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}& = &\left(x_{1} \, a + z_{1} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{1} \, b \, \mathbf{\hat{y}}+ z_{1} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se I} \\ \mathbf{B}_{2} & = &- x_{1} \, \mathbf{a}_{1}+ \left(\frac12 + y_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{1}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 - z_{1}\right) \, c \, \cos\beta - x_{1} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 + y_{1}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{1}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se I} \\ \mathbf{B}_{3} & = &- x_{1} \, \mathbf{a}_{1}- y_{1} \, \mathbf{a}_{2}- z_{1} \, \mathbf{a}_{3}& = &- \left(x_{1} \, a + z_{1} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{1} \, b \, \mathbf{\hat{y}}- z_{1} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se I} \\ \mathbf{B}_{4} & = &x_{1} \, \mathbf{a}_{1}+ \left(\frac12 - y_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{1}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 + z_{1}\right) \, c \, \cos\beta + x_{1} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 - y_{1}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{1}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se I} \\ \mathbf{B}_{5} & = &x_{2} \, \mathbf{a}_{1}+ y_{2} \, \mathbf{a}_{2}+ z_{2} \, \mathbf{a}_{3}& = &\left(x_{2} \, a + z_{2} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ z_{2} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se II} \\ \mathbf{B}_{6} & = &- x_{2} \, \mathbf{a}_{1}+ \left(\frac12 + y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{2}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 - z_{2}\right) \, c \, \cos\beta - x_{2} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 + y_{2}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{2}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se II} \\ \mathbf{B}_{7} & = &- x_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}& = &- \left(x_{2} \, a + z_{2} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}- z_{2} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se II} \\ \mathbf{B}_{8} & = &x_{2} \, \mathbf{a}_{1}+ \left(\frac12 - y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 + z_{2}\right) \, c \, \cos\beta + x_{2} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 - y_{2}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{2}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se II} \\ \mathbf{B}_{9} & = &x_{3} \, \mathbf{a}_{1}+ y_{3} \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &\left(x_{3} \, a + z_{3} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{3} \, b \, \mathbf{\hat{y}}+ z_{3} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se III} \\ \mathbf{B}_{10} & = &- x_{3} \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 - z_{3}\right) \, c \, \cos\beta - x_{3} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 + y_{3}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se III} \\ \mathbf{B}_{11} & = &- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& = &- \left(x_{3} \, a + z_{3} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}- z_{3} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se III} \\ \mathbf{B}_{12} & = &x_{3} \, \mathbf{a}_{1}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 + z_{3}\right) \, c \, \cos\beta + x_{3} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 - y_{3}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se III} \\ \mathbf{B}_{13} & = &x_{4} \, \mathbf{a}_{1}+ y_{4} \, \mathbf{a}_{2}+ z_{4} \, \mathbf{a}_{3}& = &\left(x_{4} \, a + z_{4} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{4} \, b \, \mathbf{\hat{y}}+ z_{4} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se IV} \\ \mathbf{B}_{14} & = &- x_{4} \, \mathbf{a}_{1}+ \left(\frac12 + y_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{4}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 - z_{4}\right) \, c \, \cos\beta - x_{4} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 + y_{4}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{4}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se IV} \\ \mathbf{B}_{15} & = &- x_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}& = &- \left(x_{4} \, a + z_{4} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{4} \, b \, \mathbf{\hat{y}}- z_{4} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se IV} \\ \mathbf{B}_{16} & = &x_{4} \, \mathbf{a}_{1}+ \left(\frac12 - y_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{4}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 + z_{4}\right) \, c \, \cos\beta + x_{4} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 - y_{4}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{4}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se IV} \\ \mathbf{B}_{17} & = &x_{5} \, \mathbf{a}_{1}+ y_{5} \, \mathbf{a}_{2}+ z_{5} \, \mathbf{a}_{3}& = &\left(x_{5} \, a + z_{5} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{5} \, b \, \mathbf{\hat{y}}+ z_{5} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se V} \\ \mathbf{B}_{18} & = &- x_{5} \, \mathbf{a}_{1}+ \left(\frac12 + y_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{5}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 - z_{5}\right) \, c \, \cos\beta - x_{5} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 + y_{5}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{5}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se V} \\ \mathbf{B}_{19} & = &- x_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}& = &- \left(x_{5} \, a + z_{5} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{5} \, b \, \mathbf{\hat{y}}- z_{5} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se V} \\ \mathbf{B}_{20} & = &x_{5} \, \mathbf{a}_{1}+ \left(\frac12 - y_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{5}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 + z_{5}\right) \, c \, \cos\beta + x_{5} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 - y_{5}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{5}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se V} \\ \mathbf{B}_{21} & = &x_{6} \, \mathbf{a}_{1}+ y_{6} \, \mathbf{a}_{2}+ z_{6} \, \mathbf{a}_{3}& = &\left(x_{6} \, a + z_{6} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{6} \, b \, \mathbf{\hat{y}}+ z_{6} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VI} \\ \mathbf{B}_{22} & = &- x_{6} \, \mathbf{a}_{1}+ \left(\frac12 + y_{6}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{6}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 - z_{6}\right) \, c \, \cos\beta - x_{6} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 + y_{6}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{6}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VI} \\ \mathbf{B}_{23} & = &- x_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}& = &- \left(x_{6} \, a + z_{6} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{6} \, b \, \mathbf{\hat{y}}- z_{6} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VI} \\ \mathbf{B}_{24} & = &x_{6} \, \mathbf{a}_{1}+ \left(\frac12 - y_{6}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{6}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 + z_{6}\right) \, c \, \cos\beta + x_{6} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 - y_{6}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{6}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VI} \\ \mathbf{B}_{25} & = &x_{7} \, \mathbf{a}_{1}+ y_{7} \, \mathbf{a}_{2}+ z_{7} \, \mathbf{a}_{3}& = &\left(x_{7} \, a + z_{7} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{7} \, b \, \mathbf{\hat{y}}+ z_{7} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VII} \\ \mathbf{B}_{26} & = &- x_{7} \, \mathbf{a}_{1}+ \left(\frac12 + y_{7}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{7}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 - z_{7}\right) \, c \, \cos\beta - x_{7} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 + y_{7}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{7}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VII} \\ \mathbf{B}_{27} & = &- x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}- z_{7} \, \mathbf{a}_{3}& = &- \left(x_{7} \, a + z_{7} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{7} \, b \, \mathbf{\hat{y}}- z_{7} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VII} \\ \mathbf{B}_{28} & = &x_{7} \, \mathbf{a}_{1}+ \left(\frac12 - y_{7}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{7}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 + z_{7}\right) \, c \, \cos\beta + x_{7} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 - y_{7}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{7}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VII} \\ \mathbf{B}_{29} & = &x_{8} \, \mathbf{a}_{1}+ y_{8} \, \mathbf{a}_{2}+ z_{8} \, \mathbf{a}_{3}& = &\left(x_{8} \, a + z_{8} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{8} \, b \, \mathbf{\hat{y}}+ z_{8} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VIII} \\ \mathbf{B}_{30} & = &- x_{8} \, \mathbf{a}_{1}+ \left(\frac12 + y_{8}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{8}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 - z_{8}\right) \, c \, \cos\beta - x_{8} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 + y_{8}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{8}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VIII} \\ \mathbf{B}_{31} & = &- x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}& = &- \left(x_{8} \, a + z_{8} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{8} \, b \, \mathbf{\hat{y}}- z_{8} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VIII} \\ \mathbf{B}_{32} & = &x_{8} \, \mathbf{a}_{1}+ \left(\frac12 - y_{8}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{8}\right) \, \mathbf{a}_{3}& = &\left(\left(\frac12 + z_{8}\right) \, c \, \cos\beta + x_{8} \, a\right) \, \mathbf{\hat{x}}+ \left(\frac12 - y_{8}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{8}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Se VIII} \\ \end{array} \]