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} + \frac{1}{6} \, \mathbf{a}_{3} & = & -\sqrt{3}x_{1}a \, \mathbf{\hat{y}} + \frac{1}{6}c \, \mathbf{\hat{z}} & \left(3b\right) & \mbox{Cr I} \\ \mathbf{B}_{2} & = & x_{1} \, \mathbf{a}_{1} + 2x_{1} \, \mathbf{a}_{2} + \frac{5}{6} \, \mathbf{a}_{3} & = & \frac{3}{2}x_{1}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{1}a \, \mathbf{\hat{y}} + \frac{5}{6}c \, \mathbf{\hat{z}} & \left(3b\right) & \mbox{Cr I} \\ \mathbf{B}_{3} & = & -2x_{1} \, \mathbf{a}_{1}-x_{1} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{1}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{1}a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(3b\right) & \mbox{Cr I} \\ \mathbf{B}_{4} & = & x_{2} \, \mathbf{a}_{1}-x_{2} \, \mathbf{a}_{2} + \frac{1}{6} \, \mathbf{a}_{3} & = & -\sqrt{3}x_{2}a \, \mathbf{\hat{y}} + \frac{1}{6}c \, \mathbf{\hat{z}} & \left(3b\right) & \mbox{Cr II} \\ \mathbf{B}_{5} & = & x_{2} \, \mathbf{a}_{1} + 2x_{2} \, \mathbf{a}_{2} + \frac{5}{6} \, \mathbf{a}_{3} & = & \frac{3}{2}x_{2}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{2}a \, \mathbf{\hat{y}} + \frac{5}{6}c \, \mathbf{\hat{z}} & \left(3b\right) & \mbox{Cr II} \\ \mathbf{B}_{6} & = & -2x_{2} \, \mathbf{a}_{1}-x_{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & -\frac{3}{2}x_{2}a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{2}a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(3b\right) & \mbox{Cr II} \\ \mathbf{B}_{7} & = & x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}+y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{3}+y_{3}\right)a \, \mathbf{\hat{y}} + z_{3}c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl I} \\ \mathbf{B}_{8} & = & -y_{3} \, \mathbf{a}_{1} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{2}{3} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{3}-y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{3}a \, \mathbf{\hat{y}} + \left(\frac{2}{3} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl I} \\ \mathbf{B}_{9} & = & \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{3} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(-x_{3}+\frac{1}{2}y_{3}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{3} +z_{3}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl I} \\ \mathbf{B}_{10} & = & -y_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2} + \left(\frac{1}{3} - z_{3}\right) \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{3}+y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{3}+y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{3} - z_{3}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl I} \\ \mathbf{B}_{11} & = & \left(-x_{3}+y_{3}\right) \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + \left(\frac{2}{3} - z_{3}\right) \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{3}+y_{3}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{3}a \, \mathbf{\hat{y}} + \left(\frac{2}{3} - z_{3}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl I} \\ \mathbf{B}_{12} & = & x_{3} \, \mathbf{a}_{1} + \left(x_{3}-y_{3}\right) \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & \left(x_{3}-\frac{1}{2}y_{3}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{3}a \, \mathbf{\hat{y}}-z_{3}c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl I} \\ \mathbf{B}_{13} & = & x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{4}+y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{4}+y_{4}\right)a \, \mathbf{\hat{y}} + z_{4}c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl II} \\ \mathbf{B}_{14} & = & -y_{4} \, \mathbf{a}_{1} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{2} + \left(\frac{2}{3} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{4}-y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{4}a \, \mathbf{\hat{y}} + \left(\frac{2}{3} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl II} \\ \mathbf{B}_{15} & = & \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{3} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(-x_{4}+\frac{1}{2}y_{4}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{3} +z_{4}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl II} \\ \mathbf{B}_{16} & = & -y_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2} + \left(\frac{1}{3} - z_{4}\right) \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{4}+y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{4}+y_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{3} - z_{4}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl II} \\ \mathbf{B}_{17} & = & \left(-x_{4}+y_{4}\right) \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + \left(\frac{2}{3} - z_{4}\right) \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{4}+y_{4}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{4}a \, \mathbf{\hat{y}} + \left(\frac{2}{3} - z_{4}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl II} \\ \mathbf{B}_{18} & = & x_{4} \, \mathbf{a}_{1} + \left(x_{4}-y_{4}\right) \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & \left(x_{4}-\frac{1}{2}y_{4}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{4}a \, \mathbf{\hat{y}}-z_{4}c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl II} \\ \mathbf{B}_{19} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{5}+y_{5}\right)a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl III} \\ \mathbf{B}_{20} & = & -y_{5} \, \mathbf{a}_{1} + \left(x_{5}-y_{5}\right) \, \mathbf{a}_{2} + \left(\frac{2}{3} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}x_{5}-y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{5}a \, \mathbf{\hat{y}} + \left(\frac{2}{3} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl III} \\ \mathbf{B}_{21} & = & \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \left(\frac{1}{3} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(-x_{5}+\frac{1}{2}y_{5}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{5}a \, \mathbf{\hat{y}} + \left(\frac{1}{3} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl III} \\ \mathbf{B}_{22} & = & -y_{5} \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \left(\frac{1}{3} - z_{5}\right) \, \mathbf{a}_{3} & = & -\frac{1}{2}\left(x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}\left(-x_{5}+y_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{3} - z_{5}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl III} \\ \mathbf{B}_{23} & = & \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + \left(\frac{2}{3} - z_{5}\right) \, \mathbf{a}_{3} & = & \left(-\frac{1}{2}x_{5}+y_{5}\right)a \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2}x_{5}a \, \mathbf{\hat{y}} + \left(\frac{2}{3} - z_{5}\right)c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl III} \\ \mathbf{B}_{24} & = & x_{5} \, \mathbf{a}_{1} + \left(x_{5}-y_{5}\right) \, \mathbf{a}_{2}-z_{5} \, \mathbf{a}_{3} & = & \left(x_{5}-\frac{1}{2}y_{5}\right)a \, \mathbf{\hat{x}}-\frac{\sqrt{3}}{2}y_{5}a \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(6c\right) & \mbox{Cl III} \\ \end{array} \]