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}+ x_{1} \, \mathbf{a}_{2}+ x_{1} \, \mathbf{a}_{3}& = &x_{1} \, a \, \mathbf{\hat{x}}+ x_{1} \, a \, \mathbf{\hat{y}}+ x_{1} \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Ni} \\ \mathbf{B}_{2} & = &\left(\frac12 - x_{1}\right) \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{3}& = &\left(\frac12 - x_{1}\right) \, a \, \mathbf{\hat{x}}- x_{1} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + x_{1}\right) \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Ni} \\ \mathbf{B}_{3} & = &- x_{1} \, \mathbf{a}_{1}+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{1}\right) \, \mathbf{a}_{3}& = &- x_{1} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{1}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac12 - x_{1}\right) \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Ni} \\ \mathbf{B}_{4} & = &+ \left(\frac12 + x_{1}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{1}\right) \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}& = &+ \left(\frac12 + x_{1}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{1}\right) \, a \, \mathbf{\hat{y}}- x_{1} \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Ni} \\ \mathbf{B}_{5} & = &x_{2} \, \mathbf{a}_{1}+ x_{2} \, \mathbf{a}_{2}+ x_{2} \, \mathbf{a}_{3}& = &x_{2} \, a \, \mathbf{\hat{x}}+ x_{2} \, a \, \mathbf{\hat{y}}+ x_{2} \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S} \\ \mathbf{B}_{6} & = &\left(\frac12 - x_{2}\right) \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{3}& = &\left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{x}}- x_{2} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S} \\ \mathbf{B}_{7} & = &- x_{2} \, \mathbf{a}_{1}+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{2}\right) \, \mathbf{a}_{3}& = &- x_{2} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S} \\ \mathbf{B}_{8} & = &+ \left(\frac12 + x_{2}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{2}\right) \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}& = &+ \left(\frac12 + x_{2}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{2}\right) \, a \, \mathbf{\hat{y}}- x_{2} \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{S} \\ \mathbf{B}_{9} & = &x_{3} \, \mathbf{a}_{1}+ x_{3} \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& = &x_{3} \, a \, \mathbf{\hat{x}}+ x_{3} \, a \, \mathbf{\hat{y}}+ x_{3} \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Sb} \\ \mathbf{B}_{10} & = &\left(\frac12 - x_{3}\right) \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{3}& = &\left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{x}}- x_{3} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Sb} \\ \mathbf{B}_{11} & = &- x_{3} \, \mathbf{a}_{1}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{3}\right) \, \mathbf{a}_{3}& = &- x_{3} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Sb} \\ \mathbf{B}_{12} & = &+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{3}\right) \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}& = &+ \left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac12 - x_{3}\right) \, a \, \mathbf{\hat{y}}- x_{3} \, a \, \mathbf{\hat{z}}& \left(4a\right) & \mbox{Sb} \\ \end{array} \]