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