Mayenite (12CaO·7Al2O3, $K7_{4}$, C12A7) Structure : A7B12C19_cI152_220_bc_2d_ace
| Prototype | : | Al14Ca12O33 |
| AFLOW prototype label | : | A7B12C19_cI152_220_bc_2d_ace |
| Strukturbericht designation | : | $K7_{4}$ |
| Pearson symbol | : | cI152 |
| Space group number | : | 220 |
| Space group symbol | : | $I\bar{4}3d$ |
| AFLOW prototype command | : | aflow --proto=A7B12C19_cI152_220_bc_2d_ace --params=$a$,$x_{3}$,$x_{4}$,$x_{5}$,$x_{6}$,$x_{7}$,$y_{7}$,$z_{7}$ |
- We present the structure determined by (Boysen, 2007) with data taken at 293 K. This slightly differs from the original determination of (Büssem, 1936), which was given the $K7_{4}$ designation by (Gottfried, 1938). In the original work, the calcium atoms were thought to be located at a single ($24d$) site. Newer findings show that calcium is split between two ($24d$) sites, with the site we have labeled Ca–I having 87.5% of the atoms and Ca–II the remainder, although presumably only one of the two sites is occupied in any pair.
- In all works the O–I ($12a$) site is only partially occupied: if this is occupied 1/6 of the time, we get the proper stoichiometry, though (Boysen, 2007) found the occupation was 0.251 at 293 K, dropping as the temperature decreased.
- This structure is often referred to in the literature as C12A7, to distinguish it from other CaO/Al2O3 compounds.
Body-centered Cubic primitive vectors:
\[ \begin{array}{ccc} \mathbf{a}_1 & = & - \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{y}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac12 \, a \, \mathbf{\hat{y}} + \frac12 \, a \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{y}} - \frac12 \, a \, \mathbf{\hat{z}} \\ \end{array} \]
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{5}{8} \, \mathbf{a}_{2} + \frac{3}{8} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(12a\right) & \text{O I} \\ \mathbf{B}_{2} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{3}{4}a \, \mathbf{\hat{z}} & \left(12a\right) & \text{O I} \\ \mathbf{B}_{3} & = & \frac{3}{8} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{5}{8} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} & \left(12a\right) & \text{O I} \\ \mathbf{B}_{4} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{3}{4}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} & \left(12a\right) & \text{O I} \\ \mathbf{B}_{5} & = & \frac{5}{8} \, \mathbf{a}_{1} + \frac{3}{8} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(12a\right) & \text{O I} \\ \mathbf{B}_{6} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{3}{4}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(12a\right) & \text{O I} \\ \mathbf{B}_{7} & = & \frac{1}{4} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{7}{8} \, \mathbf{a}_{3} & = & \frac{3}{8}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}}- \frac{1}{4}a \, \mathbf{\hat{z}} & \left(12b\right) & \text{Al I} \\ \mathbf{B}_{8} & = & \frac{3}{4} \, \mathbf{a}_{1} + \frac{3}{8} \, \mathbf{a}_{2} + \frac{5}{8} \, \mathbf{a}_{3} & = & \frac{1}{8}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(12b\right) & \text{Al I} \\ \mathbf{B}_{9} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \frac{1}{8} \, \mathbf{a}_{3} & = & - \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{3}{8}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(12b\right) & \text{Al I} \\ \mathbf{B}_{10} & = & \frac{5}{8} \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \frac{3}{8} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \frac{1}{8}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(12b\right) & \text{Al I} \\ \mathbf{B}_{11} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{7}{8} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{3}{8}a \, \mathbf{\hat{z}} & \left(12b\right) & \text{Al I} \\ \mathbf{B}_{12} & = & \frac{3}{8} \, \mathbf{a}_{1} + \frac{5}{8} \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{8}a \, \mathbf{\hat{z}} & \left(12b\right) & \text{Al I} \\ \mathbf{B}_{13} & = & 2x_{3} \, \mathbf{a}_{1} + 2x_{3} \, \mathbf{a}_{2} + 2x_{3} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(16c\right) & \text{Al II} \\ \mathbf{B}_{14} & = & \frac{1}{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 2x_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{y}} + x_{3}a \, \mathbf{\hat{z}} & \left(16c\right) & \text{Al II} \\ \mathbf{B}_{15} & = & \left(\frac{1}{2} - 2x_{3}\right) \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{x}} + x_{3}a \, \mathbf{\hat{y}}-x_{3}a \, \mathbf{\hat{z}} & \left(16c\right) & \text{Al II} \\ \mathbf{B}_{16} & = & \left(\frac{1}{2} - 2x_{3}\right) \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & x_{3}a \, \mathbf{\hat{x}}-x_{3}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{Al II} \\ \mathbf{B}_{17} & = & \left(\frac{1}{2} +2x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +2x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +2x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{Al II} \\ \mathbf{B}_{18} & = & \frac{1}{2} \, \mathbf{a}_{1}-2x_{3} \, \mathbf{a}_{3} & = & -\left(x_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{Al II} \\ \mathbf{B}_{19} & = & -2x_{3} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{x}}-\left(x_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{Al II} \\ \mathbf{B}_{20} & = & -2x_{3} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}}-\left(x_{3}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{Al II} \\ \mathbf{B}_{21} & = & 2x_{4} \, \mathbf{a}_{1} + 2x_{4} \, \mathbf{a}_{2} + 2x_{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(16c\right) & \text{O II} \\ \mathbf{B}_{22} & = & \frac{1}{2} \, \mathbf{a}_{1} + \left(\frac{1}{2} - 2x_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{y}} + x_{4}a \, \mathbf{\hat{z}} & \left(16c\right) & \text{O II} \\ \mathbf{B}_{23} & = & \left(\frac{1}{2} - 2x_{4}\right) \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{x}} + x_{4}a \, \mathbf{\hat{y}}-x_{4}a \, \mathbf{\hat{z}} & \left(16c\right) & \text{O II} \\ \mathbf{B}_{24} & = & \left(\frac{1}{2} - 2x_{4}\right) \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & x_{4}a \, \mathbf{\hat{x}}-x_{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{O II} \\ \mathbf{B}_{25} & = & \left(\frac{1}{2} +2x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +2x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +2x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{O II} \\ \mathbf{B}_{26} & = & \frac{1}{2} \, \mathbf{a}_{1}-2x_{4} \, \mathbf{a}_{3} & = & -\left(x_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{O II} \\ \mathbf{B}_{27} & = & -2x_{4} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} & = & \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{x}}-\left(x_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{O II} \\ \mathbf{B}_{28} & = & -2x_{4} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}}-\left(x_{4}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(16c\right) & \text{O II} \\ \mathbf{B}_{29} & = & \frac{1}{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{5}\right) \, \mathbf{a}_{2} + x_{5} \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{30} & = & \frac{3}{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{31} & = & x_{5} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + x_{5}a \, \mathbf{\hat{y}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{32} & = & \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}}-x_{5}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{33} & = & \left(\frac{1}{4} +x_{5}\right) \, \mathbf{a}_{1} + x_{5} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + x_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{34} & = & \left(\frac{1}{4} - x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{5}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}}-x_{5}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{35} & = & \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{36} & = & \left(\frac{3}{4} - x_{5}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2}-x_{5} \, \mathbf{a}_{3} & = & - \frac{1}{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{37} & = & \frac{3}{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{38} & = & \frac{1}{4} \, \mathbf{a}_{1}-x_{5} \, \mathbf{a}_{2} + \left(\frac{3}{4} - x_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}}- \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{39} & = & \left(\frac{1}{2} +x_{5}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{5}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{40} & = & -x_{5} \, \mathbf{a}_{1} + \left(\frac{3}{4} - x_{5}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca I} \\ \mathbf{B}_{41} & = & \frac{1}{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{6}\right) \, \mathbf{a}_{2} + x_{6} \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{42} & = & \frac{3}{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{43} & = & x_{6} \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{1}{4} +x_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + x_{6}a \, \mathbf{\hat{y}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{44} & = & \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{4} - x_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}}-x_{6}a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{45} & = & \left(\frac{1}{4} +x_{6}\right) \, \mathbf{a}_{1} + x_{6} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + x_{6}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{46} & = & \left(\frac{1}{4} - x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}}-x_{6}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{47} & = & \left(\frac{3}{4} +x_{6}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{48} & = & \left(\frac{3}{4} - x_{6}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2}-x_{6} \, \mathbf{a}_{3} & = & - \frac{1}{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{6}\right)a \, \mathbf{\hat{y}} + \frac{1}{2}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{49} & = & \frac{3}{4} \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +x_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{50} & = & \frac{1}{4} \, \mathbf{a}_{1}-x_{6} \, \mathbf{a}_{2} + \left(\frac{3}{4} - x_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{6}\right)a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}}- \frac{1}{4}a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{51} & = & \left(\frac{1}{2} +x_{6}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{6}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}} + \frac{1}{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{52} & = & -x_{6} \, \mathbf{a}_{1} + \left(\frac{3}{4} - x_{6}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{2}a \, \mathbf{\hat{x}}- \frac{1}{4}a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{6}\right)a \, \mathbf{\hat{z}} & \left(24d\right) & \text{Ca II} \\ \mathbf{B}_{53} & = & \left(y_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}+y_{7}\right) \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}} + y_{7}a \, \mathbf{\hat{y}} + z_{7}a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{54} & = & \left(\frac{1}{2} - y_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{7} - y_{7}\right) \, \mathbf{a}_{3} & = & -x_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{7}\right)a \, \mathbf{\hat{y}} + z_{7}a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{55} & = & \left(y_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{7} + y_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{7}\right)a \, \mathbf{\hat{x}} + y_{7}a \, \mathbf{\hat{y}}-z_{7}a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{56} & = & \left(\frac{1}{2} - y_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}-y_{7}\right) \, \mathbf{a}_{3} & = & x_{7}a \, \mathbf{\hat{x}}-y_{7}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{57} & = & \left(x_{7}+y_{7}\right) \, \mathbf{a}_{1} + \left(y_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & z_{7}a \, \mathbf{\hat{x}} + x_{7}a \, \mathbf{\hat{y}} + y_{7}a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{58} & = & \left(\frac{1}{2} - x_{7} - y_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & z_{7}a \, \mathbf{\hat{x}}-x_{7}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-y_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{59} & = & \left(\frac{1}{2} - x_{7} + y_{7}\right) \, \mathbf{a}_{1} + \left(y_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & -z_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-x_{7}\right)a \, \mathbf{\hat{y}} + y_{7}a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{60} & = & \left(x_{7}-y_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-z_{7}\right)a \, \mathbf{\hat{x}} + x_{7}a \, \mathbf{\hat{y}}-y_{7}a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{61} & = & \left(x_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}+y_{7}\right) \, \mathbf{a}_{2} + \left(y_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & y_{7}a \, \mathbf{\hat{x}} + z_{7}a \, \mathbf{\hat{y}} + x_{7}a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{62} & = & \left(-x_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{7} - y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-y_{7}\right)a \, \mathbf{\hat{x}} + z_{7}a \, \mathbf{\hat{y}}-x_{7}a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{63} & = & \left(\frac{1}{2} - x_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{7} + y_{7}\right) \, \mathbf{a}_{2} + \left(y_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & y_{7}a \, \mathbf{\hat{x}}-z_{7}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-x_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{64} & = & \left(\frac{1}{2} +x_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}-y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - y_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & -y_{7}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-z_{7}\right)a \, \mathbf{\hat{y}} + x_{7}a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{65} & = & \left(\frac{1}{2} +x_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{7} + y_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{66} & = & \left(\frac{1}{2} - x_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(-y_{7}+z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}-y_{7}\right) \, \mathbf{a}_{3} & = & -\left(y_{7}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{67} & = & \left(-x_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{7} - z_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}+y_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{x}}-\left(x_{7}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{68} & = & \left(x_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(-y_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{7} - y_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{y}}-\left(z_{7}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{69} & = & \left(\frac{1}{2} +y_{7} + z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{7} + y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{70} & = & \left(-y_{7}+z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}-y_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-x_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{y}}-\left(y_{7}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{71} & = & \left(\frac{1}{2} +y_{7} - z_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}+y_{7}\right) \, \mathbf{a}_{2} + \left(-x_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & -\left(x_{7}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-z_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{72} & = & \left(-y_{7}-z_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{7} - y_{7}\right) \, \mathbf{a}_{2} + \left(x_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{x}}-\left(z_{7}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-y_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{73} & = & \left(\frac{1}{2} +x_{7} + y_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{7} + z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{74} & = & \left(-x_{7}-y_{7}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{7} + z_{7}\right) \, \mathbf{a}_{2} + \left(-y_{7}+z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +z_{7}\right)a \, \mathbf{\hat{x}}-\left(y_{7}+\frac{1}{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-x_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{75} & = & \left(-x_{7}+y_{7}\right) \, \mathbf{a}_{1} + \left(-x_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +y_{7} - z_{7}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-z_{7}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +y_{7}\right)a \, \mathbf{\hat{y}}-\left(x_{7}+\frac{1}{4}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \mathbf{B}_{76} & = & \left(\frac{1}{2} +x_{7} - y_{7}\right) \, \mathbf{a}_{1} + \left(x_{7}-z_{7}\right) \, \mathbf{a}_{2} + \left(-y_{7}-z_{7}\right) \, \mathbf{a}_{3} & = & -\left(z_{7}+\frac{1}{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-y_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +x_{7}\right)a \, \mathbf{\hat{z}} & \left(48e\right) & \text{O III} \\ \end{array} \]References
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