Library of Crystallographic Prototypes

Basis vectors:

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