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}& =&x_{1} \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}& \left(4c\right) & \mbox{K} \\ \mathbf{B}_{2} & =&- x_{1} \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&- x_{1} \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{K} \\ \mathbf{B}_{3} & =&- x_{1} \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}& =&- x_{1} \, a \, \mathbf{\hat{x}}+ \frac34 \, b \, \mathbf{\hat{y}}& \left(4c\right) & \mbox{K} \\ \mathbf{B}_{4} & =&x_{1} \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \frac12 \, \mathbf{a}_{3}& =&x_{1} \, a \, \mathbf{\hat{x}}+ \frac14 \, b \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{K} \\ \mathbf{B}_{5} & =&x_{2} \, \mathbf{a}_{1}+ y_{2} \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{C} \\ \mathbf{B}_{6} & =&- x_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{C} \\ \mathbf{B}_{7} & =&- x_{2} \, \mathbf{a}_{1}+ \left(\frac12 + y_{2}\right) \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{2}\right) \, b \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{C} \\ \mathbf{B}_{8} & =&x_{2} \, \mathbf{a}_{1}+ \left(\frac12 - y_{2}\right) \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{2}\right) \, b \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{C} \\ \mathbf{B}_{9} & =&x_{3} \, \mathbf{a}_{1}+ y_{3} \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ y_{3} \, b \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{N} \\ \mathbf{B}_{10} & =&- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{N} \\ \mathbf{B}_{11} & =&- x_{3} \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{3}\right) \, b \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{N} \\ \mathbf{B}_{12} & =&x_{3} \, \mathbf{a}_{1}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{3}\right) \, b \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{N} \\ \mathbf{B}_{13} & =&x_{4} \, \mathbf{a}_{1}+ y_{4} \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&x_{4} \, a \, \mathbf{\hat{x}}+ y_{4} \, b \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{S} \\ \mathbf{B}_{14} & =&- x_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&- x_{4} \, a \, \mathbf{\hat{x}}- y_{4} \, b \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{S} \\ \mathbf{B}_{15} & =&- x_{4} \, \mathbf{a}_{1}+ \left(\frac12 + y_{4}\right) \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&- x_{4} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{4}\right) \, b \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{S} \\ \mathbf{B}_{16} & =&x_{4} \, \mathbf{a}_{1}+ \left(\frac12 - y_{4}\right) \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&x_{4} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{4}\right) \, b \, \mathbf{\hat{y}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(4d\right) & \mbox{S} \\ \end{array} \]