Library of Crystallographic Prototypes

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =&\frac58 \, \mathbf{a}_{1}+ \frac38 \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&\frac14 \, a \, \mathbf{\hat{y}}+ \frac38 \, c \, \mathbf{\hat{z}}& \left(8a\right) & \mbox{Pr I} \\ \mathbf{B}_{2} & =&\frac38 \, \mathbf{a}_{1}+ \frac58 \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&\frac12 \, a \, \mathbf{\hat{x}}+ \frac14 \, a \, \mathbf{\hat{y}}+ \frac18 \, c \, \mathbf{\hat{z}}& \left(8a\right) & \mbox{Pr I} \\ \mathbf{B}_{3} & =&\frac78 \, \mathbf{a}_{1}+ \frac18 \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&\frac34 \, a \, \mathbf{\hat{y}}+ \frac18 \, c \, \mathbf{\hat{z}}& \left(8a\right) & \mbox{Pr I} \\ \mathbf{B}_{4} & =&\frac18 \, \mathbf{a}_{1}+ \frac78 \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&\frac12 \, a \, \mathbf{\hat{x}}+ \frac34 \, a \, \mathbf{\hat{y}}+ \frac38 \, c \, \mathbf{\hat{z}}& \left(8a\right) & \mbox{Pr I} \\ \mathbf{B}_{5} & =&\frac38 \, \mathbf{a}_{1}+ \frac18 \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&\frac14 \, a \, \mathbf{\hat{y}}+ \frac18 \, c \, \mathbf{\hat{z}}& \left(8b\right) & \mbox{Pr II} \\ \mathbf{B}_{6} & =&\frac18 \, \mathbf{a}_{1}+ \frac38 \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&\frac12 \, a \, \mathbf{\hat{x}}+ \frac14 \, a \, \mathbf{\hat{y}}+ \frac78 \, c \, \mathbf{\hat{z}}& \left(8b\right) & \mbox{Pr II} \\ \mathbf{B}_{7} & =&\frac58 \, \mathbf{a}_{1}+ \frac78 \, \mathbf{a}_{2}+ \frac34 \, \mathbf{a}_{3}& =&\frac12 \, a \, \mathbf{\hat{x}}+ \frac14 \, a \, \mathbf{\hat{y}}+ \frac38 \, c \, \mathbf{\hat{z}}& \left(8b\right) & \mbox{Pr II} \\ \mathbf{B}_{8} & =&\frac78 \, \mathbf{a}_{1}+ \frac58 \, \mathbf{a}_{2}+ \frac14 \, \mathbf{a}_{3}& =&\frac14 \, a \, \mathbf{\hat{y}}+ \frac58 \, c \, \mathbf{\hat{z}}& \left(8b\right) & \mbox{Pr II} \\ \mathbf{B}_{9} & =&\frac14 \, \mathbf{a}_{1}+ \left(\frac14 + x_{3}\right) \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& =&x_{3} \, a \, \mathbf{\hat{x}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(16e\right) & \mbox{P} \\ \mathbf{B}_{10} & =&\frac34 \, \mathbf{a}_{1}+ \left(\frac14 - x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{3}\right) \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}+ \frac12 \, a \, \mathbf{\hat{y}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(16e\right) & \mbox{P} \\ \mathbf{B}_{11} & =&\left(\frac14 + x_{3}\right) \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ x_{3} \, \mathbf{a}_{3}& =&\frac14 \, a \, \mathbf{\hat{x}}+ \left(\frac34 + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(16e\right) & \mbox{P} \\ \mathbf{B}_{12} & =&\left(\frac14 - x_{3}\right) \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}+ \left(\frac12 - x_{3}\right) \, \mathbf{a}_{3}& =&\frac14 \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{y}}& \left(16e\right) & \mbox{P} \\ \mathbf{B}_{13} & =&\frac34 \, \mathbf{a}_{1}+ \left(\frac34 - x_{3}\right) \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}& =&- x_{3} \, a \, \mathbf{\hat{x}}+ \frac34 \, c \, \mathbf{\hat{z}}& \left(16e\right) & \mbox{P} \\ \mathbf{B}_{14} & =&\frac14 \, \mathbf{a}_{1}+ \left(\frac34 + x_{3}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{3}\right) \, a \, \mathbf{\hat{x}}+ \frac14 \, c \, \mathbf{\hat{z}}& \left(16e\right) & \mbox{P} \\ \mathbf{B}_{15} & =&\left(\frac34 - x_{3}\right) \, \mathbf{a}_{1}+ \frac14 \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}& =&\frac34 \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(16e\right) & \mbox{P} \\ \mathbf{B}_{16} & =&\left(\frac34 + x_{3}\right) \, \mathbf{a}_{1}+ \frac34 \, \mathbf{a}_{2}+ \left(\frac12 + x_{3}\right) \, \mathbf{a}_{3}& =&\frac14 \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{3}\right) \, a \, \mathbf{\hat{y}}+ \frac12 \, c \, \mathbf{\hat{z}}& \left(16e\right) & \mbox{P} \\ \mathbf{B}_{17} & =&\left(y_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(x_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(x_{4} + y_{4}\right) \, \mathbf{a}_{3}& =&x_{4} \, a \, \mathbf{\hat{x}}+ y_{4} \, a \, \mathbf{\hat{y}}+ z_{4} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{18} & =&\left(\frac12 - y_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(z_{4} - x_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{4} - y_{4}\right) \, \mathbf{a}_{3}& =&- x_{4} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{4}\right) \, a \, \mathbf{\hat{y}}+ z_{4} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{19} & =&\left(x_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(x_{4} - y_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac34 + x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{20} & =&\left(z_{4} - x_{4}\right) \, \mathbf{a}_{1}+ \left(y_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{4} + y_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{21} & =&\left(y_{4} - z_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{4} - z_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{4} + y_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{4}\right) \, a \, \mathbf{\hat{x}}+ y_{4} \, a \, \mathbf{\hat{y}}- z_{4} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{22} & =&\left(\frac12 - y_{4} - z_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{4} - z_{4}\right) \, \mathbf{a}_{2}+ \left(x_{4} - y_{4}\right) \, \mathbf{a}_{3}& =&x_{4} \, a \, \mathbf{\hat{x}}- y_{4} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{23} & =&\left(\frac12 + x_{4} - z_{4}\right) \, \mathbf{a}_{1}+ \left(y_{4} - z_{4}\right) \, \mathbf{a}_{2}+ \left(x_{4} + y_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac34 + y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{24} & =&\left(\frac12 - x_{4} - z_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{4} - z_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{4} - y_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{25} & =&- \left(y_{4} + z_{4}\right) \, \mathbf{a}_{1}- \left(x_{4} + z_{4}\right) \, \mathbf{a}_{2}- \left(x_{4} + y_{4}\right) \, \mathbf{a}_{3}& =&- x_{4} \, a \, \mathbf{\hat{x}}- y_{4} \, a \, \mathbf{\hat{y}}- z_{4} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{26} & =&\left(\frac12 + y_{4} - z_{4}\right) \, \mathbf{a}_{1}+ \left(x_{4} - z_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{4} + y_{4}\right) \, \mathbf{a}_{3}& =&x_{4} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{4}\right) \, a \, \mathbf{\hat{y}}- z_{4} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{27} & =&- \left(x_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{4} - z_{4}\right) \, \mathbf{a}_{2}+ \left(y_{4} - x_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac34 - x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{28} & =&\left(x_{4} - z_{4}\right) \, \mathbf{a}_{1}- \left(y_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{4} - y_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 - z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{29} & =&\left(z_{4} - y_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{4} - y_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{4}\right) \, a \, \mathbf{\hat{x}}- y_{4} \, a \, \mathbf{\hat{y}}+ z_{4} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{30} & =&\left(\frac12 + y_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(y_{4} - x_{4}\right) \, \mathbf{a}_{3}& =&- x_{4} \, a \, \mathbf{\hat{x}}+ y_{4} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{31} & =&\left(\frac12 - x_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(z_{4} - y_{4}\right) \, \mathbf{a}_{2}- \left(x_{4} + y_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac34 - y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{32} & =&\left(\frac12 + x_{4} + z_{4}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{4} + z_{4}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{4} + y_{4}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + y_{4}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{4}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S I} \\ \mathbf{B}_{33} & =&\left(y_{5} + z_{5}\right) \, \mathbf{a}_{1}+ \left(x_{5} + z_{5}\right) \, \mathbf{a}_{2}+ \left(x_{5} + y_{5}\right) \, \mathbf{a}_{3}& =&x_{5} \, a \, \mathbf{\hat{x}}+ y_{5} \, a \, \mathbf{\hat{y}}+ z_{5} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{34} & =&\left(\frac12 - y_{5} + z_{5}\right) \, \mathbf{a}_{1}+ \left(z_{5} - x_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{5} - y_{5}\right) \, \mathbf{a}_{3}& =&- x_{5} \, a \, \mathbf{\hat{x}}+ \left(\frac12 - y_{5}\right) \, a \, \mathbf{\hat{y}}+ z_{5} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{35} & =&\left(x_{5} + z_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{5} + z_{5}\right) \, \mathbf{a}_{2}+ \left(x_{5} - y_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac34 + x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{36} & =&\left(z_{5} - x_{5}\right) \, \mathbf{a}_{1}+ \left(y_{5} + z_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{5} + y_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{37} & =&\left(y_{5} - z_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{5} - z_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{5} + y_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 - x_{5}\right) \, a \, \mathbf{\hat{x}}+ y_{5} \, a \, \mathbf{\hat{y}}- z_{5} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{38} & =&\left(\frac12 - y_{5} - z_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{5} - z_{5}\right) \, \mathbf{a}_{2}+ \left(x_{5} - y_{5}\right) \, \mathbf{a}_{3}& =&x_{5} \, a \, \mathbf{\hat{x}}- y_{5} \, a \, \mathbf{\hat{y}}+ \left(\frac12 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{39} & =&\left(\frac12 + x_{5} - z_{5}\right) \, \mathbf{a}_{1}+ \left(y_{5} - z_{5}\right) \, \mathbf{a}_{2}+ \left(x_{5} + y_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac34 + y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{40} & =&\left(\frac12 - x_{5} - z_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 - y_{5} - z_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 - x_{5} - y_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{41} & =&- \left(y_{5} + z_{5}\right) \, \mathbf{a}_{1}- \left(x_{5} + z_{5}\right) \, \mathbf{a}_{2}- \left(x_{5} + y_{5}\right) \, \mathbf{a}_{3}& =&- x_{5} \, a \, \mathbf{\hat{x}}- y_{5} \, a \, \mathbf{\hat{y}}- z_{5} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{42} & =&\left(\frac12 + y_{5} - z_{5}\right) \, \mathbf{a}_{1}+ \left(x_{5} - z_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{5} + y_{5}\right) \, \mathbf{a}_{3}& =&x_{5} \, a \, \mathbf{\hat{x}}+ \left(\frac12 + y_{5}\right) \, a \, \mathbf{\hat{y}}- z_{5} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{43} & =&- \left(x_{5} + z_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{5} - z_{5}\right) \, \mathbf{a}_{2}+ \left(y_{5} - x_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac34 - x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{44} & =&\left(x_{5} - z_{5}\right) \, \mathbf{a}_{1}- \left(y_{5} + z_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{5} - y_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac14 - y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac34 - z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{45} & =&\left(z_{5} - y_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 + x_{5} + z_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{5} - y_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac12 + x_{5}\right) \, a \, \mathbf{\hat{x}}- y_{5} \, a \, \mathbf{\hat{y}}+ z_{5} \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{46} & =&\left(\frac12 + y_{5} + z_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 - x_{5} + z_{5}\right) \, \mathbf{a}_{2}+ \left(y_{5} - x_{5}\right) \, \mathbf{a}_{3}& =&- x_{5} \, a \, \mathbf{\hat{x}}+ y_{5} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{47} & =&\left(\frac12 - x_{5} + z_{5}\right) \, \mathbf{a}_{1}+ \left(z_{5} - y_{5}\right) \, \mathbf{a}_{2}- \left(x_{5} + y_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac34 - y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 - x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \mathbf{B}_{48} & =&\left(\frac12 + x_{5} + z_{5}\right) \, \mathbf{a}_{1}+ \left(\frac12 + y_{5} + z_{5}\right) \, \mathbf{a}_{2}+ \left(\frac12 + x_{5} + y_{5}\right) \, \mathbf{a}_{3}& =&\left(\frac14 + y_{5}\right) \, a \, \mathbf{\hat{x}}+ \left(\frac14 + x_{5}\right) \, a \, \mathbf{\hat{y}}+ \left(\frac14 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(32g\right) & \mbox{S II} \\ \end{array} \]