Library of Crystallographic Prototypes

Basis vectors:

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