Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =&\frac78 \, \mathbf{a}_{1}+ \frac18 \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&\frac34 \, a \, \mathbf{\hat{y}}+ \frac18 \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Mn I} \\ \mathbf{B}_{2} & =&\frac18 \, \mathbf{a}_{1}+ \frac78 \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&\frac12 \, a \, \mathbf{\hat{x}}+ \frac34 \, a \, \mathbf{\hat{y}}+ \frac38 \, c \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Mn I} \\ \mathbf{B}_{3} & =&\frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{2}& =&\frac12 \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Mn II} \\ \mathbf{B}_{4} & =&\frac12 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\frac12 \, a \, \mathbf{\hat{x}}& \left(8d\right) & \mbox{Mn II} \\ \mathbf{B}_{5} & =&\frac12 \, \mathbf{a}_{1}& =&\frac34 \, a \, \mathbf{\hat{x}}+ \frac14 \, a \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Mn II} \\ \mathbf{B}_{6} & =&\frac12 \, \mathbf{a}_{1}+ \frac12 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&\frac14 \, a \, \mathbf{\hat{x}}+ \frac14 \, a \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(8d\right) & \mbox{Mn II} \\ \mathbf{B}_{7} & =&\left(y_{3} + z_{3}\right) \, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ y_{3} \, \mathbf{a}_{3}& =&y_{3} \, a \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(16h\right) & \mbox{O} \\ \mathbf{B}_{8} & =&\left(\frac12 - y_{3} + z_{3}\right)\, \mathbf{a}_{1}+ z_{3} \, \mathbf{a}_{2}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - y_{3}\right) \, a \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(16h\right) & \mbox{O} \\ \mathbf{B}_{9} & =&z_{3}\, \mathbf{a}_{1}+ \left(\frac12 - y_{3} + z_{3}\right) \, \mathbf{a}_{2}- y_{3} \, \mathbf{a}_{3}& =&\left(\frac14 - y_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac34 \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(16h\right) & \mbox{O} \\ \mathbf{B}_{10} & =&z_{3}\, \mathbf{a}_{1}+ \left(y_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + y_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac14 \, a \, \mathbf{\hat{y}}+ \left(\frac34 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(16h\right) & \mbox{O} \\ \mathbf{B}_{11} & =&\left(\frac12 + y_{3} - z_{3}\right)\, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + y_{3}\right) \, a \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(16h\right) & \mbox{O} \\ \mathbf{B}_{12} & =&- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}- y_{3} \, \mathbf{a}_{3}& =&- y_{3} \, a \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(16h\right) & \mbox{O} \\ \mathbf{B}_{13} & =&- z_{3}\, \mathbf{a}_{1}+ \left(\frac12 + y_{3} - z_{3}\right) \, \mathbf{a}_{2}+ y_{3} \, \mathbf{a}_{3}& =&\left(\frac14 + y_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac34 \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(16h\right) & \mbox{O} \\ \mathbf{B}_{14} & =&- z_{3}\, \mathbf{a}_{1}- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - y_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac14 \, a \, \mathbf{\hat{y}}+ \left(\frac34 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(16h\right) & \mbox{O} \\ \end{array} \]