Library of Crystallographic Prototypes

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} & = & x_{1}a \, \mathbf{\hat{x}} + y_{1}a \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As I} \\ \mathbf{B}_{2} & = & -x_{1} \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & -x_{1}a \, \mathbf{\hat{x}}-y_{1}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As I} \\ \mathbf{B}_{3} & = & -y_{1} \, \mathbf{a}_{1} + x_{1} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{1}\right) \, \mathbf{a}_{3} & = & -y_{1}a \, \mathbf{\hat{x}} + x_{1}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As I} \\ \mathbf{B}_{4} & = & y_{1} \, \mathbf{a}_{1}-x_{1} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{1}\right) \, \mathbf{a}_{3} & = & y_{1}a \, \mathbf{\hat{x}}-x_{1}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{1}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As I} \\ \mathbf{B}_{5} & = & x_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + y_{2}a \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As II} \\ \mathbf{B}_{6} & = & -x_{2} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}}-y_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As II} \\ \mathbf{B}_{7} & = & -y_{2} \, \mathbf{a}_{1} + x_{2} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{2}\right) \, \mathbf{a}_{3} & = & -y_{2}a \, \mathbf{\hat{x}} + x_{2}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As II} \\ \mathbf{B}_{8} & = & y_{2} \, \mathbf{a}_{1}-x_{2} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{2}\right) \, \mathbf{a}_{3} & = & y_{2}a \, \mathbf{\hat{x}}-x_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{2}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As II} \\ \mathbf{B}_{9} & = & x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As III} \\ \mathbf{B}_{10} & = & -x_{3} \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}}-y_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As III} \\ \mathbf{B}_{11} & = & -y_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{3}\right) \, \mathbf{a}_{3} & = & -y_{3}a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As III} \\ \mathbf{B}_{12} & = & y_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{3}\right) \, \mathbf{a}_{3} & = & y_{3}a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As III} \\ \mathbf{B}_{13} & = & x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + y_{4}a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As IV} \\ \mathbf{B}_{14} & = & -x_{4} \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}}-y_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As IV} \\ \mathbf{B}_{15} & = & -y_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{4}\right) \, \mathbf{a}_{3} & = & -y_{4}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As IV} \\ \mathbf{B}_{16} & = & y_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{4}\right) \, \mathbf{a}_{3} & = & y_{4}a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{As IV} \\ \mathbf{B}_{17} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + y_{5}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{18} & = & -x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-y_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{19} & = & -y_{5} \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{5}\right) \, \mathbf{a}_{3} & = & -y_{5}a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{20} & = & y_{5} \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{5}\right) \, \mathbf{a}_{3} & = & y_{5}a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{21} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O II} \\ \mathbf{B}_{22} & = & -x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O II} \\ \mathbf{B}_{23} & = & -y_{6} \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{6}\right) \, \mathbf{a}_{3} & = & -y_{6}a \, \mathbf{\hat{x}} + x_{6}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O II} \\ \mathbf{B}_{24} & = & y_{6} \, \mathbf{a}_{1}-x_{6} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{6}\right) \, \mathbf{a}_{3} & = & y_{6}a \, \mathbf{\hat{x}}-x_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O II} \\ \mathbf{B}_{25} & = & x_{7} \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + y_{7}a \, \mathbf{\hat{y}} + z_{7}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O III} \\ \mathbf{B}_{26} & = & -x_{7} \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}}-y_{7}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O III} \\ \mathbf{B}_{27} & = & -y_{7} \, \mathbf{a}_{1} + x_{7} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{7}\right) \, \mathbf{a}_{3} & = & -y_{7}a \, \mathbf{\hat{x}} + x_{7}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O III} \\ \mathbf{B}_{28} & = & y_{7} \, \mathbf{a}_{1}-x_{7} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{7}\right) \, \mathbf{a}_{3} & = & y_{7}a \, \mathbf{\hat{x}}-x_{7}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O III} \\ \mathbf{B}_{29} & = & x_{8} \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + y_{8}a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O IV} \\ \mathbf{B}_{30} & = & -x_{8} \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}}-y_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O IV} \\ \mathbf{B}_{31} & = & -y_{8} \, \mathbf{a}_{1} + x_{8} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{8}\right) \, \mathbf{a}_{3} & = & -y_{8}a \, \mathbf{\hat{x}} + x_{8}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O IV} \\ \mathbf{B}_{32} & = & y_{8} \, \mathbf{a}_{1}-x_{8} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{8}\right) \, \mathbf{a}_{3} & = & y_{8}a \, \mathbf{\hat{x}}-x_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O IV} \\ \mathbf{B}_{33} & = & x_{9} \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & x_{9}a \, \mathbf{\hat{x}} + y_{9}a \, \mathbf{\hat{y}} + z_{9}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O V} \\ \mathbf{B}_{34} & = & -x_{9} \, \mathbf{a}_{1}-y_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & -x_{9}a \, \mathbf{\hat{x}}-y_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O V} \\ \mathbf{B}_{35} & = & -y_{9} \, \mathbf{a}_{1} + x_{9} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{9}\right) \, \mathbf{a}_{3} & = & -y_{9}a \, \mathbf{\hat{x}} + x_{9}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O V} \\ \mathbf{B}_{36} & = & y_{9} \, \mathbf{a}_{1}-x_{9} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{9}\right) \, \mathbf{a}_{3} & = & y_{9}a \, \mathbf{\hat{x}}-x_{9}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{9}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O V} \\ \mathbf{B}_{37} & = & x_{10} \, \mathbf{a}_{1} + y_{10} \, \mathbf{a}_{2} + z_{10} \, \mathbf{a}_{3} & = & x_{10}a \, \mathbf{\hat{x}} + y_{10}a \, \mathbf{\hat{y}} + z_{10}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VI} \\ \mathbf{B}_{38} & = & -x_{10} \, \mathbf{a}_{1}-y_{10} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{10}\right) \, \mathbf{a}_{3} & = & -x_{10}a \, \mathbf{\hat{x}}-y_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VI} \\ \mathbf{B}_{39} & = & -y_{10} \, \mathbf{a}_{1} + x_{10} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{10}\right) \, \mathbf{a}_{3} & = & -y_{10}a \, \mathbf{\hat{x}} + x_{10}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VI} \\ \mathbf{B}_{40} & = & y_{10} \, \mathbf{a}_{1}-x_{10} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{10}\right) \, \mathbf{a}_{3} & = & y_{10}a \, \mathbf{\hat{x}}-x_{10}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{10}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VI} \\ \mathbf{B}_{41} & = & x_{11} \, \mathbf{a}_{1} + y_{11} \, \mathbf{a}_{2} + z_{11} \, \mathbf{a}_{3} & = & x_{11}a \, \mathbf{\hat{x}} + y_{11}a \, \mathbf{\hat{y}} + z_{11}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VII} \\ \mathbf{B}_{42} & = & -x_{11} \, \mathbf{a}_{1}-y_{11} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{11}\right) \, \mathbf{a}_{3} & = & -x_{11}a \, \mathbf{\hat{x}}-y_{11}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VII} \\ \mathbf{B}_{43} & = & -y_{11} \, \mathbf{a}_{1} + x_{11} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{11}\right) \, \mathbf{a}_{3} & = & -y_{11}a \, \mathbf{\hat{x}} + x_{11}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VII} \\ \mathbf{B}_{44} & = & y_{11} \, \mathbf{a}_{1}-x_{11} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{11}\right) \, \mathbf{a}_{3} & = & y_{11}a \, \mathbf{\hat{x}}-x_{11}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{11}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VII} \\ \mathbf{B}_{45} & = & x_{12} \, \mathbf{a}_{1} + y_{12} \, \mathbf{a}_{2} + z_{12} \, \mathbf{a}_{3} & = & x_{12}a \, \mathbf{\hat{x}} + y_{12}a \, \mathbf{\hat{y}} + z_{12}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VIII} \\ \mathbf{B}_{46} & = & -x_{12} \, \mathbf{a}_{1}-y_{12} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{12}\right) \, \mathbf{a}_{3} & = & -x_{12}a \, \mathbf{\hat{x}}-y_{12}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VIII} \\ \mathbf{B}_{47} & = & -y_{12} \, \mathbf{a}_{1} + x_{12} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{12}\right) \, \mathbf{a}_{3} & = & -y_{12}a \, \mathbf{\hat{x}} + x_{12}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VIII} \\ \mathbf{B}_{48} & = & y_{12} \, \mathbf{a}_{1}-x_{12} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{12}\right) \, \mathbf{a}_{3} & = & y_{12}a \, \mathbf{\hat{x}}-x_{12}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{12}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O VIII} \\ \mathbf{B}_{49} & = & x_{13} \, \mathbf{a}_{1} + y_{13} \, \mathbf{a}_{2} + z_{13} \, \mathbf{a}_{3} & = & x_{13}a \, \mathbf{\hat{x}} + y_{13}a \, \mathbf{\hat{y}} + z_{13}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O IX} \\ \mathbf{B}_{50} & = & -x_{13} \, \mathbf{a}_{1}-y_{13} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{13}\right) \, \mathbf{a}_{3} & = & -x_{13}a \, \mathbf{\hat{x}}-y_{13}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O IX} \\ \mathbf{B}_{51} & = & -y_{13} \, \mathbf{a}_{1} + x_{13} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{13}\right) \, \mathbf{a}_{3} & = & -y_{13}a \, \mathbf{\hat{x}} + x_{13}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O IX} \\ \mathbf{B}_{52} & = & y_{13} \, \mathbf{a}_{1}-x_{13} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{13}\right) \, \mathbf{a}_{3} & = & y_{13}a \, \mathbf{\hat{x}}-x_{13}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{13}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O IX} \\ \mathbf{B}_{53} & = & x_{14} \, \mathbf{a}_{1} + y_{14} \, \mathbf{a}_{2} + z_{14} \, \mathbf{a}_{3} & = & x_{14}a \, \mathbf{\hat{x}} + y_{14}a \, \mathbf{\hat{y}} + z_{14}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O X} \\ \mathbf{B}_{54} & = & -x_{14} \, \mathbf{a}_{1}-y_{14} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{14}\right) \, \mathbf{a}_{3} & = & -x_{14}a \, \mathbf{\hat{x}}-y_{14}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O X} \\ \mathbf{B}_{55} & = & -y_{14} \, \mathbf{a}_{1} + x_{14} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{14}\right) \, \mathbf{a}_{3} & = & -y_{14}a \, \mathbf{\hat{x}} + x_{14}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O X} \\ \mathbf{B}_{56} & = & y_{14} \, \mathbf{a}_{1}-x_{14} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{14}\right) \, \mathbf{a}_{3} & = & y_{14}a \, \mathbf{\hat{x}}-x_{14}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{14}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O X} \\ \mathbf{B}_{57} & = & x_{15} \, \mathbf{a}_{1} + y_{15} \, \mathbf{a}_{2} + z_{15} \, \mathbf{a}_{3} & = & x_{15}a \, \mathbf{\hat{x}} + y_{15}a \, \mathbf{\hat{y}} + z_{15}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XI} \\ \mathbf{B}_{58} & = & -x_{15} \, \mathbf{a}_{1}-y_{15} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{15}\right) \, \mathbf{a}_{3} & = & -x_{15}a \, \mathbf{\hat{x}}-y_{15}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XI} \\ \mathbf{B}_{59} & = & -y_{15} \, \mathbf{a}_{1} + x_{15} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{15}\right) \, \mathbf{a}_{3} & = & -y_{15}a \, \mathbf{\hat{x}} + x_{15}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XI} \\ \mathbf{B}_{60} & = & y_{15} \, \mathbf{a}_{1}-x_{15} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{15}\right) \, \mathbf{a}_{3} & = & y_{15}a \, \mathbf{\hat{x}}-x_{15}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{15}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XI} \\ \mathbf{B}_{61} & = & x_{16} \, \mathbf{a}_{1} + y_{16} \, \mathbf{a}_{2} + z_{16} \, \mathbf{a}_{3} & = & x_{16}a \, \mathbf{\hat{x}} + y_{16}a \, \mathbf{\hat{y}} + z_{16}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XII} \\ \mathbf{B}_{62} & = & -x_{16} \, \mathbf{a}_{1}-y_{16} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{16}\right) \, \mathbf{a}_{3} & = & -x_{16}a \, \mathbf{\hat{x}}-y_{16}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XII} \\ \mathbf{B}_{63} & = & -y_{16} \, \mathbf{a}_{1} + x_{16} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{16}\right) \, \mathbf{a}_{3} & = & -y_{16}a \, \mathbf{\hat{x}} + x_{16}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XII} \\ \mathbf{B}_{64} & = & y_{16} \, \mathbf{a}_{1}-x_{16} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{16}\right) \, \mathbf{a}_{3} & = & y_{16}a \, \mathbf{\hat{x}}-x_{16}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{16}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XII} \\ \mathbf{B}_{65} & = & x_{17} \, \mathbf{a}_{1} + y_{17} \, \mathbf{a}_{2} + z_{17} \, \mathbf{a}_{3} & = & x_{17}a \, \mathbf{\hat{x}} + y_{17}a \, \mathbf{\hat{y}} + z_{17}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XIII} \\ \mathbf{B}_{66} & = & -x_{17} \, \mathbf{a}_{1}-y_{17} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{17}\right) \, \mathbf{a}_{3} & = & -x_{17}a \, \mathbf{\hat{x}}-y_{17}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XIII} \\ \mathbf{B}_{67} & = & -y_{17} \, \mathbf{a}_{1} + x_{17} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{17}\right) \, \mathbf{a}_{3} & = & -y_{17}a \, \mathbf{\hat{x}} + x_{17}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XIII} \\ \mathbf{B}_{68} & = & y_{17} \, \mathbf{a}_{1}-x_{17} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{17}\right) \, \mathbf{a}_{3} & = & y_{17}a \, \mathbf{\hat{x}}-x_{17}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{17}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XIII} \\ \mathbf{B}_{69} & = & x_{18} \, \mathbf{a}_{1} + y_{18} \, \mathbf{a}_{2} + z_{18} \, \mathbf{a}_{3} & = & x_{18}a \, \mathbf{\hat{x}} + y_{18}a \, \mathbf{\hat{y}} + z_{18}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XIV} \\ \mathbf{B}_{70} & = & -x_{18} \, \mathbf{a}_{1}-y_{18} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{18}\right) \, \mathbf{a}_{3} & = & -x_{18}a \, \mathbf{\hat{x}}-y_{18}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XIV} \\ \mathbf{B}_{71} & = & -y_{18} \, \mathbf{a}_{1} + x_{18} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{18}\right) \, \mathbf{a}_{3} & = & -y_{18}a \, \mathbf{\hat{x}} + x_{18}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XIV} \\ \mathbf{B}_{72} & = & y_{18} \, \mathbf{a}_{1}-x_{18} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{18}\right) \, \mathbf{a}_{3} & = & y_{18}a \, \mathbf{\hat{x}}-x_{18}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{18}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{O XIV} \\ \mathbf{B}_{73} & = & x_{19} \, \mathbf{a}_{1} + y_{19} \, \mathbf{a}_{2} + z_{19} \, \mathbf{a}_{3} & = & x_{19}a \, \mathbf{\hat{x}} + y_{19}a \, \mathbf{\hat{y}} + z_{19}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr I} \\ \mathbf{B}_{74} & = & -x_{19} \, \mathbf{a}_{1}-y_{19} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{19}\right) \, \mathbf{a}_{3} & = & -x_{19}a \, \mathbf{\hat{x}}-y_{19}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr I} \\ \mathbf{B}_{75} & = & -y_{19} \, \mathbf{a}_{1} + x_{19} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{19}\right) \, \mathbf{a}_{3} & = & -y_{19}a \, \mathbf{\hat{x}} + x_{19}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr I} \\ \mathbf{B}_{76} & = & y_{19} \, \mathbf{a}_{1}-x_{19} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{19}\right) \, \mathbf{a}_{3} & = & y_{19}a \, \mathbf{\hat{x}}-x_{19}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{19}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr I} \\ \mathbf{B}_{77} & = & x_{20} \, \mathbf{a}_{1} + y_{20} \, \mathbf{a}_{2} + z_{20} \, \mathbf{a}_{3} & = & x_{20}a \, \mathbf{\hat{x}} + y_{20}a \, \mathbf{\hat{y}} + z_{20}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr II} \\ \mathbf{B}_{78} & = & -x_{20} \, \mathbf{a}_{1}-y_{20} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{20}\right) \, \mathbf{a}_{3} & = & -x_{20}a \, \mathbf{\hat{x}}-y_{20}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{20}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr II} \\ \mathbf{B}_{79} & = & -y_{20} \, \mathbf{a}_{1} + x_{20} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{20}\right) \, \mathbf{a}_{3} & = & -y_{20}a \, \mathbf{\hat{x}} + x_{20}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{20}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr II} \\ \mathbf{B}_{80} & = & y_{20} \, \mathbf{a}_{1}-x_{20} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{20}\right) \, \mathbf{a}_{3} & = & y_{20}a \, \mathbf{\hat{x}}-x_{20}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{20}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr II} \\ \mathbf{B}_{81} & = & x_{21} \, \mathbf{a}_{1} + y_{21} \, \mathbf{a}_{2} + z_{21} \, \mathbf{a}_{3} & = & x_{21}a \, \mathbf{\hat{x}} + y_{21}a \, \mathbf{\hat{y}} + z_{21}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr III} \\ \mathbf{B}_{82} & = & -x_{21} \, \mathbf{a}_{1}-y_{21} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{21}\right) \, \mathbf{a}_{3} & = & -x_{21}a \, \mathbf{\hat{x}}-y_{21}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{21}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr III} \\ \mathbf{B}_{83} & = & -y_{21} \, \mathbf{a}_{1} + x_{21} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{21}\right) \, \mathbf{a}_{3} & = & -y_{21}a \, \mathbf{\hat{x}} + x_{21}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{21}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr III} \\ \mathbf{B}_{84} & = & y_{21} \, \mathbf{a}_{1}-x_{21} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{21}\right) \, \mathbf{a}_{3} & = & y_{21}a \, \mathbf{\hat{x}}-x_{21}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{21}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr III} \\ \mathbf{B}_{85} & = & x_{22} \, \mathbf{a}_{1} + y_{22} \, \mathbf{a}_{2} + z_{22} \, \mathbf{a}_{3} & = & x_{22}a \, \mathbf{\hat{x}} + y_{22}a \, \mathbf{\hat{y}} + z_{22}c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr IV} \\ \mathbf{B}_{86} & = & -x_{22} \, \mathbf{a}_{1}-y_{22} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{22}\right) \, \mathbf{a}_{3} & = & -x_{22}a \, \mathbf{\hat{x}}-y_{22}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{22}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr IV} \\ \mathbf{B}_{87} & = & -y_{22} \, \mathbf{a}_{1} + x_{22} \, \mathbf{a}_{2} + \left(\frac{3}{4} +z_{22}\right) \, \mathbf{a}_{3} & = & -y_{22}a \, \mathbf{\hat{x}} + x_{22}a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{22}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr IV} \\ \mathbf{B}_{88} & = & y_{22} \, \mathbf{a}_{1}-x_{22} \, \mathbf{a}_{2} + \left(\frac{1}{4} +z_{22}\right) \, \mathbf{a}_{3} & = & y_{22}a \, \mathbf{\hat{x}}-x_{22}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{22}\right)c \, \mathbf{\hat{z}} & \left(4a\right) & \mbox{Sr IV} \\ \end{array} \]