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} + x_{1} \, \mathbf{a}_{2}& =& x_{1} \, a \, \mathbf{\hat{x}}& \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{2} & =& - x_{1} \, \mathbf{a}_{1} - x_{1} \, \mathbf{a}_{2} + \frac12 \, \mathbf{a}_{3}& =& - x_{1} \, a \, \mathbf{\hat{x}} + \frac12 \, c \mathbf{\hat{z}}& \left(4a\right) & \mbox{O I} \\ \mathbf{B}_{3} & =& - y_{2} \, \mathbf{a}_{1} + y_{2} \, \mathbf{a}_{2} + \frac14 \, \mathbf{a}_{3}& =& y_{2} \, b \, \mathbf{\hat{y}} + \frac14 \, c \mathbf{\hat{z}}& \left(4b\right) & \mbox{O II} \\ \mathbf{B}_{4} & =& y_{2} \, \mathbf{a}_{1} - y_{2} \, \mathbf{a}_{2} + \frac34 \, \mathbf{a}_{3}& =& - y_{2} \, b \, \mathbf{\hat{y}} + \frac34 \, c \mathbf{\hat{z}}& \left(4b\right) & \mbox{O II} \\ \mathbf{B}_{5} & =& \left(x_{3} - y_{3}\right) \, \mathbf{a}_{1} + \left(x_{3} + y_{3}\right) \, \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(8c\right) & \mbox{O III} \\ \mathbf{B}_{6} & =& \left(y_{3} - x_{3}\right) \, \mathbf{a}_{1} - \left(x_{3} + y_{3}\right) \, \mathbf{a}_{2} +\left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& =& - x_{3} \, a \mathbf{\hat{x}} - y_{3} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{3}\right) \, c \mathbf{\hat{z}}& \left(8c\right) & \mbox{O III} \\ \mathbf{B}_{7} & =& - \left(x_{3} + y_{3}\right) \, \mathbf{a}_{1} + \left(y_{3} - x_{3}\right) \, \mathbf{a}_{2} +\left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& =& - x_{3} \, a \mathbf{\hat{x}} + y_{3} \, b \, \mathbf{\hat{y}} + \left(\frac12 - z_{3}\right) \, c \mathbf{\hat{z}}& \left(8c\right) & \mbox{O III} \\ \mathbf{B}_{8} & =& \left(x_{3} + y_{3}\right) \, \mathbf{a}_{1} + \left(x_{3} - y_{3}\right) \, \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(8c\right) & \mbox{O III} \\ \mathbf{B}_{9} & =& \left(x_{4} - y_{4}\right) \, \mathbf{a}_{1} + \left(x_{4} + y_{4}\right) \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3}& =& x_{4} \, a \mathbf{\hat{x}} + y_{4} \, b \, \mathbf{\hat{y}} + z_{4} \, c \mathbf{\hat{z}}& \left(8c\right) & \mbox{Si} \\ \mathbf{B}_{10} & =& \left(y_{4} - x_{4}\right) \, \mathbf{a}_{1} - \left(x_{4} + y_{4}\right) \, \mathbf{a}_{2} +\left(\frac12 + z_{4}\right) \, \mathbf{a}_{3}& =& - x_{4} \, a \mathbf{\hat{x}} - y_{4} \, b \, \mathbf{\hat{y}} + \left(\frac12 + z_{4}\right) \, c \mathbf{\hat{z}}& \left(8c\right) & \mbox{Si} \\ \mathbf{B}_{11} & =& - \left(x_{4} + y_{4}\right) \, \mathbf{a}_{1} + \left(y_{4} - x_{4}\right) \, \mathbf{a}_{2} +\left(\frac12 - z_{4}\right) \, \mathbf{a}_{3}& =& - x_{4} \, a \mathbf{\hat{x}} + y_{4} \, b \, \mathbf{\hat{y}} + \left(\frac12 - z_{4}\right) \, c \mathbf{\hat{z}}& \left(8c\right) & \mbox{Si} \\ \mathbf{B}_{12} & =& \left(x_{4} + y_{4}\right) \, \mathbf{a}_{1} + \left(x_{4} - y_{4}\right) \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}& =& x_{4} \, a \mathbf{\hat{x}} - y_{4} \, b \, \mathbf{\hat{y}} - z_{4} \, c \mathbf{\hat{z}}& \left(8c\right) & \mbox{Si} \\ \end{array} \]