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(4a\right) & \mbox{As} \\ \mathbf{B}_{2} & =& - x_{1} \, \mathbf{a}_{1} - y_{1} \, \mathbf{a}_{2} + \left(\frac12 + z_{1}\right) \, \mathbf{a}_{3}& =& - x_{1} \, a \, \mathbf{\hat{x}} - y_{1} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{As} \\ \mathbf{B}_{3} & =& \left(\frac12 + x_{1}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{1}\right) \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{1}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{1}\right) \, b \, \mathbf{\hat{y}} + z_{1} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{As} \\ \mathbf{B}_{4} & =& \left(\frac12 - x_{1}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{1}\right) \, \mathbf{a}_{2} + \left(\frac12 + z_{1}\right) \, \mathbf{a}_{3}& =& \left(\frac12 - x_{1}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{1}\right) \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{As} \\ \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} \, b \, \mathbf{\hat{y}} + z_{2} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K I} \\ \mathbf{B}_{6} & =& - x_{2} \, \mathbf{a}_{1} - y_{2} \, \mathbf{a}_{2} + \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& =& - x_{2} \, a \, \mathbf{\hat{x}} - y_{2} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K I} \\ \mathbf{B}_{7} & =& \left(\frac12 + x_{2}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{2}\right) \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{2}\right) \, b \, \mathbf{\hat{y}} + z_{2} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K I} \\ \mathbf{B}_{8} & =& \left(\frac12 - x_{2}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{2}\right) \, \mathbf{a}_{2} + \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& =& \left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{2}\right) \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K I} \\ \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} \, b \, \mathbf{\hat{y}} + z_{3} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K II} \\ \mathbf{B}_{10} & =& - x_{3} \, \mathbf{a}_{1} - y_{3} \, \mathbf{a}_{2} + \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =& - x_{3} \, a \, \mathbf{\hat{x}} - y_{3} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K II} \\ \mathbf{B}_{11} & =& \left(\frac12 + x_{3}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{3}\right) \, b \, \mathbf{\hat{y}} + z_{3} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K II} \\ \mathbf{B}_{12} & =& \left(\frac12 - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2} + \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =& \left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{3}\right) \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K II} \\ \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} \, b \, \mathbf{\hat{y}} + z_{4} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K III} \\ \mathbf{B}_{14} & =& - x_{4} \, \mathbf{a}_{1} - y_{4} \, \mathbf{a}_{2} + \left(\frac12 + z_{4}\right) \, \mathbf{a}_{3}& =& - x_{4} \, a \, \mathbf{\hat{x}} - y_{4} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K III} \\ \mathbf{B}_{15} & =& \left(\frac12 + x_{4}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{4}\right) \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{4}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{4}\right) \, b \, \mathbf{\hat{y}} + z_{4} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K III} \\ \mathbf{B}_{16} & =& \left(\frac12 - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{4}\right) \, \mathbf{a}_{2} + \left(\frac12 + z_{4}\right) \, \mathbf{a}_{3}& =& \left(\frac12 - x_{4}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{4}\right) \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{K III} \\ \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} \, b \, \mathbf{\hat{y}} + z_{5} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S I} \\ \mathbf{B}_{18} & =& - x_{5} \, \mathbf{a}_{1} - y_{5} \, \mathbf{a}_{2} + \left(\frac12 + z_{5}\right) \, \mathbf{a}_{3}& =& - x_{5} \, a \, \mathbf{\hat{x}} - y_{5} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S I} \\ \mathbf{B}_{19} & =& \left(\frac12 + x_{5}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{5}\right) \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{5}\right) \, b \, \mathbf{\hat{y}} + z_{5} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S I} \\ \mathbf{B}_{20} & =& \left(\frac12 - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{5}\right) \, \mathbf{a}_{2} + \left(\frac12 + z_{5}\right) \, \mathbf{a}_{3}& =& \left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{5}\right) \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S 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} \, b \, \mathbf{\hat{y}} + z_{6} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S II} \\ \mathbf{B}_{22} & =& - x_{6} \, \mathbf{a}_{1} - y_{6} \, \mathbf{a}_{2} + \left(\frac12 + z_{6}\right) \, \mathbf{a}_{3}& =& - x_{6} \, a \, \mathbf{\hat{x}} - y_{6} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S II} \\ \mathbf{B}_{23} & =& \left(\frac12 + x_{6}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{6}\right) \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{6}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{6}\right) \, b \, \mathbf{\hat{y}} + z_{6} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S II} \\ \mathbf{B}_{24} & =& \left(\frac12 - x_{6}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{6}\right) \, \mathbf{a}_{2} + \left(\frac12 + z_{6}\right) \, \mathbf{a}_{3}& =& \left(\frac12 - x_{6}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{6}\right) \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S 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} \, b \, \mathbf{\hat{y}} + z_{7} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S III} \\ \mathbf{B}_{26} & =& - x_{7} \, \mathbf{a}_{1} - y_{7} \, \mathbf{a}_{2} + \left(\frac12 + z_{7}\right) \, \mathbf{a}_{3}& =& - x_{7} \, a \, \mathbf{\hat{x}} - y_{7} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S III} \\ \mathbf{B}_{27} & =& \left(\frac12 + x_{7}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{7}\right) \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{7}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{7}\right) \, b \, \mathbf{\hat{y}} + z_{7} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S III} \\ \mathbf{B}_{28} & =& \left(\frac12 - x_{7}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{7}\right) \, \mathbf{a}_{2} + \left(\frac12 + z_{7}\right) \, \mathbf{a}_{3}& =& \left(\frac12 - x_{7}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{7}\right) \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S 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} \, b \, \mathbf{\hat{y}} + z_{8} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S IV} \\ \mathbf{B}_{30} & =& - x_{8} \, \mathbf{a}_{1} - y_{8} \, \mathbf{a}_{2} + \left(\frac12 + z_{8}\right) \, \mathbf{a}_{3}& =& - x_{8} \, a \, \mathbf{\hat{x}} - y_{8} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{8}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S IV} \\ \mathbf{B}_{31} & =& \left(\frac12 + x_{8}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{8}\right) \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{8}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{8}\right) \, b \, \mathbf{\hat{y}} + z_{8} \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S IV} \\ \mathbf{B}_{32} & =& \left(\frac12 - x_{8}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{8}\right) \, \mathbf{a}_{2} + \left(\frac12 + z_{8}\right) \, \mathbf{a}_{3}& =& \left(\frac12 - x_{8}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{8}\right) \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{8}\right) \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S IV} \\ \end{array} \]