Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B_1} & = & 0 \, \mathbf{a}_{1} + 0 \, \mathbf{a}_{2} + 0 \, \mathbf{a}_{3} & = & 0 \mathbf{\hat{x}} + 0 \mathbf{\hat{y}} + 0 \mathbf{\hat{z}} & \left(2a\right) & \mbox{N I} \\ \mathbf{B_2} & = & \frac12 \mathbf{a}_{3} & = & \frac12 \, c \, \mathbf{\hat{z}}& \left(2a\right) & \mbox{N I} \\ \mathbf{B_3} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + \frac14 \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac14 \, c \, \mathbf{\hat{z}}& \left(2c\right) & \mbox{Al} \\ \mathbf{B_4} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} + \frac34 \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac34 \, c \, \mathbf{\hat{z}}& \left(2c\right) & \mbox{Al} \\ \mathbf{B_5} & =& z_3 \, \mathbf{a}_{3}& =& z_3 \, c \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Ti I} \\ \mathbf{B_6} & =& \left(\frac12 + z_3\right) \, \mathbf{a}_{3}& =& \left(\frac12 + z_3\right) \, c \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Ti I} \\ \mathbf{B_7} & =& - z_3 \, \mathbf{a}_{3}& =& - z_3 \, c \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Ti I} \\ \mathbf{B_8} & =& \left(\frac12 - z_3\right) \, \mathbf{a}_{3}& =& \left(\frac12 - z_3\right) \, c \, \mathbf{\hat{z}}& \left(4e\right) & \mbox{Ti I} \\ \mathbf{B_9} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + z_4 \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + z_4 \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{N II} \\ \mathbf{B}_{10} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} + \left(\frac12 + z_4\right) \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_4\right) \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{N II} \\ \mathbf{B}_{11} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} - z_4 \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} - z_4 \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{N II} \\ \mathbf{B}_{12} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + \left(\frac12 - z_4\right) \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_4\right) \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{N II} \\ \mathbf{B_13} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + z_5 \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + z_5 \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Ti II} \\ \mathbf{B}_{14} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} + \left(\frac12 + z_5\right) \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_5\right) \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Ti II} \\ \mathbf{B}_{15} & =& \frac23 \mathbf{a}_{1} + \frac13 \mathbf{a}_{2} - z_5 \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} - z_5 \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Ti II} \\ \mathbf{B}_{16} & =& \frac13 \mathbf{a}_{1} + \frac23 \mathbf{a}_{2} + \left(\frac12 - z_5\right) \, \mathbf{a}_{3}& =& \frac12 \, a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_5\right) \, c \, \mathbf{\hat{z}}& \left(4f\right) & \mbox{Ti II} \\ \end{array} \]