Manganese–leonite 110 K [K2Mn(SO4)2·4H2O] Structure : A8B2CD12E2_mP100_14_8e_2e_ad_12e_2e

Picture of Structure; Click for Big Picture
Prototype : H8K2MnO12S2
AFLOW prototype label : A8B2CD12E2_mP100_14_8e_2e_ad_12e_2e
Strukturbericht designation : None
Pearson symbol : mP100
Space group number : 14
Space group symbol : $P2_{1}/c$
AFLOW prototype command : aflow --proto=A8B2CD12E2_mP100_14_8e_2e_ad_12e_2e
--params=
$a$,$b/a$,$c/a$,$\beta$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$,$x_{10}$,$y_{10}$,$z_{10}$,$x_{11}$,$y_{11}$,$z_{11}$,$x_{12}$,$y_{12}$,$z_{12}$,$x_{13}$,$y_{13}$,$z_{13}$,$x_{14}$,$y_{14}$,$z_{14}$,$x_{15}$,$y_{15}$,$z_{15}$,$x_{16}$,$y_{16}$,$z_{16}$,$x_{17}$,$y_{17}$,$z_{17}$,$x_{18}$,$y_{18}$,$z_{18}$,$x_{19}$,$y_{19}$,$z_{19}$,$x_{20}$,$y_{20}$,$z_{20}$,$x_{21}$,$y_{21}$,$z_{21}$,$x_{22}$,$y_{22}$,$z_{22}$,$x_{23}$,$y_{23}$,$z_{23}$,$x_{24}$,$y_{24}$,$z_{24}$,$x_{25}$,$y_{25}$,$z_{25}$,$x_{26}$,$y_{26}$,$z_{26}$


Other compounds with this structure

  • K2Mg(SO4)2·4H2O (leonite)

  • This is the intermediate–low temperature structure of leonite. For Mn–leonite, discussed here, it is stable below 168 K. The structure shown here was determined at 110 K.
  • The room temperature structure is Strukturbericht $H4_{23}$, space group $C2/m$ #13. The current structure doubles the size of the unit cell and orders all of the SO4 radicals.
  • The intermediate temperature structure (168–205 K) has space group $P2_1/c$ #14.
  • (Hertweck, 2001) give crystallographic information of this phase in the $P2_1/a$ setting of space group #14. We used FINDSYM to transform this to the $P2_1/c$ setting, which resulted in a switching of the $a$– and $c$–axes.

Simple Monoclinic primitive vectors:

