & \left(x_{7}+y_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}-y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{7}a + z_{7}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{7}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{8}a+z_{8}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{8}b \, \mathbf{\hat{y}} + z_{8}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H IV} \\ \mathbf{B}_{22} & = & \left(-x_{8}-y_{8}\right) \, \mathbf{a}_{1} + \left(-x_{8}+y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{8}a - z_{8}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{8}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{8}a-z_{8}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{8}b \, \mathbf{\hat{y}}-z_{8}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H IV} \\ \mathbf{B}_{24} & = & \left(x_{8}+y_{8}\right) \, \mathbf{a}_{1} + \left(x_{8}-y_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{8}a + z_{8}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{8}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{9}a+z_{9}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{9}b \, \mathbf{\hat{y}} + z_{9}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H V} \\ \mathbf{B}_{26} & = & \left(-x_{9}-y_{9}\right) \, \mathbf{a}_{1} + \left(-x_{9}+y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{9}a - z_{9}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{9}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{9}a-z_{9}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{9}b \, \mathbf{\hat{y}}-z_{9}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H V} \\ \mathbf{B}_{28} & = & \left(x_{9}+y_{9}\right) \, \mathbf{a}_{1} + \left(x_{9}-y_{9}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{9}a + z_{9}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{9}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{10}a+z_{10}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{10}b \, \mathbf{\hat{y}} + z_{10}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H VI} \\ \mathbf{B}_{30} & = & \left(-x_{10}-y_{10}\right) \, \mathbf{a}_{1} + \left(-x_{10}+y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{10}a - z_{10}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{10}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{10}a-z_{10}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{10}b \, \mathbf{\hat{y}}-z_{10}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H VI} \\ \mathbf{B}_{32} & = & \left(x_{10}+y_{10}\right) \, \mathbf{a}_{1} + \left(x_{10}-y_{10}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{10}a + z_{10}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{10}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{11}a+z_{11}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{11}b \, \mathbf{\hat{y}} + z_{11}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H VII} \\ \mathbf{B}_{34} & = & \left(-x_{11}-y_{11}\right) \, \mathbf{a}_{1} + \left(-x_{11}+y_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{11}a - z_{11}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{11}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{11}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{11}a-z_{11}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{11}b \, \mathbf{\hat{y}}-z_{11}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H VII} \\ \mathbf{B}_{36} & = & \left(x_{11}+y_{11}\right) \, \mathbf{a}_{1} + \left(x_{11}-y_{11}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{11}a + z_{11}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{11}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{12}a+z_{12}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{12}b \, \mathbf{\hat{y}} + z_{12}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H VIII} \\ \mathbf{B}_{38} & = & \left(-x_{12}-y_{12}\right) \, \mathbf{a}_{1} + \left(-x_{12}+y_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{12}a - z_{12}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{12}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{12}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{12}a-z_{12}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{12}b \, \mathbf{\hat{y}}-z_{12}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{H VIII} \\ \mathbf{B}_{40} & = & \left(x_{12}+y_{12}\right) \, \mathbf{a}_{1} + \left(x_{12}-y_{12}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{12}a + z_{12}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{12}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{13}a+z_{13}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{13}b \, \mathbf{\hat{y}} + z_{13}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{K I} \\ \mathbf{B}_{42} & = & \left(-x_{13}-y_{13}\right) \, \mathbf{a}_{1} + \left(-x_{13}+y_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{13}a - z_{13}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{13}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{13}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{13}a-z_{13}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{13}b \, \mathbf{\hat{y}}-z_{13}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{K I} \\ \mathbf{B}_{44} & = & \left(x_{13}+y_{13}\right) \, \mathbf{a}_{1} + \left(x_{13}-y_{13}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{13}a + z_{13}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{13}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{14}a+z_{14}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{14}b \, \mathbf{\hat{y}} + z_{14}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{K II} \\ \mathbf{B}_{46} & = & \left(-x_{14}-y_{14}\right) \, \mathbf{a}_{1} + \left(-x_{14}+y_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{14}a - z_{14}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{14}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{14}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{14}a-z_{14}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{14}b \, \mathbf{\hat{y}}-z_{14}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{K II} \\ \mathbf{B}_{48} & = & \left(x_{14}+y_{14}\right) \, \mathbf{a}_{1} + \left(x_{14}-y_{14}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{14}a + z_{14}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{14}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{15}a+z_{15}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{15}b \, \mathbf{\hat{y}} + z_{15}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O III} \\ \mathbf{B}_{50} & = & \left(-x_{15}-y_{15}\right) \, \mathbf{a}_{1} + \left(-x_{15}+y_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{15}a - z_{15}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{15}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{15}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{15}a-z_{15}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{15}b \, \mathbf{\hat{y}}-z_{15}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O III} \\ \mathbf{B}_{52} & = & \left(x_{15}+y_{15}\right) \, \mathbf{a}_{1} + \left(x_{15}-y_{15}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{15}a + z_{15}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{15}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{16}a+z_{16}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{16}b \, \mathbf{\hat{y}} + z_{16}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O IV} \\ \mathbf{B}_{54} & = & \left(-x_{16}-y_{16}\right) \, \mathbf{a}_{1} + \left(-x_{16}+y_{16}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{16}a - z_{16}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{16}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{16}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{16}a-z_{16}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{16}b \, \mathbf{\hat{y}}-z_{16}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O IV} \\ \mathbf{B}_{56} & = & \left(x_{16}+y_{16}\right) \, \mathbf{a}_{1} + \left(x_{16}-y_{16}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{16}a + z_{16}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{16}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{17}a+z_{17}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{17}b \, \mathbf{\hat{y}} + z_{17}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O V} \\ \mathbf{B}_{58} & = & \left(-x_{17}-y_{17}\right) \, \mathbf{a}_{1} + \left(-x_{17}+y_{17}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{17}a - z_{17}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{17}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{17}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{17}a-z_{17}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{17}b \, \mathbf{\hat{y}}-z_{17}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O V} \\ \mathbf{B}_{60} & = & \left(x_{17}+y_{17}\right) \, \mathbf{a}_{1} + \left(x_{17}-y_{17}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{17}a + z_{17}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{17}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{18}a+z_{18}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{18}b \, \mathbf{\hat{y}} + z_{18}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O VI} \\ \mathbf{B}_{62} & = & \left(-x_{18}-y_{18}\right) \, \mathbf{a}_{1} + \left(-x_{18}+y_{18}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{18}a - z_{18}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{18}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{18}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{18}a-z_{18}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{18}b \, \mathbf{\hat{y}}-z_{18}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O VI} \\ \mathbf{B}_{64} & = & \left(x_{18}+y_{18}\right) \, \mathbf{a}_{1} + \left(x_{18}-y_{18}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{18}a + z_{18}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{18}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{19}a+z_{19}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{19}b \, \mathbf{\hat{y}} + z_{19}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O VII} \\ \mathbf{B}_{66} & = & \left(-x_{19}-y_{19}\right) \, \mathbf{a}_{1} + \left(-x_{19}+y_{19}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{19}a - z_{19}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{19}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{19}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{19}a-z_{19}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{19}b \, \mathbf{\hat{y}}-z_{19}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O VII} \\ \mathbf{B}_{68} & = & \left(x_{19}+y_{19}\right) \, \mathbf{a}_{1} + \left(x_{19}-y_{19}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{19}a + z_{19}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{19}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{20}a+z_{20}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{20}b \, \mathbf{\hat{y}} + z_{20}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O VIII} \\ \mathbf{B}_{70} & = & \left(-x_{20}-y_{20}\right) \, \mathbf{a}_{1} + \left(-x_{20}+y_{20}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{20}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{20}a - z_{20}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{20}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{20}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{20}a-z_{20}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{20}b \, \mathbf{\hat{y}}-z_{20}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O VIII} \\ \mathbf{B}_{72} & = & \left(x_{20}+y_{20}\right) \, \mathbf{a}_{1} + \left(x_{20}-y_{20}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{20}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{20}a + z_{20}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{20}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{20}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{21}a+z_{21}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{21}b \, \mathbf{\hat{y}} + z_{21}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O IX} \\ \mathbf{B}_{74} & = & \left(-x_{21}-y_{21}\right) \, \mathbf{a}_{1} + \left(-x_{21}+y_{21}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{21}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{21}a - z_{21}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{21}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{21}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{21}a-z_{21}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{21}b \, \mathbf{\hat{y}}-z_{21}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O IX} \\ \mathbf{B}_{76} & = & \left(x_{21}+y_{21}\right) \, \mathbf{a}_{1} + \left(x_{21}-y_{21}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{21}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{21}a + z_{21}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{21}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{21}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{22}a+z_{22}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{22}b \, \mathbf{\hat{y}} + z_{22}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O X} \\ \mathbf{B}_{78} & = & \left(-x_{22}-y_{22}\right) \, \mathbf{a}_{1} + \left(-x_{22}+y_{22}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{22}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{22}a - z_{22}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{22}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{22}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{22}a-z_{22}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{22}b \, \mathbf{\hat{y}}-z_{22}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O X} \\ \mathbf{B}_{80} & = & \left(x_{22}+y_{22}\right) \, \mathbf{a}_{1} + \left(x_{22}-y_{22}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{22}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{22}a + z_{22}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{22}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{22}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{23}a+z_{23}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{23}b \, \mathbf{\hat{y}} + z_{23}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O XI} \\ \mathbf{B}_{82} & = & \left(-x_{23}-y_{23}\right) \, \mathbf{a}_{1} + \left(-x_{23}+y_{23}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{23}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{23}a - z_{23}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{23}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{23}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{23}a-z_{23}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{23}b \, \mathbf{\hat{y}}-z_{23}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O XI} \\ \mathbf{B}_{84} & = & \left(x_{23}+y_{23}\right) \, \mathbf{a}_{1} + \left(x_{23}-y_{23}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{23}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{23}a + z_{23}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{23}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{23}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{24}a+z_{24}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{24}b \, \mathbf{\hat{y}} + z_{24}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O XII} \\ \mathbf{B}_{86} & = & \left(-x_{24}-y_{24}\right) \, \mathbf{a}_{1} + \left(-x_{24}+y_{24}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{24}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{24}a - z_{24}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{24}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{24}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{24}a-z_{24}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{24}b \, \mathbf{\hat{y}}-z_{24}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O XII} \\ \mathbf{B}_{88} & = & \left(x_{24}+y_{24}\right) \, \mathbf{a}_{1} + \left(x_{24}-y_{24}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{24}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{24}a + z_{24}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{24}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{24}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{25}a+z_{25}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{25}b \, \mathbf{\hat{y}} + z_{25}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O XIII} \\ \mathbf{B}_{90} & = & \left(-x_{25}-y_{25}\right) \, \mathbf{a}_{1} + \left(-x_{25}+y_{25}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{25}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{25}a - z_{25}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{25}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{25}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{25}a-z_{25}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{25}b \, \mathbf{\hat{y}}-z_{25}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{O XIII} \\ \mathbf{B}_{92} & = & \left(x_{25}+y_{25}\right) \, \mathbf{a}_{1} + \left(x_{25}-y_{25}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{25}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{25}a + z_{25}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{25}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{25}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{26}a+z_{26}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{26}b \, \mathbf{\hat{y}} + z_{26}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{S I} \\ \mathbf{B}_{94} & = & \left(-x_{26}-y_{26}\right) \, \mathbf{a}_{1} + \left(-x_{26}+y_{26}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{26}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{26}a - z_{26}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{26}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{26}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{26}a-z_{26}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{26}b \, \mathbf{\hat{y}}-z_{26}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{S I} \\ \mathbf{B}_{96} & = & \left(x_{26}+y_{26}\right) \, \mathbf{a}_{1} + \left(x_{26}-y_{26}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{26}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{26}a + z_{26}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{26}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{26}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(x_{27}a+z_{27}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{27}b \, \mathbf{\hat{y}} + z_{27}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{S II} \\ \mathbf{B}_{98} & = & \left(-x_{27}-y_{27}\right) \, \mathbf{a}_{1} + \left(-x_{27}+y_{27}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{27}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{27}a - z_{27}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{27}b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{27}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{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(-x_{27}a-z_{27}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{27}b \, \mathbf{\hat{y}}-z_{27}c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{S II} \\ \mathbf{B}_{100} & = & \left(x_{27}+y_{27}\right) \, \mathbf{a}_{1} + \left(x_{27}-y_{27}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{27}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{27}a + z_{27}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{27}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{27}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(8f\right) & \mbox{S II} \\ \end{array}$

### References

• B. Hertweck, G. Giester, and E. Libowitzky, The crystal structures of the low–temperature phases of leonite–type compounds, K2Me(SO4)2·4H2O (Me2+ = Mg, Mn, Fe), Am. Mineral. 86, 1282–1292 (2001), doi:10.2138/am-2001-1016.

### Prototype Generator

aflow --proto=A8B2CD12E2_mC200_15_8f_2f_ae_2e11f_2f --params=