Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = &\frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& = &\frac14 \, \, a \, \mathbf{\hat{x}}+ \frac14 \, \, a \, \mathbf{\hat{y}}+ \frac14 \, \, a \, \mathbf{\hat{z}}& \left(8a\right) & \mbox{Fe} \\ \mathbf{B}_{2} & = &\frac12 \, \mathbf{a}_{1}& = &\frac34 \, \, a \, \mathbf{\hat{x}}+ \frac14 \, \, a \, \mathbf{\hat{y}}+ \frac14 \, \, a \, \mathbf{\hat{z}}& \left(8a\right) & \mbox{Fe} \\ \mathbf{B}_{3} & = &\frac12 \, \mathbf{a}_{2}& = &\frac14 \, \, a \, \mathbf{\hat{x}}+ \frac34 \, \, a \, \mathbf{\hat{y}}+ \frac14 \, \, a \, \mathbf{\hat{z}}& \left(8a\right) & \mbox{Fe} \\ \mathbf{B}_{4} & = &\frac12 \, \mathbf{a}_{3}& = &\frac14 \, \, a \, \mathbf{\hat{x}}+ \frac14 \, \, a \, \mathbf{\hat{y}}+ \frac34 \, \, a \, \mathbf{\hat{z}}& \left(8a\right) & \mbox{Fe} \\ \mathbf{B}_{5} & = &\frac14 \, \mathbf{a}_{1}+ \left(\frac14 + x_{2}\right) \, \mathbf{a}_{2}+ x_{2} \, \mathbf{a}_{3}& = &x_{2} \, \, a \, \mathbf{\hat{x}}+ \frac14 \, \, a \, \mathbf{\hat{z}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{6} & = &\frac34 \, \mathbf{a}_{1}+ \left(\frac14 - x_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{2}\right) \, \mathbf{a}_{3}& = &- x_{2} \, \, a \, \mathbf{\hat{x}}+ \frac12 \, \, a \, \mathbf{\hat{y}}+ \frac14 \, \, a \, \mathbf{\hat{z}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{7} & = &x_{2} \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \left(\frac14 + x_{2}\right) \, \mathbf{a}_{3}& = &\frac14 \, \, a \, \mathbf{\hat{x}}+ x_{2} \, \, a \, \mathbf{\hat{y}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{8} & = &\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \left(\frac14 - x_{2}\right) \, \mathbf{a}_{3}& = &\frac14 \, \, a \, \mathbf{\hat{x}}- x_{2} \, \, a \, \mathbf{\hat{y}}+ \frac12 \, \, a \, \mathbf{\hat{z}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{9} & = &\left(\frac14 + x_{2}\right) \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& = &\frac14 \, \, a \, \mathbf{\hat{y}}+ x_{2} \, \, a \, \mathbf{\hat{z}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{10} & = &\left(\frac14 - x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{2}\right) \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& = &\frac12 \, \, a \, \mathbf{\hat{x}}\frac14 \, \, a \, \mathbf{\hat{y}}- x_{2} \, \, a \, \mathbf{\hat{z}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{11} & = &\frac34 \, \mathbf{a}_{1}+ \left(\frac34 - x_{2}\right) \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}& = &- x_{2} \, \, a \, \mathbf{\hat{x}}+ \frac34 \, \, a \, \mathbf{\hat{z}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{12} & = &\frac14 \, \mathbf{a}_{1}+ \left(\frac34 + x_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{3}& = &\left(\frac12 + x_{2}\right) \, \, a \, \mathbf{\hat{x}}+ \frac14 \, \, a \, \mathbf{\hat{z}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{13} & = &- x_{2} \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \left(\frac34 - x_{2}\right) \, \mathbf{a}_{3}& = &\frac34 \, \, a \, \mathbf{\hat{x}}- x_{2} \, \, a \, \mathbf{\hat{y}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{14} & = &\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \left(\frac34 + x_{2}\right) \, \mathbf{a}_{3}& = &\frac14 \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{2}\right) \, \, a \, \mathbf{\hat{y}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{15} & = &\left(\frac34 - x_{2}\right) \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& = &\frac34 \, \, a \, \mathbf{\hat{y}}- x_{2} \, \, a \, \mathbf{\hat{z}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{16} & = &\left(\frac34 + x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& = &\frac14 \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 + x_{2}\right) \, \, a \, \mathbf{\hat{z}}& \left(24d\right) & \mbox{Mn} \\ \mathbf{B}_{17} & = &\left(y_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(x_{3} + y_{3}\right) \, \mathbf{a}_{3}& = &x_{3} \, \, a \, \mathbf{\hat{x}}+ y_{3} \, \, a \, \mathbf{\hat{y}}+ z_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{18} & = &\left(\frac12 - y_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(z_{3} - x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{3} - y_{3}\right) \, \mathbf{a}_{3}& = &- x_{3} \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{3}\right) \, \, a \, \mathbf{\hat{y}}+ z_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{19} & = &\left(y_{3} - z_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{3} - z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{3} + y_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac12 - x_{3}\right) \, \, a \, \mathbf{\hat{x}}+ y_{3} \, \, a \, \mathbf{\hat{y}}- z_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{20} & = &\left(\frac12 - y_{3} - z_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{3} - z_{3}\right) \, \mathbf{a}_{2}+ \left(x_{3} - y_{3}\right) \, \mathbf{a}_{3}& = &x_{3} \, \, a \, \mathbf{\hat{x}}- y_{3} \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{21} & = &\left(x_{3} + y_{3}\right) \, \mathbf{a}_{1}+ \left(y_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(z_{3} + x_{3}\right) \, \mathbf{a}_{3}& = &z_{3} \, \, a \, \mathbf{\hat{x}}+ x_{3} \, \, a \, \mathbf{\hat{y}}+ y_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{22} & = &\left(\frac12 - x_{3} + y_{3}\right) \, \mathbf{a}_{1}+ \left(y_{3} - z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{3} - x_{3}\right) \, \mathbf{a}_{3}& = &- z_{3} \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{3}\right) \, \, a \, \mathbf{\hat{y}}+ y_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{23} & = &\left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - z_{3} - y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{3} + x_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac12 - z_{3}\right) \, \, a \, \mathbf{\hat{x}}+ x_{3} \, \, a \, \mathbf{\hat{y}}- y_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{24} & = &\left(\frac12 - x_{3} - y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + z_{3} - y_{3}\right) \, \mathbf{a}_{2}+ \left(z_{3} - x_{3}\right) \, \mathbf{a}_{3}& = &z_{3} \, \, a \, \mathbf{\hat{x}}- x_{3} \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 - y_{3}\right) \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{25} & = &\left(z_{3} + x_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} + y_{3}\right) \, \mathbf{a}_{2}+ \left(y_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &y_{3} \, \, a \, \mathbf{\hat{x}}+ z_{3} \, \, a \, \mathbf{\hat{y}}+ x_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{26} & = &\left(\frac12 - z_{3} + x_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - y_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &- y_{3} \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 - z_{3}\right) \, \, a \, \mathbf{\hat{y}}+ x_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{27} & = &\left(z_{3} - x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{3} - x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - y_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac12 - y_{3}\right) \, \, a \, \mathbf{\hat{x}}+ z_{3} \, \, a \, \mathbf{\hat{y}}- x_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{28} & = &\left(\frac12 - z_{3} - x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{3} - x_{3}\right) \, \mathbf{a}_{2}+ \left(y_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &y_{3} \, \, a \, \mathbf{\hat{x}}- z_{3} \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 - x_{3}\right) \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{29} & = &- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{2}- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{3}& = &- x_{3} \, \, a \, \mathbf{\hat{x}}- y_{3} \, \, a \, \mathbf{\hat{y}}- z_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{30} & = &\left(\frac12 + y_{3} - z_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} - z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{3} + y_{3}\right) \, \mathbf{a}_{3}& = &x_{3} \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{3}\right) \, \, a \, \mathbf{\hat{y}}- z_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{31} & = &\left(z_{3} - y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{3} - y_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac12 + x_{3}\right) \, \, a \, \mathbf{\hat{x}}- y_{3} \, \, a \, \mathbf{\hat{y}}+ z_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{32} & = &\left(\frac12 + y_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(y_{3} - x_{3}\right) \, \mathbf{a}_{3}& = &- x_{3} \, \, a \, \mathbf{\hat{x}}+ y_{3} \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{33} & = &- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{1}- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{2}- \left(z_{3} + x_{3}\right) \, \mathbf{a}_{3}& = &- z_{3} \, \, a \, \mathbf{\hat{x}}- x_{3} \, \, a \, \mathbf{\hat{y}}- y_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{34} & = &\left(\frac12 + x_{3} - y_{3}\right) \, \mathbf{a}_{1}+ \left(z_{3} - y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &z_{3} \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{3}\right) \, \, a \, \mathbf{\hat{y}}- y_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{35} & = &\left(y_{3} - x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{3} - x_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac12 + z_{3}\right) \, \, a \, \mathbf{\hat{x}}- x_{3} \, \, a \, \mathbf{\hat{y}}+ y_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{36} & = &\left(\frac12 + x_{3} + y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - z_{3} + y_{3}\right) \, \mathbf{a}_{2}+ \left(x_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &- z_{3} \, \, a \, \mathbf{\hat{x}}+ x_{3} \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 + y_{3}\right) \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{37} & = &- \left(x_{3} + z_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{2}- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &- y_{3} \, \, a \, \mathbf{\hat{x}}- z_{3} \, \, a \, \mathbf{\hat{y}}- x_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{38} & = &\left(\frac12 + z_{3} - x_{3}\right) \, \mathbf{a}_{1}+ \left(y_{3} - x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + y_{3} + z_{3}\right) \, \mathbf{a}_{3}& = &y_{3} \, \, a \, \mathbf{\hat{x}}+ \left(\frac12 + z_{3}\right) \, \, a \, \mathbf{\hat{y}}- x_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{39} & = &\left(x_{3} - z_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{3} + x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + y_{3} - z_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac12 + y_{3}\right) \, \, a \, \mathbf{\hat{x}}- z_{3} \, \, a \, \mathbf{\hat{y}}+ x_{3} \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \mathbf{B}_{40} & = &\left(\frac12 + x_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{3} - y_{3}\right) \, \mathbf{a}_{2}+ \left(z_{3} - y_{3}\right) \, \mathbf{a}_{3}& = &- y_{3} \, \, a \, \mathbf{\hat{x}}+ z_{3} \, \, a \, \mathbf{\hat{y}}+ \left(\frac12 + x_{3}\right) \, \, a \, \mathbf{\hat{z}}& \left(48e\right) & \mbox{O} \\ \end{array} \]