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

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(2a\right) & \mbox{Mn I} \\ \mathbf{B}_{2} & = & \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}c\cos\beta \, \mathbf{\hat{x}} + \frac{1}{2}b \, \mathbf{\hat{y}} + \frac{1}{2}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Mn I} \\ \mathbf{B}_{3} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(a+c\cos\beta\right) \, \mathbf{\hat{x}} + \frac{1}{2}c\sin\beta \, \mathbf{\hat{z}} & \left(2d\right) & \mbox{Mn II} \\ \mathbf{B}_{4} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{2}b \, \mathbf{\hat{y}} & \left(2d\right) & \mbox{Mn II} \\ \mathbf{B}_{5} & = & x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \left(x_{3}a+z_{3}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{3}b \, \mathbf{\hat{y}} + z_{3}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H I} \\ \mathbf{B}_{6} & = & -x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{3}a - z_{3}c\cos\beta\right) \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{3}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{3}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H I} \\ \mathbf{B}_{7} & = & -x_{3} \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & \left(-x_{3}a-z_{3}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{3}b \, \mathbf{\hat{y}}-z_{3}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H I} \\ \mathbf{B}_{8} & = & x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{3}a + z_{3}c\cos\beta\right) \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{3}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H I} \\ \mathbf{B}_{9} & = & x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \left(x_{4}a+z_{4}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{4}b \, \mathbf{\hat{y}} + z_{4}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H II} \\ \mathbf{B}_{10} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta - x_{4}a - z_{4}c\cos\beta\right) \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{4}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{4}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H II} \\ \mathbf{B}_{11} & = & -x_{4} \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & \left(-x_{4}a-z_{4}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{4}b \, \mathbf{\hat{y}}-z_{4}c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H II} \\ \mathbf{B}_{12} & = & x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{4}a + z_{4}c\cos\beta\right) \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{4}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H II} \\ \mathbf{B}_{13} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \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(4e\right) & \mbox{H III} \\ \mathbf{B}_{14} & = & -x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{5}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{5}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H III} \\ \mathbf{B}_{15} & = & -x_{5} \, \mathbf{a}_{1}-y_{5} \, \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(4e\right) & \mbox{H III} \\ \mathbf{B}_{16} & = & x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{5}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H III} \\ \mathbf{B}_{17} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \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(4e\right) & \mbox{H IV} \\ \mathbf{B}_{18} & = & -x_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{6}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H IV} \\ \mathbf{B}_{19} & = & -x_{6} \, \mathbf{a}_{1}-y_{6} \, \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(4e\right) & \mbox{H IV} \\ \mathbf{B}_{20} & = & x_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{6}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H IV} \\ \mathbf{B}_{21} & = & x_{7} \, \mathbf{a}_{1} + y_{7} \, \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(4e\right) & \mbox{H V} \\ \mathbf{B}_{22} & = & -x_{7} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{7}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H V} \\ \mathbf{B}_{23} & = & -x_{7} \, \mathbf{a}_{1}-y_{7} \, \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(4e\right) & \mbox{H V} \\ \mathbf{B}_{24} & = & x_{7} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{7}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H V} \\ \mathbf{B}_{25} & = & x_{8} \, \mathbf{a}_{1} + y_{8} \, \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(4e\right) & \mbox{H VI} \\ \mathbf{B}_{26} & = & -x_{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{8}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H VI} \\ \mathbf{B}_{27} & = & -x_{8} \, \mathbf{a}_{1}-y_{8} \, \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(4e\right) & \mbox{H VI} \\ \mathbf{B}_{28} & = & x_{8} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{8}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H VI} \\ \mathbf{B}_{29} & = & x_{9} \, \mathbf{a}_{1} + y_{9} \, \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(4e\right) & \mbox{H VII} \\ \mathbf{B}_{30} & = & -x_{9} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{9}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{9}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H VII} \\ \mathbf{B}_{31} & = & -x_{9} \, \mathbf{a}_{1}-y_{9} \, \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(4e\right) & \mbox{H VII} \\ \mathbf{B}_{32} & = & x_{9} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{9}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H VII} \\ \mathbf{B}_{33} & = & x_{10} \, \mathbf{a}_{1} + y_{10} \, \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(4e\right) & \mbox{H VIII} \\ \mathbf{B}_{34} & = & -x_{10} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{10}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{10}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H VIII} \\ \mathbf{B}_{35} & = & -x_{10} \, \mathbf{a}_{1}-y_{10} \, \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(4e\right) & \mbox{H VIII} \\ \mathbf{B}_{36} & = & x_{10} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{10}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{H VIII} \\ \mathbf{B}_{37} & = & x_{11} \, \mathbf{a}_{1} + y_{11} \, \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(4e\right) & \mbox{K I} \\ \mathbf{B}_{38} & = & -x_{11} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{11}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{11}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{K I} \\ \mathbf{B}_{39} & = & -x_{11} \, \mathbf{a}_{1}-y_{11} \, \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(4e\right) & \mbox{K I} \\ \mathbf{B}_{40} & = & x_{11} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{11}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{K I} \\ \mathbf{B}_{41} & = & x_{12} \, \mathbf{a}_{1} + y_{12} \, \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(4e\right) & \mbox{K II} \\ \mathbf{B}_{42} & = & -x_{12} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{12}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{12}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{K II} \\ \mathbf{B}_{43} & = & -x_{12} \, \mathbf{a}_{1}-y_{12} \, \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(4e\right) & \mbox{K II} \\ \mathbf{B}_{44} & = & x_{12} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{12}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{K II} \\ \mathbf{B}_{45} & = & x_{13} \, \mathbf{a}_{1} + y_{13} \, \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(4e\right) & \mbox{O I} \\ \mathbf{B}_{46} & = & -x_{13} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{13}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{13}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O I} \\ \mathbf{B}_{47} & = & -x_{13} \, \mathbf{a}_{1}-y_{13} \, \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(4e\right) & \mbox{O I} \\ \mathbf{B}_{48} & = & x_{13} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{13}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O I} \\ \mathbf{B}_{49} & = & x_{14} \, \mathbf{a}_{1} + y_{14} \, \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(4e\right) & \mbox{O II} \\ \mathbf{B}_{50} & = & -x_{14} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{14}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{14}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O II} \\ \mathbf{B}_{51} & = & -x_{14} \, \mathbf{a}_{1}-y_{14} \, \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(4e\right) & \mbox{O