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}+ \frac14 \, \mathbf{a}_{2}+ z_{1} \, \mathbf{a}_{3}& =&x_{1} \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}+ z_{1} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{C I} \\ \mathbf{B}_{2} & =&\left(\frac12 - x_{1}\right) \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \left(\frac12 + z_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{1}\right) \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{C I} \\ \mathbf{B}_{3} & =&- x_{1} \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}- z_{1} \, \mathbf{a}_{3}& =&- x_{1} \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}- z_{1} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{C I} \\ \mathbf{B}_{4} & =&\left(\frac12 + x_{1}\right) \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \left(\frac12 - z_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{1}\right) \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{C I} \\ \mathbf{B}_{5} & =&x_{2} \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ z_{2} \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}+ z_{2} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr I} \\ \mathbf{B}_{6} & =&\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr I} \\ \mathbf{B}_{7} & =&- x_{2} \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}- z_{2} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr I} \\ \mathbf{B}_{8} & =&\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \left(\frac12 - z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr I} \\ \mathbf{B}_{9} & =&x_{3} \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr II} \\ \mathbf{B}_{10} & =&\left(\frac12 - x_{3}\right) \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr II} \\ \mathbf{B}_{11} & =&- x_{3} \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr II} \\ \mathbf{B}_{12} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr II} \\ \mathbf{B}_{13} & =&x_{4} \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ z_{4} \, \mathbf{a}_{3}& =&x_{4} \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}+ z_{4} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr III} \\ \mathbf{B}_{14} & =&\left(\frac12 - x_{4}\right) \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \left(\frac12 + z_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{4}\right) \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr III} \\ \mathbf{B}_{15} & =&- x_{4} \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}& =&- x_{4} \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}- z_{4} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr III} \\ \mathbf{B}_{16} & =&\left(\frac12 + x_{4}\right) \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \left(\frac12 - z_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{4}\right) \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{Cr 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(8d\right) & \mbox{C II} \\ \mathbf{B}_{18} & =&\left(\frac12 - x_{5}\right) \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}+ \left(\frac12 + z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{x}}- y_{5} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{19} & =&- x_{5} \, \mathbf{a}_{1}+ \left(\frac12 + y_{5}\right) \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}& =&- x_{5} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{5}\right) \, b \, \mathbf{\hat{y}}- z_{5} \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{C II} \\ \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(8d\right) & \mbox{C II} \\ \mathbf{B}_{21} & =&- 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(8d\right) & \mbox{C II} \\ \mathbf{B}_{22} & =&\left(\frac12 + x_{5}\right) \, \mathbf{a}_{1}+ y_{5} \, \mathbf{a}_{2}+ \left(\frac12 - z_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{x}}+ y_{5} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{23} & =&x_{5} \, \mathbf{a}_{1}+ \left(\frac12 - y_{5}\right) \, \mathbf{a}_{2}+ z_{5} \, \mathbf{a}_{3}& =&x_{5} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{5}\right) \, b \, \mathbf{\hat{y}}+ z_{5} \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{C II} \\ \mathbf{B}_{24} & =&\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(8d\right) & \mbox{C II} \\ \mathbf{B}_{25} & =&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(8d\right) & \mbox{Cr IV} \\ \mathbf{B}_{26} & =&\left(\frac12 - x_{6}\right) \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}+ \left(\frac12 + z_{6}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{6}\right) \, a \, \mathbf{\hat{x}}- y_{6} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Cr IV} \\ \mathbf{B}_{27} & =&- x_{6} \, \mathbf{a}_{1}+ \left(\frac12 + y_{6}\right) \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}& =&- x_{6} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{6}\right) \, b \, \mathbf{\hat{y}}- z_{6} \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Cr IV} \\ \mathbf{B}_{28} & =&\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(8d\right) & \mbox{Cr IV} \\ \mathbf{B}_{29} & =&- 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(8d\right) & \mbox{Cr IV} \\ \mathbf{B}_{30} & =&\left(\frac12 + x_{6}\right) \, \mathbf{a}_{1}+ y_{6} \, \mathbf{a}_{2}+ \left(\frac12 - z_{6}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{6}\right) \, a \, \mathbf{\hat{x}}+ y_{6} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Cr IV} \\ \mathbf{B}_{31} & =&x_{6} \, \mathbf{a}_{1}+ \left(\frac12 - y_{6}\right) \, \mathbf{a}_{2}+ z_{6} \, \mathbf{a}_{3}& =&x_{6} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{6}\right) \, b \, \mathbf{\hat{y}}+ z_{6} \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Cr IV} \\ \mathbf{B}_{32} & =&\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(8d\right) & \mbox{Cr IV} \\ \mathbf{B}_{33} & =&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(8d\right) & \mbox{Cr V} \\ \mathbf{B}_{34} & =&\left(\frac12 - x_{7}\right) \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}+ \left(\frac12 + z_{7}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{7}\right) \, a \, \mathbf{\hat{x}}- y_{7} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Cr V} \\ \mathbf{B}_{35} & =&- x_{7} \, \mathbf{a}_{1}+ \left(\frac12 + y_{7}\right) \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}& =&- x_{7} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{7}\right) \, b \, \mathbf{\hat{y}}- z_{7} \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Cr V} \\ \mathbf{B}_{36} & =&\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(8d\right) & \mbox{Cr V} \\ \mathbf{B}_{37} & =&- 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(8d\right) & \mbox{Cr V} \\ \mathbf{B}_{38} & =&\left(\frac12 + x_{7}\right) \, \mathbf{a}_{1}+ y_{7} \, \mathbf{a}_{2}+ \left(\frac12 - z_{7}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{7}\right) \, a \, \mathbf{\hat{x}}+ y_{7} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{7}\right) \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Cr V} \\ \mathbf{B}_{39} & =&x_{7} \, \mathbf{a}_{1}+ \left(\frac12 - y_{7}\right) \, \mathbf{a}_{2}+ z_{7} \, \mathbf{a}_{3}& =&x_{7} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{7}\right) \, b \, \mathbf{\hat{y}}+ z_{7} \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Cr V} \\ \mathbf{B}_{40} & =&\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(8d\right) & \mbox{Cr V} \\ \end{array} \]