Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =& - y_{1} \, \mathbf{a}_{1} + y_{1} \, \mathbf{a}_{2} + \frac14 \, \mathbf{a}_{3}& = &\frac14 \, c \, \cos\beta \, \mathbf{\hat{x}}+ y_{1} \, b \, \mathbf{\hat{y}}+ \frac14 \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Pd I} \\ \mathbf{B}_{2} & =& y_{1} \, \mathbf{a}_{1} - y_{1} \, \mathbf{a}_{2} + \frac34 \, \mathbf{a}_{3}& = &\frac34 \, c \, \cos\beta \, \mathbf{\hat{x}}- y_{1} \, b \, \mathbf{\hat{y}}+ \frac34 \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Pd I} \\ \mathbf{B}_{3} & =&\left(x_{2} - y_{2}\right) \, \mathbf{a}_{1}+ \left(x_{2} + y_{2}\right) \, \mathbf{a}_{2}+ z_{2} \, \mathbf{a}_{3}& = &\left(x_{2} \, a + z_{2} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ z_{2} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{B} \\ \mathbf{B}_{4} & =&- \left(x_{2} + y_{2}\right) \, \mathbf{a}_{1}+ \left(y_{2} - x_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{2}\right) \, \mathbf{a}_{3}& = &\left( - x_{2} \, a + \left(\frac12 - z_{2}\right) \, c \, \cos\beta \right) \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{2}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{B} \\ \mathbf{B}_{5} & =&\left(y_{2} - x_{2}\right) \, \mathbf{a}_{1}- \left(x_{2} + y_{2}\right) \, \mathbf{a}_{2}- z_{2} \, \mathbf{a}_{3}& = &- \left(x_{2} \, a + z_{2} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}- z_{2} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{B} \\ \mathbf{B}_{6} & =&\left(x_{2} + y_{2}\right) \, \mathbf{a}_{1}+ \left(x_{2} - y_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& = &\left(x_{2} \, a + \left(\frac12 + z_{2}\right) \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{2}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{B} \\ \mathbf{B}_{7} & =&\left(x_{3} - y_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} + y_{3}\right) \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &\left(x_{3} \, a + z_{3} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{3} \, b \, \mathbf{\hat{y}}+ z_{3} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{Pd II} \\ \mathbf{B}_{8} & =&- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{1}+ \left(y_{3} - x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{3}\right) \, \mathbf{a}_{3}& = &\left( - x_{3} \, a + \left(\frac12 - z_{3}\right) \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{3} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{3}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{Pd II} \\ \mathbf{B}_{9} & =&\left(y_{3} - x_{3}\right) \, \mathbf{a}_{1}- \left(x_{3} + y_{3}\right) \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}& = &- \left(x_{3} \, a + z_{3} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}- z_{3} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{Pd II} \\ \mathbf{B}_{10} & =&\left(x_{3} + y_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} - y_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& = &\left(x_{3} \, a + \left(\frac12 + z_{3}\right) \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{Pd II} \\ \mathbf{B}_{11} & =&\left(x_{4} - y_{4}\right) \, \mathbf{a}_{1}+ \left(x_{4} + y_{4}\right) \, \mathbf{a}_{2}+ z_{4} \, \mathbf{a}_{3}& = &\left(x_{4} \, a + z_{4} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{4} \, b \, \mathbf{\hat{y}}+ z_{4} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{Pd III} \\ \mathbf{B}_{12} & =&- \left(x_{4} + y_{4}\right) \, \mathbf{a}_{1}+ \left(y_{4} - x_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 - z_{4}\right) \, \mathbf{a}_{3}& = &\left( - x_{4} \, a + \left(\frac12 - z_{4}\right) \, c \, \cos\beta\right) \, \mathbf{\hat{x}}+ y_{4} \, b \, \mathbf{\hat{y}}+ \left(\frac12 - z_{4}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{Pd III} \\ \mathbf{B}_{13} & =&\left(y_{4} - x_{4}\right) \, \mathbf{a}_{1}- \left(x_{4} + y_{4}\right) \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}& = &- \left(x_{4} \, a + z_{4} \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{4} \, b \, \mathbf{\hat{y}}- z_{4} \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{Pd III} \\ \mathbf{B}_{14} & =&\left(x_{4} + y_{4}\right) \, \mathbf{a}_{1}+ \left(x_{4} - y_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 + z_{4}\right) \, \mathbf{a}_{3}& = &\left(x_{4} \, a + \left(\frac12 + z_{4}\right) \, c \, \cos\beta\right) \, \mathbf{\hat{x}}- y_{4} \, b \, \mathbf{\hat{y}}+ \left(\frac12 + z_{4}\right) \, c \, \sin\beta \, \mathbf{\hat{z}}& \left(8f\right) & \mbox{Pd III} \\ \end{array} \]