Low–Temperature Mo8O23 Structure : A8B23_mP124_7_16a_46a

Picture of Structure; Click for Big Picture
Prototype : Mo8O23
AFLOW prototype label : A8B23_mP124_7_16a_46a
Strukturbericht designation : None
Pearson symbol : mP124
Space group number : 7
Space group symbol : $Pc$
AFLOW prototype command : aflow --proto=A8B23_mP124_7_16a_46a
--params=
$a$,$b/a$,$c/a$,$\beta$,$x_{1}$,$y_{1}$,$z_{1}$,$x_{2}$,$y_{2}$,$z_{2}$,$x_{3}$,$y_{3}$,$z_{3}$,$x_{4}$,$y_{4}$,$z_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$,$x_{9}$,$y_{9}$,$z_{9}$,$x_{10}$,$y_{10}$,$z_{10}$,$x_{11}$,$y_{11}$,$z_{11}$,$x_{12}$,$y_{12}$,$z_{12}$,$x_{13}$,$y_{13}$,$z_{13}$,$x_{14}$,$y_{14}$,$z_{14}$,$x_{15}$,$y_{15}$,$z_{15}$,$x_{16}$,$y_{16}$,$z_{16}$,$x_{17}$,$y_{17}$,$z_{17}$,$x_{18}$,$y_{18}$,$z_{18}$,$x_{19}$,$y_{19}$,$z_{19}$,$x_{20}$,$y_{20}$,$z_{20}$,$x_{21}$,$y_{21}$,$z_{21}$,$x_{22}$,$y_{22}$,$z_{22}$,$x_{23}$,$y_{23}$,$z_{23}$,$x_{24}$,$y_{24}$,$z_{24}$,$x_{25}$,$y_{25}$,$z_{25}$,$x_{26}$,$y_{26}$,$z_{26}$,$x_{27}$,$y_{27}$,$z_{27}$,$x_{28}$,$y_{28}$,$z_{28}$,$x_{29}$,$y_{29}$,$z_{29}$,$x_{30}$,$y_{30}$,$z_{30}$,$x_{31}$,$y_{31}$,$z_{31}$,$x_{32}$,$y_{32}$,$z_{32}$,$x_{33}$,$y_{33}$,$z_{33}$,$x_{34}$,$y_{34}$,$z_{34}$,$x_{35}$,$y_{35}$,$z_{35}$,$x_{36}$,$y_{36}$,$z_{36}$,$x_{37}$,$y_{37}$,$z_{37}$,$x_{38}$,$y_{38}$,$z_{38}$,$x_{39}$,$y_{39}$,$z_{39}$,$x_{40}$,$y_{40}$,$z_{40}$,$x_{41}$,$y_{41}$,$z_{41}$,$x_{42}$,$y_{42}$,$z_{42}$,$x_{43}$,$y_{43}$,$z_{43}$,$x_{44}$,$y_{44}$,$z_{44}$,$x_{45}$,$y_{45}$,$z_{45}$,$x_{46}$,$y_{46}$,$z_{46}$,$x_{47}$,$y_{47}$,$z_{47}$,$x_{48}$,$y_{48}$,$z_{48}$,$x_{49}$,$y_{49}$,$z_{49}$,$x_{50}$,$y_{50}$,$z_{50}$,$x_{51}$,$y_{51}$,$z_{51}$,$x_{52}$,$y_{52}$,$z_{52}$,$x_{53}$,$y_{53}$,$z_{53}$,$x_{54}$,$y_{54}$,$z_{54}$,$x_{55}$,$y_{55}$,$z_{55}$,$x_{56}$,$y_{56}$,$z_{56}$,$x_{57}$,$y_{57}$,$z_{57}$,$x_{58}$,$y_{58}$,$z_{58}$,$x_{59}$,$y_{59}$,$z_{59}$,$x_{60}$,$y_{60}$,$z_{60}$,$x_{61}$,$y_{61}$,$z_{61}$,$x_{62}$,$y_{62}$,$z_{62}$


  • This data was taken at 100 K. Above 285 K the structure exhibits an incommensurate charge density wave, which is approximated by the high temperature Mo8O23 structure.

