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}_{3}& =& z_{1} \, c \, \mathbf{\hat{z}}& \left(2a\right) & \mbox{Ba I} \\ \mathbf{B}_{2} & =& \frac12 \, \mathbf{a}_{1} + \frac12 \, \mathbf{a}_{2} - z_{1} \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, b \, \mathbf{\hat{y}} - z_{1} \, c \, \mathbf{\hat{z}}& \left(2a\right) & \mbox{Ba I} \\ \mathbf{B}_{3} & =& \frac12 \, \mathbf{a}_{2} + z_{2} \, \mathbf{a}_{3}& =& \frac12 \, b \, \hat{y} + z_{2} \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{Ba II} \\ \mathbf{B}_{4} & =& \frac12 \, \mathbf{a}_{1} - z_{2} \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - z_{2} \, c \, \mathbf{\hat{z}}& \left(2b\right) & \mbox{Ba II} \\ \mathbf{B}_{5} & =& x_{3} \, \mathbf{a}_{1} + y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3}& =& x_{3} \, a \, \mathbf{\hat{x}} + y_{3} \, b \, \mathbf{\hat{y}} + z_{3} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S I} \\ \mathbf{B}_{6} & =& - x_{3} \, \mathbf{a}_{1} - y_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3}& =& - x_{3} \, a \, \mathbf{\hat{x}} - y_{3} \, b \, \mathbf{\hat{y}} + z_{3} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S I} \\ \mathbf{B}_{7} & =& \left(\frac12 - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{3}\right) \, \mathbf{a}_{2} - z_{3} \, \mathbf{a}_{3}& =& \left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{3}\right) \, b \, \mathbf{\hat{y}} - z_{3} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S I} \\ \mathbf{B}_{8} & =& \left(\frac12 + x_{3}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{3}\right) \, \mathbf{a}_{2} - z_{3} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{3}\right) \, b \, \mathbf{\hat{y}} - z_{3} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S I} \\ \mathbf{B}_{9} & =& x_{4} \, \mathbf{a}_{1} + y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3}& =& x_{4} \, a \, \mathbf{\hat{x}} + y_{4} \, b \, \mathbf{\hat{y}} + z_{4} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S II} \\ \mathbf{B}_{10} & =& - x_{4} \, \mathbf{a}_{1} - y_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3}& =& - x_{4} \, a \, \mathbf{\hat{x}} - y_{4} \, b \, \mathbf{\hat{y}} + z_{4} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S II} \\ \mathbf{B}_{11} & =& \left(\frac12 - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{4}\right) \, \mathbf{a}_{2} - z_{4} \, \mathbf{a}_{3}& =& \left(\frac12 - x_{4}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{4}\right) \, b \, \mathbf{\hat{y}} - z_{4} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S II} \\ \mathbf{B}_{12} & =& \left(\frac12 + x_{4}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{4}\right) \, \mathbf{a}_{2} - z_{4} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{4}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{4}\right) \, b \, \mathbf{\hat{y}} - z_{4} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S II} \\ \mathbf{B}_{13} & =& x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3}& =& x_{5} \, a \, \mathbf{\hat{x}} + y_{5} \, b \, \mathbf{\hat{y}} + z_{5} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S III} \\ \mathbf{B}_{14} & =& - x_{5} \, \mathbf{a}_{1} - y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3}& =& - x_{5} \, a \, \mathbf{\hat{x}} - y_{5} \, b \, \mathbf{\hat{y}} + z_{5} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S III} \\ \mathbf{B}_{15} & =& \left(\frac12 - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac12 + y_{5}\right) \, \mathbf{a}_{2} - z_{5} \, \mathbf{a}_{3}& =& \left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 + y_{5}\right) \, b \, \mathbf{\hat{y}} - z_{5} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S III} \\ \mathbf{B}_{16} & =& \left(\frac12 + x_{5}\right) \, \mathbf{a}_{1} + \left(\frac12 - y_{5}\right) \, \mathbf{a}_{2} - z_{5} \, \mathbf{a}_{3}& =& \left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{x}} + \left(\frac12 - y_{5}\right) \, b \, \mathbf{\hat{y}} - z_{5} \, c \, \mathbf{\hat{z}}& \left(4c\right) & \mbox{S III} \\ \end{array} \]