Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =&\left(y_{1} + z_{1} - x_{1}\right) \, \mathbf{a}_{1}+ \left(z_{1} + x_{1} - y_{1}\right) \, \mathbf{a}_{2}+ \left(x_{1} + y_{1} - z_{1}\right) \, \mathbf{a}_{3}& =&x_{1} \, a \, \mathbf{\hat{x}}+ y_{1} \, b \, \mathbf{\hat{y}}+ z_{1} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S I} \\ \mathbf{B}_{2} & =&\left(x_{1} - y_{1} + z_{1}\right) \, \mathbf{a}_{1}+ \left(y_{1} + z_{1} - x_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{1} - y_{1} - z_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - x_{1}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - y_{1}\right) \, b \, \mathbf{\hat{y}}+ z_{1} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S I} \\ \mathbf{B}_{3} & =&\left(x_{1} + y_{1} - z_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{1} - y_{1} - z_{1}\right) \, \mathbf{a}_{2}+ \left(y_{1} + z_{1} - x_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - x_{1}\right) \, a \, \mathbf{\hat{x}}+ y_{1} \, b \, \mathbf{\hat{y}}+ \left(\frac14 - z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S I} \\ \mathbf{B}_{4} & =&\left(\frac12 - x_{1} - y_{1} - z_{1}\right) \, \mathbf{a}_{1}+ \left(x_{1} + y_{1} - z_{1}\right) \, \mathbf{a}_{2}+ \left(x_{1} - y_{1} + z_{1}\right) \, \mathbf{a}_{3}& =&x_{1} \, a \, \mathbf{\hat{x}}+ \left(\frac14 - y_{1}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac14 - z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S I} \\ \mathbf{B}_{5} & =&\left(x_{1} - y_{1} - z_{1}\right) \, \mathbf{a}_{1}+ \left(y_{1} - z_{1} - x_{1}\right) \, \mathbf{a}_{2}+ \left(z_{1} - x_{1} - y_{1}\right) \, \mathbf{a}_{3}& =&- x_{1} \, a \, \mathbf{\hat{x}}- y_{1} \, b \, \mathbf{\hat{y}}- z_{1} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S I} \\ \mathbf{B}_{6} & =&\left(y_{1} - z_{1} - x_{1}\right) \, \mathbf{a}_{1}+ \left(x_{1} - y_{1} - z_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{1} + y_{1} + z_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + x_{1}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + y_{1}\right) \, b \, \mathbf{\hat{y}}- z_{1} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S I} \\ \mathbf{B}_{7} & =&\left(z_{1} - x_{1} - y_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{1} + y_{1} + z_{1}\right) \, \mathbf{a}_{2}+ \left(x_{1} - y_{1} - z_{1}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + x_{1}\right) \, a \, \mathbf{\hat{x}}- y_{1} \, b \, \mathbf{\hat{y}}+ \left(\frac14 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S I} \\ \mathbf{B}_{8} & =&\left(\frac12 + x_{1} + y_{1} + z_{1}\right) \, \mathbf{a}_{1}+ \left(z_{1} - x_{1} - y_{1}\right) \, \mathbf{a}_{2}+ \left(y_{1} - z_{1} - x_{1}\right) \, \mathbf{a}_{3}& =&- x_{1} \, a \, \mathbf{\hat{x}}+ \left(\frac14 + y_{1}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac14 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S I} \\ \mathbf{B}_{9} & =&\left(y_{2} + z_{2} - x_{2}\right) \, \mathbf{a}_{1}+ \left(z_{2} + x_{2} - y_{2}\right) \, \mathbf{a}_{2}+ \left(x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ z_{2} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S II} \\ \mathbf{B}_{10} & =&\left(x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{1}+ \left(y_{2} + z_{2} - x_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - y_{2}\right) \, b \, \mathbf{\hat{y}}+ z_{2} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S II} \\ \mathbf{B}_{11} & =&\left(x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{2}+ \left(y_{2} + z_{2} - x_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - x_{2}\right) \, a \, \mathbf{\hat{x}}+ y_{2} \, b \, \mathbf{\hat{y}}+ \left(\frac14 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S II} \\ \mathbf{B}_{12} & =&\left(\frac12 - x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{1}+ \left(x_{2} + y_{2} - z_{2}\right) \, \mathbf{a}_{2}+ \left(x_{2} - y_{2} + z_{2}\right) \, \mathbf{a}_{3}& =&x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac14 - y_{2}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac14 - z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S II} \\ \mathbf{B}_{13} & =&\left(x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{1}+ \left(y_{2} - z_{2} - x_{2}\right) \, \mathbf{a}_{2}+ \left(z_{2} - x_{2} - y_{2}\right) \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}- z_{2} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S II} \\ \mathbf{B}_{14} & =&\left(y_{2} - z_{2} - x_{2}\right) \, \mathbf{a}_{1}+ \left(x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + y_{2}\right) \, b \, \mathbf{\hat{y}}- z_{2} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S II} \\ \mathbf{B}_{15} & =&\left(z_{2} - x_{2} - y_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{2}+ \left(x_{2} - y_{2} - z_{2}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + x_{2}\right) \, a \, \mathbf{\hat{x}}- y_{2} \, b \, \mathbf{\hat{y}}+ \left(\frac14 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S II} \\ \mathbf{B}_{16} & =&\left(\frac12 + x_{2} + y_{2} + z_{2}\right) \, \mathbf{a}_{1}+ \left(z_{2} - x_{2} - y_{2}\right) \, \mathbf{a}_{2}+ \left(y_{2} - z_{2} - x_{2}\right) \, \mathbf{a}_{3}& =&- x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac14 + y_{2}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac14 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S II} \\ \mathbf{B}_{17} & =&\left(y_{3} + z_{3} - x_{3}\right) \, \mathbf{a}_{1}+ \left(z_{3} + x_{3} - y_{3}\right) \, \mathbf{a}_{2}+ \left(x_{3} + 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(32h\right) & \mbox{S III} \\ \mathbf{B}_{18} & =&\left(x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(y_{3} + z_{3} - x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - y_{3}\right) \, b \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S III} \\ \mathbf{B}_{19} & =&\left(x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{2}+ \left(y_{3} + z_{3} - x_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{x}}+ y_{3} \, b \, \mathbf{\hat{y}}+ \left(\frac14 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S III} \\ \mathbf{B}_{20} & =&\left(\frac12 - x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} + y_{3} - z_{3}\right) \, \mathbf{a}_{2}+ \left(x_{3} - y_{3} + z_{3}\right) \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac14 - y_{3}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac14 - z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S III} \\ \mathbf{B}_{21} & =&\left(x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{1}+ \left(y_{3} - z_{3} - x_{3}\right) \, \mathbf{a}_{2}+ \left(z_{3} - x_{3} - y_{3}\right) \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S III} \\ \mathbf{B}_{22} & =&\left(y_{3} - z_{3} - x_{3}\right) \, \mathbf{a}_{1}+ \left(x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + y_{3}\right) \, b \, \mathbf{\hat{y}}- z_{3} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S III} \\ \mathbf{B}_{23} & =&\left(z_{3} - x_{3} - y_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{2}+ \left(x_{3} - y_{3} - z_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{x}}- y_{3} \, b \, \mathbf{\hat{y}}+ \left(\frac14 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S III} \\ \mathbf{B}_{24} & =&\left(\frac12 + x_{3} + y_{3} + z_{3}\right) \, \mathbf{a}_{1}+ \left(z_{3} - x_{3} - y_{3}\right) \, \mathbf{a}_{2}+ \left(y_{3} - z_{3} - x_{3}\right) \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac14 + y_{3}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac14 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S III} \\ \mathbf{B}_{25} & =&\left(y_{4} + z_{4} - x_{4}\right) \, \mathbf{a}_{1}+ \left(z_{4} + x_{4} - y_{4}\right) \, \mathbf{a}_{2}+ \left(x_{4} + 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(32h\right) & \mbox{S IV} \\ \mathbf{B}_{26} & =&\left(x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(y_{4} + z_{4} - x_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - x_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - y_{4}\right) \, b \, \mathbf{\hat{y}}+ z_{4} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S IV} \\ \mathbf{B}_{27} & =&\left(x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{2}+ \left(y_{4} + z_{4} - x_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - x_{4}\right) \, a \, \mathbf{\hat{x}}+ y_{4} \, b \, \mathbf{\hat{y}}+ \left(\frac14 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S IV} \\ \mathbf{B}_{28} & =&\left(\frac12 - x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{1}+ \left(x_{4} + y_{4} - z_{4}\right) \, \mathbf{a}_{2}+ \left(x_{4} - y_{4} + z_{4}\right) \, \mathbf{a}_{3}& =&x_{4} \, a \, \mathbf{\hat{x}}+ \left(\frac14 - y_{4}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac14 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S IV} \\ \mathbf{B}_{29} & =&\left(x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{1}+ \left(y_{4} - z_{4} - x_{4}\right) \, \mathbf{a}_{2}+ \left(z_{4} - x_{4} - y_{4}\right) \, \mathbf{a}_{3}& =&- x_{4} \, a \, \mathbf{\hat{x}}- y_{4} \, b \, \mathbf{\hat{y}}- z_{4} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S IV} \\ \mathbf{B}_{30} & =&\left(y_{4} - z_{4} - x_{4}\right) \, \mathbf{a}_{1}+ \left(x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + x_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + y_{4}\right) \, b \, \mathbf{\hat{y}}- z_{4} \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S IV} \\ \mathbf{B}_{31} & =&\left(z_{4} - x_{4} - y_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(x_{4} - y_{4} - z_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + x_{4}\right) \, a \, \mathbf{\hat{x}}- y_{4} \, b \, \mathbf{\hat{y}}+ \left(\frac14 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S IV} \\ \mathbf{B}_{32} & =&\left(\frac12 + x_{4} + y_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(z_{4} - x_{4} - y_{4}\right) \, \mathbf{a}_{2}+ \left(y_{4} - z_{4} - x_{4}\right) \, \mathbf{a}_{3}& =&- x_{4} \, a \, \mathbf{\hat{x}}+ \left(\frac14 + y_{4}\right) \, b \, \mathbf{\hat{y}}+ \left(\frac14 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32h\right) & \mbox{S IV} \\ \end{array} \]