Encyclopedia of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \text{P} \\ \mathbf{B}_{2} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{3} & = & - \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \text{P} \\ \mathbf{B}_{3} & = & \frac{1}{4} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}}- \frac{1}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \text{P} \\ \mathbf{B}_{4} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \frac{1}{8}a \, \mathbf{\hat{x}}- \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(8a\right) & \text{P} \\ \mathbf{B}_{5} & = & \left(\frac{1}{4} +2y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} +y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{8} +y_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + y_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{6} & = & \frac{3}{4} \, \mathbf{a}_{1} + \left(\frac{1}{8} +y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{3}{8} - y_{2}\right) \, \mathbf{a}_{3} & = & - \frac{1}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{7} & = & \frac{3}{4} \, \mathbf{a}_{1} + \left(\frac{1}{8} - y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{3}{8} +y_{2}\right) \, \mathbf{a}_{3} & = & \frac{7}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{2}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{8} & = & \left(\frac{1}{4} - 2y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} - y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{8} - y_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}}-y_{2}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{9} & = & \left(\frac{1}{8} +y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} +2y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{3}{8} +y_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + y_{2}a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{10} & = & \left(\frac{3}{8} - y_{2}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{8} +y_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{x}}- \frac{1}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - y_{2}\right)a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{11} & = & \left(\frac{3}{8} +y_{2}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{8} - y_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{x}} + \frac{7}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{2}\right)a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{12} & = & \left(\frac{1}{8} - y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{4} - 2y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{3}{8} - y_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}}-y_{2}a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{13} & = & \left(\frac{3}{8} +y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} +y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +2y_{2}\right) \, \mathbf{a}_{3} & = & y_{2}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{14} & = & \left(\frac{1}{8} +y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} - y_{2}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{2}\right)a \, \mathbf{\hat{y}}- \frac{1}{8}a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{15} & = & \left(\frac{1}{8} - y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} +y_{2}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{2}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{y}} + \frac{7}{8}a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{16} & = & \left(\frac{3}{8} - y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} - y_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - 2y_{2}\right) \, \mathbf{a}_{3} & = & -y_{2}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(24g\right) & \text{Ca} \\ \mathbf{B}_{17} & = & \frac{1}{4} \, \mathbf{a}_{1} + \left(\frac{3}{8} - y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{8} +y_{3}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + y_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{18} & = & \left(\frac{3}{4} - 2y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} - y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{3}{8} - y_{3}\right) \, \mathbf{a}_{3} & = & - \frac{1}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} - y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{19} & = & \left(\frac{3}{4} +2y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} +y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{3}{8} +y_{3}\right) \, \mathbf{a}_{3} & = & \frac{7}{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{20} & = & \frac{1}{4} \, \mathbf{a}_{1} + \left(\frac{3}{8} +y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{8} - y_{3}\right) \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}}-y_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{21} & = & \left(\frac{1}{8} +y_{3}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{3}{8} - y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + y_{3}a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{22} & = & \left(\frac{3}{8} - y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} - 2y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{8} - y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{x}}- \frac{1}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} - y_{3}\right)a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{23} & = & \left(\frac{3}{8} +y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +2y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{8} +y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \frac{7}{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{3}\right)a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{24} & = & \left(\frac{1}{8} - y_{3}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{3}{8} +y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}}-y_{3}a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{25} & = & \left(\frac{3}{8} - y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} +y_{3}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & y_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{26} & = & \left(\frac{1}{8} - y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} - y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} - 2y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} - y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} - y_{3}\right)a \, \mathbf{\hat{y}}- \frac{1}{8}a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{27} & = & \left(\frac{1}{8} +y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} +y_{3}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +2y_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +y_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{y}} + \frac{7}{8}a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \mathbf{B}_{28} & = & \left(\frac{3}{8} +y_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} - y_{3}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & -y_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(24h\right) & \text{I} \\ \end{array} \]