Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =& \frac14 \, \mathbf{a}_{1} + \frac14 \, \mathbf{a}_{2} \, + z_{1} \, \mathbf{a}_{3}& =& \frac14 \, a \, \mathbf{\hat{x}} + \frac14 \, b \, \mathbf{\hat{y}} + z_{1} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{O I} \\ \mathbf{B}_{2} & =& \frac34 \, \mathbf{a}_{1} + \frac34 \, \mathbf{a}_{2} \, + \left(\frac12 - z_{1}\right) \, \mathbf{a}_{3}& =& \frac34 \, a \, \mathbf{\hat{x}} + \frac34 \, b \, \mathbf{\hat{y}} + \left(\frac12 - z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{O I} \\ \mathbf{B}_{3} & =& \frac34 \, \mathbf{a}_{1} + \frac34 \, \mathbf{a}_{2} \, - z_{1} \, \mathbf{a}_{3}& =& \frac34 \, a \, \mathbf{\hat{x}} + \frac34 \, b \, \mathbf{\hat{y}} - z_{1} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{O I} \\ \mathbf{B}_{4} & =& \frac14 \, \mathbf{a}_{1} + \frac14 \, \mathbf{a}_{2} \, + \left(\frac12 + z_{1}\right) \, \mathbf{a}_{3}& =& \frac14 \, a \, \mathbf{\hat{x}} + \frac14 \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{O I} \\ \mathbf{B}_{5} & =&x_{2} \, \mathbf{a}_{1}+ y_{2} \, \mathbf{a}_{2} \,+ z_{2} \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ z_{2} \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{O II} \\ \mathbf{B}_{6} & =&\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{2}\right) \, \mathbf{a}_{2}+ z_{2} \, \mathbf{a}_{3}& =&\left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{2}\right) \, b \, \mathbf{\hat{y}}+ z_{2} \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{O II} \\ \mathbf{B}_{7} & =&- x_{2} \, \mathbf{a}_{1}+ \left(\frac12 + y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{2}\right) \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{2}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{O II} \\ \mathbf{B}_{8} & =&\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+ \left(\frac12 - z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{O II} \\ \mathbf{B}_{9} & =&- x_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2} \,- z_{2} \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}- z_{2} \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{O II} \\ \mathbf{B}_{10} & =&\left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{2}\right) \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}& =&\left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{2}\right) \, b \, \mathbf{\hat{y}}- z_{2} \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{O II} \\ \mathbf{B}_{11} & =&x_{2} \, \mathbf{a}_{1}+ \left(\frac12 - y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{2}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{O II} \\ \mathbf{B}_{12} & =&\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}+ y_{2} \, \mathbf{a}_{2}+ \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{O II} \\ \mathbf{B}_{13} & =&x_{3} \, \mathbf{a}_{1}+ y_{3} \, \mathbf{a}_{2} \,+ z_{3} \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ y_{3} \, b \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{Sb} \\ \mathbf{B}_{14} & =&\left(\frac12 - x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& =&\left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{3}\right) \, b \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{Sb} \\ \mathbf{B}_{15} & =&- x_{3} \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{3}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{Sb} \\ \mathbf{B}_{16} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+ \left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{Sb} \\ \mathbf{B}_{17} & =&- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2} \,- z_{3} \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{Sb} \\ \mathbf{B}_{18} & =&\left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& =&\left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{3}\right) \, b \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{Sb} \\ \mathbf{B}_{19} & =&x_{3} \, \mathbf{a}_{1}+ \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{3}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{Sb} \\ \mathbf{B}_{20} & =&\left(\frac12 - x_{3}\right) \, \mathbf{a}_{1}+ y_{3} \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ y_{3} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(8e\right) & \mbox{Sb} \\ \end{array} \]