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} + y_{1} \, \mathbf{a}_{2} + z_{1} \, \mathbf{a}_{3} & = & \left(x_{1}a+z_{1}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{1}b \, \mathbf{\hat{y}} + z_{1}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{As I} \\ \mathbf{B}_{2} & = & x_{1} \, \mathbf{a}_{1}-y_{1} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{1}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{1}a + z_{1}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{1}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{1}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{As I} \\ \mathbf{B}_{3} & = & x_{2} \, \mathbf{a}_{1} + y_{2} \, \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(2a\right) & \mbox{As II} \\ \mathbf{B}_{4} & = & x_{2} \, \mathbf{a}_{1}-y_{2} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{2}a + z_{2}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{2}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{2}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{As II} \\ \mathbf{B}_{5} & = & x_{3} \, \mathbf{a}_{1} + y_{3} \, \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(2a\right) & \mbox{As III} \\ \mathbf{B}_{6} & = & x_{3} \, \mathbf{a}_{1}-y_{3} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{3}a + z_{3}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{3}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{3}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{As III} \\ \mathbf{B}_{7} & = & x_{4} \, \mathbf{a}_{1} + y_{4} \, \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(2a\right) & \mbox{As IV} \\ \mathbf{B}_{8} & = & x_{4} \, \mathbf{a}_{1}-y_{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{4}a + z_{4}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{4}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{4}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{As IV} \\ \mathbf{B}_{9} & = & x_{5} \, \mathbf{a}_{1} + y_{5} \, \mathbf{a}_{2} + z_{5} \, \mathbf{a}_{3} & = & \left(x_{5}a+z_{5}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{5}b \, \mathbf{\hat{y}} + z_{5}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{As V} \\ \mathbf{B}_{10} & = & x_{5} \, \mathbf{a}_{1}-y_{5} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{5}a + z_{5}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{5}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{5}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{As V} \\ \mathbf{B}_{11} & = & x_{6} \, \mathbf{a}_{1} + y_{6} \, \mathbf{a}_{2} + z_{6} \, \mathbf{a}_{3} & = & \left(x_{6}a+z_{6}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{6}b \, \mathbf{\hat{y}} + z_{6}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{As VI} \\ \mathbf{B}_{12} & = & x_{6} \, \mathbf{a}_{1}-y_{6} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{6}a + z_{6}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{6}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{As VI} \\ \mathbf{B}_{13} & = & x_{7} \, \mathbf{a}_{1} + y_{7} \, \mathbf{a}_{2} + z_{7} \, \mathbf{a}_{3} & = & \left(x_{7}a+z_{7}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{7}b \, \mathbf{\hat{y}} + z_{7}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Ba I} \\ \mathbf{B}_{14} & = & x_{7} \, \mathbf{a}_{1}-y_{7} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{7}a + z_{7}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{7}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{7}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Ba I} \\ \mathbf{B}_{15} & = & x_{8} \, \mathbf{a}_{1} + y_{8} \, \mathbf{a}_{2} + z_{8} \, \mathbf{a}_{3} & = & \left(x_{8}a+z_{8}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{8}b \, \mathbf{\hat{y}} + z_{8}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Ba II} \\ \mathbf{B}_{16} & = & x_{8} \, \mathbf{a}_{1}-y_{8} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{8}a + z_{8}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{8}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Ba II} \\ \mathbf{B}_{17} & = & x_{9} \, \mathbf{a}_{1} + y_{9} \, \mathbf{a}_{2} + z_{9} \, \mathbf{a}_{3} & = & \left(x_{9}a+z_{9}c\cos\beta\right) \, \mathbf{\hat{x}} + y_{9}b \, \mathbf{\hat{y}} + z_{9}c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Ba III} \\ \mathbf{B}_{18} & = & x_{9} \, \mathbf{a}_{1}-y_{9} \, \mathbf{a}_{2} + \left(\frac{1}{2} +z_{9}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}c\cos\beta +x_{9}a + z_{9}c\cos\beta\right) \, \mathbf{\hat{x}}-y_{9}b \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{9}\right)c\sin\beta \, \mathbf{\hat{z}} & \left(2a\right) & \mbox{Ba III} \\ \end{array} \]