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}b \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O I} \\ \mathbf{B}_{2} & = & \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{1}\right) \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{1}\right)b \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O I} \\ \mathbf{B}_{3} & = & \left(\frac{1}{2} - x_{1}\right) \, \mathbf{a}_{1} + y_{1} \, \mathbf{a}_{2}-z_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{1}\right)a \, \mathbf{\hat{x}} + y_{1}b \, \mathbf{\hat{y}}-z_{1}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O I} \\ \mathbf{B}_{4} & = & x_{1} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{1}\right) \, \mathbf{a}_{2}-z_{1} \, \mathbf{a}_{3} & = & x_{1}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{1}\right)b \, \mathbf{\hat{y}}-z_{1}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O I} \\ \mathbf{B}_{5} & = & -x_{1} \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2}-z_{1} \, \mathbf{a}_{3} & = & -x_{1}a \, \mathbf{\hat{x}}-y_{1}b \, \mathbf{\hat{y}}-z_{1}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O I} \\ \mathbf{B}_{6} & = & \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{1}\right) \, \mathbf{a}_{2}-z_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{1}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{1}\right)b \, \mathbf{\hat{y}}-z_{1}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O I} \\ \mathbf{B}_{7} & = & \left(\frac{1}{2} +x_{1}\right) \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{1}\right)a \, \mathbf{\hat{x}}-y_{1}b \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O I} \\ \mathbf{B}_{8} & = & -x_{1} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{1}\right) \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & -x_{1}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{1}\right)b \, \mathbf{\hat{y}} + z_{1}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O I} \\ \mathbf{B}_{9} & = & x_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + y_{2}b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O II} \\ \mathbf{B}_{10} & = & \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{2}\right) \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{2}\right)b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O II} \\ \mathbf{B}_{11} & = & \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2}-z_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{2}\right)a \, \mathbf{\hat{x}} + y_{2}b \, \mathbf{\hat{y}}-z_{2}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O II} \\ \mathbf{B}_{12} & = & x_{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{2}\right) \, \mathbf{a}_{2}-z_{2} \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{2}\right)b \, \mathbf{\hat{y}}-z_{2}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O II} \\ \mathbf{B}_{13} & = & -x_{2} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2}-z_{2} \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}}-y_{2}b \, \mathbf{\hat{y}}-z_{2}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O II} \\ \mathbf{B}_{14} & = & \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{2}-z_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{2}\right)b \, \mathbf{\hat{y}}-z_{2}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O II} \\ \mathbf{B}_{15} & = & \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{2}\right)a \, \mathbf{\hat{x}}-y_{2}b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O II} \\ \mathbf{B}_{16} & = & -x_{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{2}\right) \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{2}\right)b \, \mathbf{\hat{y}} + z_{2}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O II} \\ \mathbf{B}_{17} & = & x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + y_{3}b \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O III} \\ \mathbf{B}_{18} & = & \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3}\right) \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{3}\right)b \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O III} \\ \mathbf{B}_{19} & = & \left(\frac{1}{2} - x_{3}\right) \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{3}\right)a \, \mathbf{\hat{x}} + y_{3}b \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O III} \\ \mathbf{B}_{20} & = & x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{3}\right) \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{3}\right)b \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O III} \\ \mathbf{B}_{21} & = & -x_{3} \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}}-y_{3}b \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O III} \\ \mathbf{B}_{22} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3}\right) \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{3}\right)b \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O III} \\ \mathbf{B}_{23} & = & \left(\frac{1}{2} +x_{3}\right) \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}}-y_{3}b \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O III} \\ \mathbf{B}_{24} & = & -x_{3} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{3}\right) \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{3}\right)b \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{O III} \\ \mathbf{B}_{25} & = & x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + y_{4}b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Te} \\ \mathbf{B}_{26} & = & \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4}\right) \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{4}\right)b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Te} \\ \mathbf{B}_{27} & = & \left(\frac{1}{2} - x_{4}\right) \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{4}\right)a \, \mathbf{\hat{x}} + y_{4}b \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Te} \\ \mathbf{B}_{28} & = & x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{4}\right) \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{4}\right)b \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Te} \\ \mathbf{B}_{29} & = & -x_{4} \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}}-y_{4}b \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Te} \\ \mathbf{B}_{30} & = & \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4}\right) \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{4}\right)b \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Te} \\ \mathbf{B}_{31} & = & \left(\frac{1}{2} +x_{4}\right) \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}}-y_{4}b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Te} \\ \mathbf{B}_{32} & = & -x_{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{4}\right) \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{4}\right)b \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Te} \\ \mathbf{B}_{33} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl I} \\ \mathbf{B}_{34} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{5}\right)b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl I} \\ \mathbf{B}_{35} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{5}\right)a \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl I} \\ \mathbf{B}_{36} & = & x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{5}\right)b \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl I} \\ \mathbf{B}_{37} & = & -x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl I} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)b \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl I} \\ \mathbf{B}_{39} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl I} \\ \mathbf{B}_{40} & = & -x_{5} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)b \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl I} \\ \mathbf{B}_{41} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl II} \\ \mathbf{B}_{42} & = & \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{6}\right)b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl II} \\ \mathbf{B}_{43} & = & \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - x_{6}\right)a \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl II} \\ \mathbf{B}_{44} & = & x_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6}\right) \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{6}\right)b \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl II} \\ \mathbf{B}_{45} & = & -x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl II} \\ \mathbf{B}_{46} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2}-z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)b \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl II} \\ \mathbf{B}_{47} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl II} \\ \mathbf{B}_{48} & = & -x_{6} \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)b \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(8m\right) & \mbox{Tl II} \\ \end{array} \]