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