Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = &0 \, \mathbf{a}_{1} + 0 \, \mathbf{a}_{2} + 0 \, \mathbf{a}_{3} & = &0 \mathbf{\hat{x}} + 0 \mathbf{\hat{y}} + 0 \mathbf{\hat{z}} & \left(4a\right) & \mbox{Cr I} \\ \mathbf{B}_{2} & = &\frac14 \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& = &\frac14 \, a \, \mathbf{\hat{x}}+ \frac14 \, a \, \mathbf{\hat{y}}+ \frac14 \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Cr II} \\ \mathbf{B}_{3} & = &\frac34 \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& = &\frac34 \, a \, \mathbf{\hat{x}}+ \frac34 \, a \, \mathbf{\hat{y}}+ \frac34 \, a \, \mathbf{\hat{z}}& \left(8c\right) & \mbox{Cr II} \\ \mathbf{B}_{4} & = &- x_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& = &x_{3} \, a \, \mathbf{\hat{x}}& \left(24e\right) & \mbox{C} \\ \mathbf{B}_{5} & = &x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& = &x_{3} \, a \, \mathbf{\hat{y}}& \left(24e\right) & \mbox{C} \\ \mathbf{B}_{6} & = &x_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}& = &x_{3} \, a \, \mathbf{\hat{z}}& \left(24e\right) & \mbox{C} \\ \mathbf{B}_{7} & = &x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}& = &- x_{3} \, a \, \mathbf{\hat{x}}& \left(24e\right) & \mbox{C} \\ \mathbf{B}_{8} & = &- x_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}& = &- x_{3} \, a \, \mathbf{\hat{y}}& \left(24e\right) & \mbox{C} \\ \mathbf{B}_{9} & = &- x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& = &- x_{3} \, a \, \mathbf{\hat{z}}& \left(24e\right) & \mbox{C} \\ \mathbf{B}_{10} & = &x_{4} \, \mathbf{a}_{1}+ x_{4} \, \mathbf{a}_{2}+ x_{4} \, \mathbf{a}_{3}& = &x_{4} \, a \, \mathbf{\hat{x}}+ x_{4} \, a \, \mathbf{\hat{y}}+ x_{4} \, a \, \mathbf{\hat{z}}& \left(32f\right) & \mbox{Cr III} \\ \mathbf{B}_{11} & = &x_{4} \, \mathbf{a}_{1}+ x_{4} \, \mathbf{a}_{2}- 3 \, x_{4} \, \mathbf{a}_{3}& = &- x_{4} \, a \, \mathbf{\hat{x}}- x_{4} \, a \, \mathbf{\hat{y}}+ x_{4} \, a \, \mathbf{\hat{z}}& \left(32f\right) & \mbox{Cr III} \\ \mathbf{B}_{12} & = &x_{4} \, \mathbf{a}_{1}- 3 \, x_{4} \, \mathbf{a}_{2}+ x_{4} \, \mathbf{a}_{3}& = &- x_{4} \, a \, \mathbf{\hat{x}}+ x_{4} \, a \, \mathbf{\hat{y}}- x_{4} \, a \, \mathbf{\hat{z}}& \left(32f\right) & \mbox{Cr III} \\ \mathbf{B}_{13} & = &- 3 \, x_{4} \, \mathbf{a}_{1}+ x_{4} \, \mathbf{a}_{2}+ x_{4} \, \mathbf{a}_{3}& = &x_{4} \, a \, \mathbf{\hat{x}}- x_{4} \, a \, \mathbf{\hat{y}}- x_{4} \, a \, \mathbf{\hat{z}}& \left(32f\right) & \mbox{Cr III} \\ \mathbf{B}_{14} & = &- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}+ 3 \, x_{4} \, \mathbf{a}_{3}& = &x_{4} \, a \, \mathbf{\hat{x}}+ x_{4} \, a \, \mathbf{\hat{y}}- x_{4} \, a \, \mathbf{\hat{z}}& \left(32f\right) & \mbox{Cr III} \\ \mathbf{B}_{15} & = &- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}& = &- x_{4} \, a \, \mathbf{\hat{x}}- x_{4} \, a \, \mathbf{\hat{y}}- x_{4} \, a \, \mathbf{\hat{z}}& \left(32f\right) & \mbox{Cr III} \\ \mathbf{B}_{16} & = &- x_{4} \, \mathbf{a}_{1}+ 3 \, x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}& = &x_{4} \, a \, \mathbf{\hat{x}}- x_{4} \, a \, \mathbf{\hat{y}}+ x_{4} \, a \, \mathbf{\hat{z}}& \left(32f\right) & \mbox{Cr III} \\ \mathbf{B}_{17} & = &3 \, x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}& = &- x_{4} \, a \, \mathbf{\hat{x}}+ x_{4} \, a \, \mathbf{\hat{y}}+ x_{4} \, a \, \mathbf{\hat{z}}& \left(32f\right) & \mbox{Cr III} \\ \mathbf{B}_{18} & = &2 \, y_{5} \, \mathbf{a}_{1}& = &y_{5} \, a \, \mathbf{\hat{y}}+ y_{5} \, a \, \mathbf{\hat{z}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{19} & = &2 \, y_{5} \, \mathbf{a}_{2}- 2 \, y_{5} \, \mathbf{a}_{3}& = &- y_{5} \, a \, \mathbf{\hat{y}}+ y_{5} \, a \, \mathbf{\hat{z}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{20} & = &- 2 \, y_{5} \, \mathbf{a}_{2}+ 2 \, y_{5} \, \mathbf{a}_{3}& = &y_{5} \, a \, \mathbf{\hat{y}}- y_{5} \, a \, \mathbf{\hat{z}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{21} & = &- 2 \, y_{5} \, \mathbf{a}_{1}& = &- y_{5} \, a \, \mathbf{\hat{y}}- y_{5} \, a \, \mathbf{\hat{z}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{22} & = &2 \, y_{5} \, \mathbf{a}_{2}& = &y_{5} \, a \, \mathbf{\hat{x}}+ y_{5} \, a \, \mathbf{\hat{z}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{23} & = &- 2 \, y_{5} \, \mathbf{a}_{1}+ 2 \, y_{5} \, \mathbf{a}_{3}& = &y_{5} \, a \, \mathbf{\hat{x}}- y_{5} \, a \, \mathbf{\hat{z}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{24} & = &2 \, y_{5} \, \mathbf{a}_{1}- 2 \, y_{5} \, \mathbf{a}_{3}& = &- y_{5} \, a \, \mathbf{\hat{x}}+ y_{5} \, a \, \mathbf{\hat{z}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{25} & = &- 2 \, y_{5} \, \mathbf{a}_{2}& = &- y_{5} \, a \, \mathbf{\hat{x}}- y_{5} \, a \, \mathbf{\hat{z}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{26} & = &2 \, y_{5} \, \mathbf{a}_{3}& = &y_{5} \, a \, \mathbf{\hat{x}}+ y_{5} \, a \, \mathbf{\hat{y}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{27} & = &2 \, y_{5} \, \mathbf{a}_{1}- 2 \, y_{5} \, \mathbf{a}_{2}& = &- y_{5} \, a \, \mathbf{\hat{x}}+ y_{5} \, a \, \mathbf{\hat{y}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{28} & = &- 2 \, y_{5} \, \mathbf{a}_{1}+ 2 \, y_{5} \, \mathbf{a}_{2}& = &y_{5} \, a \, \mathbf{\hat{x}}- y_{5} \, a \, \mathbf{\hat{y}}& \left(48h\right) & \mbox{Cr IV} \\ \mathbf{B}_{29} & = &- 2 \, y_{5} \, \mathbf{a}_{3}& = &- y_{5} \, a \, \mathbf{\hat{x}}- y_{5} \, a \, \mathbf{\hat{y}}& \left(48h\right) & \mbox{Cr IV} \\ \end{array} \]