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} & = & x_{1} \, \mathbf{a}_{1} + y_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(x_{1}a+z_{1}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{1}b \, \mathbf{\hat{y}} + z_{1}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Mo I} \\ \mathbf{B}_{2} & = & x_{1} \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{1}a + z_{1}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{1}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Mo I} \\ \mathbf{B}_{3} & = & x_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \left(x_{2}a+z_{2}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{2}b \, \mathbf{\hat{y}} + z_{2}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Mo II} \\ \mathbf{B}_{4} & = & x_{2} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{2}a + z_{2}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{2}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Mo 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(2a\right) & \mbox{Mo III} \\ \mathbf{B}_{6} & = & x_{3} \, \mathbf{a}_{1}-y_{3} \, \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}}-y_{3}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Mo III} \\ \mathbf{B}_{7} & = & 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(2a\right) & \mbox{Mo IV} \\ \mathbf{B}_{8} & = & x_{4} \, \mathbf{a}_{1}-y_{4} \, \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}}-y_{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Mo IV} \\ \mathbf{B}_{9} & = & 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(2a\right) & \mbox{Mo V} \\ \mathbf{B}_{10} & = & x_{5} \, \mathbf{a}_{1}-y_{5} \, \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(2a\right) & \mbox{Mo V} \\ \mathbf{B}_{11} & = & 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(2a\right) & \mbox{Mo VI} \\ \mathbf{B}_{12} & = & x_{6} \, \mathbf{a}_{1}-y_{6} \, \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(2a\right) & \mbox{Mo VI} \\ \mathbf{B}_{13} & = & 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(2a\right) & \mbox{Mo VII} \\ \mathbf{B}_{14} & = & x_{7} \, \mathbf{a}_{1}-y_{7} \, \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(2a\right) & \mbox{Mo VII} \\ \mathbf{B}_{15} & = & 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(2a\right) & \mbox{Mo VIII} \\ \mathbf{B}_{16} & = & x_{8} \, \mathbf{a}_{1}-y_{8} \, \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(2a\right) & \mbox{Mo VIII} \\ \mathbf{B}_{17} & = & 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(2a\right) & \mbox{Mo IX} \\ \mathbf{B}_{18} & = & x_{9} \, \mathbf{a}_{1}-y_{9} \, \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(2a\right) & \mbox{Mo IX} \\ \mathbf{B}_{19} & = & 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(2a\right) & \mbox{Mo X} \\ \mathbf{B}_{20} & = & x_{10} \, \mathbf{a}_{1}-y_{10} \, \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(2a\right) & \mbox{Mo X} \\ \mathbf{B}_{21} & = & 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(2a\right) & \mbox{Mo XI} \\ \mathbf{B}_{22} & = & x_{11} \, \mathbf{a}_{1}-y_{11} \, \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(2a\right) & \mbox{Mo XI} \\ \mathbf{B}_{23} & = & 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(2a\right) & \mbox{Mo XII} \\ \mathbf{B}_{24} & = & x_{12} \, \mathbf{a}_{1}-y_{12} \, \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(2a\right) & \mbox{Mo XII} \\ \mathbf{B}_{25} & = & 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(2a\right) & \mbox{Mo XIII} \\ \mathbf{B}_{26} & = & x_{13} \, \mathbf{a}_{1}-y_{13} \, \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(2a\right) & \mbox{Mo XIII} \\ \mathbf{B}_{27} & = & 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(2a\right) & \mbox{Mo XIV} \\ \mathbf{B}_{28} & = & x_{14} \, \mathbf{a}_{1}-y_{14} \, \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(2a\right) & \mbox{Mo XIV} \\ \mathbf{B}_{29} & = & 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(2a\right) & \mbox{Mo XV} \\ \mathbf{B}_{30} & = & x_{15} \, \mathbf{a}_{1}-y_{15} \, \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(2a\right) & \mbox{Mo XV} \\ \mathbf{B}_{31} & = & 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(2a\right) & \mbox{Mo XVI} \\ \mathbf{B}_{32} & = & x_{16} \, \mathbf{a}_{1}-y_{16} \, \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(2a\right) & \mbox{Mo XVI} \\ \mathbf{B}_{33} & = & 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(2a\right) & \mbox{O I} \\ \mathbf{B}_{34} & = & x_{17} \, \mathbf{a}_{1}-y_{17} \, \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(2a\right) & \mbox{O I} \\ \mathbf{B}_{35} & = & 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(2a\right) & \mbox{O II} \\ \mathbf{B}_{36} & = & x_{18} \, \mathbf{a}_{1}-y_{18} \, \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(2a\right) & \mbox{O II} \\ \mathbf{B}_{37} & = & 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(2a\right) & \mbox{O III} \\ \mathbf{B}_{38} & = & x_{19} \, \mathbf{a}_{1}-y_{19} \, \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(2a\right) & \mbox{O III} \\ \mathbf{B}_{39} & = & 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(2a\right) & \mbox{O IV} \\ \mathbf{B}_{40} & = & x_{20} \, \mathbf{a}_{1}-y_{20} \, \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(2a\right) & \mbox{O IV} \\ \mathbf{B}_{41} & = & 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(2a\right) & \mbox{O V} \\ \mathbf{B}_{42} & = & x_{21} \, \mathbf{a}_{1}-y_{21} \, \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(2a\right) & \mbox{O V} \\ \mathbf{B}_{43} & = & 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(2a\right) & \mbox{O VI} \\ \mathbf{B}_{44} & = & x_{22} \, \mathbf{a}_{1}-y_{22} \, \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(2a\right) & \mbox{O VI} \\ \mathbf{B}_{45} & = & 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(2a\right) & \mbox{O VII} \\ \mathbf{B}_{46} & = & x_{23} \, \mathbf{a}_{1}-y_{23} \, \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(2a\right) & \mbox{O VII} \\ \mathbf{B}_{47} & = & 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(2a\right) & \mbox{O VIII} \\ \mathbf{B}_{48} & = & x_{24} \, \mathbf{a}_{1}-y_{24} \, \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(2a\right) & \mbox{O VIII} \\ \mathbf{B}_{49} & = & 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(2a\right) & \mbox{O IX} \\ \mathbf{B}_{50} & = & x_{25} \, \mathbf{a}_{1}-y_{25} \, \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(2a\right) & \mbox{O IX} \\ \mathbf{B}_{51} & = & 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(2a\right) & \mbox{O X} \\ \mathbf{B}_{52} & = & x_{26} \, \mathbf{a}_{1}-y_{26} \, \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(2a\right) & \mbox{O X} \\ \mathbf{B}_{53} & = & x_{27} \, \mathbf{a}_{1} + y_{27} \, \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(2a\right) & \mbox{O XI} \\ \mathbf{B}_{54} & = & x_{27} \, \mathbf{a}_{1}-y_{27} \, \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(2a\right) & \mbox{O XI} \\ \mathbf{B}_{55} & = & x_{28} \, \mathbf{a}_{1} + y_{28} \, \mathbf{a}_{2} + z_{28} \, \mathbf{a}_{3} & = & \left(x_{28}a+z_{28}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{28}b \, \mathbf{\hat{y}} + z_{28}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XII} \\ \mathbf{B}_{56} & = & x_{28} \, \mathbf{a}_{1}-y_{28} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{28}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{28}a + z_{28}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{28}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{28}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XII} \\ \mathbf{B}_{57} & = & x_{29} \, \mathbf{a}_{1} + y_{29} \, \mathbf{a}_{2} + z_{29} \, \mathbf{a}_{3} & = & \left(x_{29}a+z_{29}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{29}b \, \mathbf{\hat{y}} + z_{29}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XIII} \\ \mathbf{B}_{58} & = & x_{29} \, \mathbf{a}_{1}-y_{29} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{29}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{29}a + z_{29}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{29}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{29}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XIII} \\ \mathbf{B}_{59} & = & x_{30} \, \mathbf{a}_{1} + y_{30} \, \mathbf{a}_{2} + z_{30} \, \mathbf{a}_{3} & = & \left(x_{30}a+z_{30}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{30}b \, \mathbf{\hat{y}} + z_{30}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XIV} \\ \mathbf{B}_{60} & = & x_{30} \, \mathbf{a}_{1}-y_{30} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{30}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{30}a + z_{30}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{30}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{30}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XIV} \\ \mathbf{B}_{61} & = & x_{31} \, \mathbf{a}_{1} + y_{31} \, \mathbf{a}_{2} + z_{31} \, \mathbf{a}_{3} & = & \left(x_{31}a+z_{31}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{31}b \, \mathbf{\hat{y}} + z_{31}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XV} \\ \mathbf{B}_{62} & = & x_{31} \, \mathbf{a}_{1}-y_{31} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{31}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{31}a + z_{31}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{31}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{31}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XV} \\ \mathbf{B}_{63} & = & x_{32} \, \mathbf{a}_{1} + y_{32} \, \mathbf{a}_{2} + z_{32} \, \mathbf{a}_{3} & = & \left(x_{32}a+z_{32}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{32}b \, \mathbf{\hat{y}} + z_{32}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XVI} \\ \mathbf{B}_{64} & = & x_{32} \, \mathbf{a}_{1}-y_{32} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{32}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{32}a + z_{32}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{32}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{32}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XVI} \\ \mathbf{B}_{65} & = & x_{33} \, \mathbf{a}_{1} + y_{33} \, \mathbf{a}_{2} + z_{33} \, \mathbf{a}_{3} & = & \left(x_{33}a+z_{33}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{33}b \, \mathbf{\hat{y}} + z_{33}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XVII} \\ \mathbf{B}_{66} & = & x_{33} \, \mathbf{a}_{1}-y_{33} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{33}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{33}a + z_{33}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{33}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{33}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XVII} \\ \mathbf{B}_{67} & = & x_{34} \, \mathbf{a}_{1} + y_{34} \, \mathbf{a}_{2} + z_{34} \, \mathbf{a}_{3} & = & \left(x_{34}a+z_{34}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{34}b \, \mathbf{\hat{y}} + z_{34}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XVIII} \\ \mathbf{B}_{68} & = & x_{34} \, \mathbf{a}_{1}-y_{34} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{34}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{34}a + z_{34}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{34}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{34}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XVIII} \\ \mathbf{B}_{69} & = & x_{35} \, \mathbf{a}_{1} + y_{35} \, \mathbf{a}_{2} + z_{35} \, \mathbf{a}_{3} & = & \left(x_{35}a+z_{35}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{35}b \, \mathbf{\hat{y}} + z_{35}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XIX} \\ \mathbf{B}_{70} & = & x_{35} \, \mathbf{a}_{1}-y_{35} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{35}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{35}a + z_{35}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{35}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{35}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XIX} \\ \mathbf{B}_{71} & = & x_{36} \, \mathbf{a}_{1} + y_{36} \, \mathbf{a}_{2} + z_{36} \, \mathbf{a}_{3} & = & \left(x_{36}a+z_{36}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{36}b \, \mathbf{\hat{y}} + z_{36}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XX} \\ \mathbf{B}_{72} & = & x_{36} \, \mathbf{a}_{1}-y_{36} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{36}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{36}a + z_{36}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{36}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{36}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XX} \\ \mathbf{B}_{73} & = & x_{37} \, \mathbf{a}_{1} + y_{37} \, \mathbf{a}_{2} + z_{37} \, \mathbf{a}_{3} & = & \left(x_{37}a+z_{37}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{37}b \, \mathbf{\hat{y}} + z_{37}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXI} \\ \mathbf{B}_{74} & = & x_{37} \, \mathbf{a}_{1}-y_{37} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{37}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{37}a + z_{37}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{37}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{37}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXI} \\ \mathbf{B}_{75} & = & x_{38} \, \mathbf{a}_{1} + y_{38} \, \mathbf{a}_{2} + z_{38} \, \mathbf{a}_{3} & = & \left(x_{38}a+z_{38}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{38}b \, \mathbf{\hat{y}} + z_{38}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXII} \\ \mathbf{B}_{76} & = & x_{38} \, \mathbf{a}_{1}-y_{38} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{38}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{38}a + z_{38}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{38}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{38}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXII} \\ \mathbf{B}_{77} & = & x_{39} \, \mathbf{a}_{1} + y_{39} \, \mathbf{a}_{2} + z_{39} \, \mathbf{a}_{3} & = & \left(x_{39}a+z_{39}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{39}b \, \mathbf{\hat{y}} + z_{39}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXIII} \\ \mathbf{B}_{78} & = & x_{39} \, \mathbf{a}_{1}-y_{39} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{39}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{39}a + z_{39}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{39}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{39}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXIII} \\ \mathbf{B}_{79} & = & x_{40} \, \mathbf{a}_{1} + y_{40} \, \mathbf{a}_{2} + z_{40} \, \mathbf{a}_{3} & = & \left(x_{40}a+z_{40}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{40}b \, \mathbf{\hat{y}} + z_{40}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXIV} \\ \mathbf{B}_{80} & = & x_{40} \, \mathbf{a}_{1}-y_{40} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{40}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{40}a + z_{40}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{40}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{40}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXIV} \\ \mathbf{B}_{81} & = & x_{41} \, \mathbf{a}_{1} + y_{41} \, \mathbf{a}_{2} + z_{41} \, \mathbf{a}_{3} & = & \left(x_{41}a+z_{41}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{41}b \, \mathbf{\hat{y}} + z_{41}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXV} \\ \mathbf{B}_{82} & = & x_{41} \, \mathbf{a}_{1}-y_{41} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{41}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{41}a + z_{41}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{41}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{41}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXV} \\ \mathbf{B}_{83} & = & x_{42} \, \mathbf{a}_{1} + y_{42} \, \mathbf{a}_{2} + z_{42} \, \mathbf{a}_{3} & = & \left(x_{42}a+z_{42}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{42}b \, \mathbf{\hat{y}} + z_{42}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXVI} \\ \mathbf{B}_{84} & = & x_{42} \, \mathbf{a}_{1}-y_{42} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{42}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{42}a + z_{42}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{42}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{42}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXVI} \\ \mathbf{B}_{85} & = & x_{43} \, \mathbf{a}_{1} + y_{43} \, \mathbf{a}_{2} + z_{43} \, \mathbf{a}_{3} & = & \left(x_{43}a+z_{43}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{43}b \, \mathbf{\hat{y}} + z_{43}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXVII} \\ \mathbf{B}_{86} & = & x_{43} \, \mathbf{a}_{1}-y_{43} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{43}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{43}a + z_{43}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{43}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{43}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXVII} \\ \mathbf{B}_{87} & = & x_{44} \, \mathbf{a}_{1} + y_{44} \, \mathbf{a}_{2} + z_{44} \, \mathbf{a}_{3} & = & \left(x_{44}a+z_{44}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{44}b \, \mathbf{\hat{y}} + z_{44}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXVIII} \\ \mathbf{B}_{88} & = & x_{44} \, \mathbf{a}_{1}-y_{44} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{44}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{44}a + z_{44}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{44}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{44}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXVIII} \\ \mathbf{B}_{89} & = & x_{45} \, \mathbf{a}_{1} + y_{45} \, \mathbf{a}_{2} + z_{45} \, \mathbf{a}_{3} & = & \left(x_{45}a+z_{45}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{45}b \, \mathbf{\hat{y}} + z_{45}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXIX} \\ \mathbf{B}_{90} & = & x_{45} \, \mathbf{a}_{1}-y_{45} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{45}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{45}a + z_{45}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{45}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{45}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXIX} \\ \mathbf{B}_{91} & = & x_{46} \, \mathbf{a}_{1} + y_{46} \, \mathbf{a}_{2} + z_{46} \, \mathbf{a}_{3} & = & \left(x_{46}a+z_{46}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{46}b \, \mathbf{\hat{y}} + z_{46}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXX} \\ \mathbf{B}_{92} & = & x_{46} \, \mathbf{a}_{1}-y_{46} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{46}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{46}a + z_{46}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{46}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{46}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXX} \\ \mathbf{B}_{93} & = & x_{47} \, \mathbf{a}_{1} + y_{47} \, \mathbf{a}_{2} + z_{47} \, \mathbf{a}_{3} & = & \left(x_{47}a+z_{47}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{47}b \, \mathbf{\hat{y}} + z_{47}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXI} \\ \mathbf{B}_{94} & = & x_{47} \, \mathbf{a}_{1}-y_{47} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{47}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{47}a + z_{47}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{47}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{47}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXI} \\ \mathbf{B}_{95} & = & x_{48} \, \mathbf{a}_{1} + y_{48} \, \mathbf{a}_{2} + z_{48} \, \mathbf{a}_{3} & = & \left(x_{48}a+z_{48}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{48}b \, \mathbf{\hat{y}} + z_{48}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXII} \\ \mathbf{B}_{96} & = & x_{48} \, \mathbf{a}_{1}-y_{48} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{48}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{48}a + z_{48}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{48}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{48}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXII} \\ \mathbf{B}_{97} & = & x_{49} \, \mathbf{a}_{1} + y_{49} \, \mathbf{a}_{2} + z_{49} \, \mathbf{a}_{3} & = & \left(x_{49}a+z_{49}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{49}b \, \mathbf{\hat{y}} + z_{49}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXIII} \\ \mathbf{B}_{98} & = & x_{49} \, \mathbf{a}_{1}-y_{49} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{49}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{49}a + z_{49}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{49}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{49}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXIII} \\ \mathbf{B}_{99} & = & x_{50} \, \mathbf{a}_{1} + y_{50} \, \mathbf{a}_{2} + z_{50} \, \mathbf{a}_{3} & = & \left(x_{50}a+z_{50}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{50}b \, \mathbf{\hat{y}} + z_{50}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXIV} \\ \mathbf{B}_{100} & = & x_{50} \, \mathbf{a}_{1}-y_{50} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{50}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{50}a + z_{50}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{50}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{50}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXIV} \\ \mathbf{B}_{101} & = & x_{51} \, \mathbf{a}_{1} + y_{51} \, \mathbf{a}_{2} + z_{51} \, \mathbf{a}_{3} & = & \left(x_{51}a+z_{51}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{51}b \, \mathbf{\hat{y}} + z_{51}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXV} \\ \mathbf{B}_{102} & = & x_{51} \, \mathbf{a}_{1}-y_{51} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{51}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{51}a + z_{51}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{51}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{51}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXV} \\ \mathbf{B}_{103} & = & x_{52} \, \mathbf{a}_{1} + y_{52} \, \mathbf{a}_{2} + z_{52} \, \mathbf{a}_{3} & = & \left(x_{52}a+z_{52}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{52}b \, \mathbf{\hat{y}} + z_{52}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXVI} \\ \mathbf{B}_{104} & = & x_{52} \, \mathbf{a}_{1}-y_{52} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{52}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{52}a + z_{52}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{52}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{52}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXVI} \\ \mathbf{B}_{105} & = & x_{53} \, \mathbf{a}_{1} + y_{53} \, \mathbf{a}_{2} + z_{53} \, \mathbf{a}_{3} & = & \left(x_{53}a+z_{53}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{53}b \, \mathbf{\hat{y}} + z_{53}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXVII} \\ \mathbf{B}_{106} & = & x_{53} \, \mathbf{a}_{1}-y_{53} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{53}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{53}a + z_{53}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{53}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{53}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXVII} \\ \mathbf{B}_{107} & = & x_{54} \, \mathbf{a}_{1} + y_{54} \, \mathbf{a}_{2} + z_{54} \, \mathbf{a}_{3} & = & \left(x_{54}a+z_{54}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{54}b \, \mathbf{\hat{y}} + z_{54}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXVIII} \\ \mathbf{B}_{108} & = & x_{54} \, \mathbf{a}_{1}-y_{54} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{54}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{54}a + z_{54}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{54}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{54}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXVIII} \\ \mathbf{B}_{109} & = & x_{55} \, \mathbf{a}_{1} + y_{55} \, \mathbf{a}_{2} + z_{55} \, \mathbf{a}_{3} & = & \left(x_{55}a+z_{55}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{55}b \, \mathbf{\hat{y}} + z_{55}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXIX} \\ \mathbf{B}_{110} & = & x_{55} \, \mathbf{a}_{1}-y_{55} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{55}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{55}a + z_{55}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{55}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{55}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XXXIX} \\ \mathbf{B}_{111} & = & x_{56} \, \mathbf{a}_{1} + y_{56} \, \mathbf{a}_{2} + z_{56} \, \mathbf{a}_{3} & = & \left(x_{56}a+z_{56}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{56}b \, \mathbf{\hat{y}} + z_{56}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XL} \\ \mathbf{B}_{112} & = & x_{56} \, \mathbf{a}_{1}-y_{56} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{56}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{56}a + z_{56}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{56}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{56}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XL} \\ \mathbf{B}_{113} & = & x_{57} \, \mathbf{a}_{1} + y_{57} \, \mathbf{a}_{2} + z_{57} \, \mathbf{a}_{3} & = & \left(x_{57}a+z_{57}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{57}b \, \mathbf{\hat{y}} + z_{57}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLI} \\ \mathbf{B}_{114} & = & x_{57} \, \mathbf{a}_{1}-y_{57} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{57}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{57}a + z_{57}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{57}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{57}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLI} \\ \mathbf{B}_{115} & = & x_{58} \, \mathbf{a}_{1} + y_{58} \, \mathbf{a}_{2} + z_{58} \, \mathbf{a}_{3} & = & \left(x_{58}a+z_{58}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{58}b \, \mathbf{\hat{y}} + z_{58}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLII} \\ \mathbf{B}_{116} & = & x_{58} \, \mathbf{a}_{1}-y_{58} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{58}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{58}a + z_{58}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{58}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{58}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLII} \\ \mathbf{B}_{117} & = & x_{59} \, \mathbf{a}_{1} + y_{59} \, \mathbf{a}_{2} + z_{59} \, \mathbf{a}_{3} & = & \left(x_{59}a+z_{59}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{59}b \, \mathbf{\hat{y}} + z_{59}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLIII} \\ \mathbf{B}_{118} & = & x_{59} \, \mathbf{a}_{1}-y_{59} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{59}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{59}a + z_{59}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{59}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{59}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLIII} \\ \mathbf{B}_{119} & = & x_{60} \, \mathbf{a}_{1} + y_{60} \, \mathbf{a}_{2} + z_{60} \, \mathbf{a}_{3} & = & \left(x_{60}a+z_{60}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{60}b \, \mathbf{\hat{y}} + z_{60}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLIV} \\ \mathbf{B}_{120} & = & x_{60} \, \mathbf{a}_{1}-y_{60} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{60}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{60}a + z_{60}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{60}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{60}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLIV} \\ \mathbf{B}_{121} & = & x_{61} \, \mathbf{a}_{1} + y_{61} \, \mathbf{a}_{2} + z_{61} \, \mathbf{a}_{3} & = & \left(x_{61}a+z_{61}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{61}b \, \mathbf{\hat{y}} + z_{61}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLV} \\ \mathbf{B}_{122} & = & x_{61} \, \mathbf{a}_{1}-y_{61} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{61}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{61}a + z_{61}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{61}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{61}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLV} \\ \mathbf{B}_{123} & = & x_{62} \, \mathbf{a}_{1} + y_{62} \, \mathbf{a}_{2} + z_{62} \, \mathbf{a}_{3} & = & \left(x_{62}a+z_{62}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{62}b \, \mathbf{\hat{y}} + z_{62}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLVI} \\ \mathbf{B}_{124} & = & x_{62} \, \mathbf{a}_{1}-y_{62} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{62}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{62}a + z_{62}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{62}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{62}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{O XLVI} \\ \end{array} \]

References

  • H. Fujishita, M. Sato, S. Sato, and S. Hoshino, Structure Determination of low–dimensional conductor Mo8O23, J. Solid State Chem. 66, 40–46 (1987), doi:10.1016/0022-4596(87)90218-0.

Geometry files


Prototype Generator

aflow --proto=A8B23_mP124_7_16a_46a --params=

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