II} \\ \mathbf{B}_{52} & = & x_{14} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{14}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O II} \\ \mathbf{B}_{53} & = & x_{15} \, \mathbf{a}_{1} + y_{15} \, \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(4e\right) & \mbox{O III} \\ \mathbf{B}_{54} & = & -x_{15} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{15}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{15}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{55} & = & -x_{15} \, \mathbf{a}_{1}-y_{15} \, \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(4e\right) & \mbox{O III} \\ \mathbf{B}_{56} & = & x_{15} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{15}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O III} \\ \mathbf{B}_{57} & = & x_{16} \, \mathbf{a}_{1} + y_{16} \, \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(4e\right) & \mbox{O IV} \\ \mathbf{B}_{58} & = & -x_{16} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{16}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{16}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IV} \\ \mathbf{B}_{59} & = & -x_{16} \, \mathbf{a}_{1}-y_{16} \, \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(4e\right) & \mbox{O IV} \\ \mathbf{B}_{60} & = & x_{16} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{16}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IV} \\ \mathbf{B}_{61} & = & x_{17} \, \mathbf{a}_{1} + y_{17} \, \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(4e\right) & \mbox{O V} \\ \mathbf{B}_{62} & = & -x_{17} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{17}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{17}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O V} \\ \mathbf{B}_{63} & = & -x_{17} \, \mathbf{a}_{1}-y_{17} \, \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(4e\right) & \mbox{O V} \\ \mathbf{B}_{64} & = & x_{17} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{17}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O V} \\ \mathbf{B}_{65} & = & x_{18} \, \mathbf{a}_{1} + y_{18} \, \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(4e\right) & \mbox{O VI} \\ \mathbf{B}_{66} & = & -x_{18} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{18}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{18}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VI} \\ \mathbf{B}_{67} & = & -x_{18} \, \mathbf{a}_{1}-y_{18} \, \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(4e\right) & \mbox{O VI} \\ \mathbf{B}_{68} & = & x_{18} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{18}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VI} \\ \mathbf{B}_{69} & = & x_{19} \, \mathbf{a}_{1} + y_{19} \, \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(4e\right) & \mbox{O VII} \\ \mathbf{B}_{70} & = & -x_{19} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{19}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{19}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VII} \\ \mathbf{B}_{71} & = & -x_{19} \, \mathbf{a}_{1}-y_{19} \, \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(4e\right) & \mbox{O VII} \\ \mathbf{B}_{72} & = & x_{19} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{19}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VII} \\ \mathbf{B}_{73} & = & x_{20} \, \mathbf{a}_{1} + y_{20} \, \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(4e\right) & \mbox{O VIII} \\ \mathbf{B}_{74} & = & -x_{20} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{20}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{20}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VIII} \\ \mathbf{B}_{75} & = & -x_{20} \, \mathbf{a}_{1}-y_{20} \, \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(4e\right) & \mbox{O VIII} \\ \mathbf{B}_{76} & = & x_{20} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{20}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{20}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O VIII} \\ \mathbf{B}_{77} & = & x_{21} \, \mathbf{a}_{1} + y_{21} \, \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(4e\right) & \mbox{O IX} \\ \mathbf{B}_{78} & = & -x_{21} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{21}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{21}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IX} \\ \mathbf{B}_{79} & = & -x_{21} \, \mathbf{a}_{1}-y_{21} \, \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(4e\right) & \mbox{O IX} \\ \mathbf{B}_{80} & = & x_{21} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{21}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{21}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O IX} \\ \mathbf{B}_{81} & = & x_{22} \, \mathbf{a}_{1} + y_{22} \, \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(4e\right) & \mbox{O X} \\ \mathbf{B}_{82} & = & -x_{22} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{22}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{22}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O X} \\ \mathbf{B}_{83} & = & -x_{22} \, \mathbf{a}_{1}-y_{22} \, \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(4e\right) & \mbox{O X} \\ \mathbf{B}_{84} & = & x_{22} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{22}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{22}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O X} \\ \mathbf{B}_{85} & = & x_{23} \, \mathbf{a}_{1} + y_{23} \, \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(4e\right) & \mbox{O XI} \\ \mathbf{B}_{86} & = & -x_{23} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{23}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{23}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XI} \\ \mathbf{B}_{87} & = & -x_{23} \, \mathbf{a}_{1}-y_{23} \, \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(4e\right) & \mbox{O XI} \\ \mathbf{B}_{88} & = & x_{23} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{23}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{23}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XI} \\ \mathbf{B}_{89} & = & x_{24} \, \mathbf{a}_{1} + y_{24} \, \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(4e\right) & \mbox{O XII} \\ \mathbf{B}_{90} & = & -x_{24} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{24}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{24}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XII} \\ \mathbf{B}_{91} & = & -x_{24} \, \mathbf{a}_{1}-y_{24} \, \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(4e\right) & \mbox{O XII} \\ \mathbf{B}_{92} & = & x_{24} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{24}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{24}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{O XII} \\ \mathbf{B}_{93} & = & x_{25} \, \mathbf{a}_{1} + y_{25} \, \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(4e\right) & \mbox{S I} \\ \mathbf{B}_{94} & = & -x_{25} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{25}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{25}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{S I} \\ \mathbf{B}_{95} & = & -x_{25} \, \mathbf{a}_{1}-y_{25} \, \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(4e\right) & \mbox{S I} \\ \mathbf{B}_{96} & = & x_{25} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{25}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{25}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{S I} \\ \mathbf{B}_{97} & = & x_{26} \, \mathbf{a}_{1} + y_{26} \, \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(4e\right) & \mbox{S II} \\ \mathbf{B}_{98} & = & -x_{26} \, \mathbf{a}_{1} + \left(\frac{1}{2} +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}} + \left(\frac{1}{2} +y_{26}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{26}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\right) & \mbox{S II} \\ \mathbf{B}_{99} & = & -x_{26} \, \mathbf{a}_{1}-y_{26} \, \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(4e\right) & \mbox{S II} \\ \mathbf{B}_{100} & = & x_{26} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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}} + \left(\frac{1}{2}-y_{26}\right)b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{26}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(4e\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, K2$Me$(SO4)2·4H2O ($Me2+ = Mg, Mn, Fe), Am. Mineral. 86, 1282–1292 (2001), doi:10.2138/am-2001-1016.

Geometry files


Prototype Generator

aflow --proto=A8B2CD12E2_mP100_14_8e_2e_ad_12e_2e --params=

Species:

Running:

Output: