Analcime (NaAlSi2O6·H2O, $S6_{1}$) Structure : A2B2C3D12E4_tI184_142_f_f_be_3g_g

Picture of Structure; Click for Big Picture
Prototype : Al(H2O)NaO6Si2
AFLOW prototype label : A2B2C3D12E4_tI184_142_f_f_be_3g_g
Strukturbericht designation : $S6_{1}$
Pearson symbol : tI184
Space group number : 142
Space group symbol : $I4_{1}/acd$
AFLOW prototype command : aflow --proto=A2B2C3D12E4_tI184_142_f_f_be_3g_g
--params=
$a$,$c/a$,$x_{2}$,$y_{2}$,$x_{3}$,$x_{4}$,$x_{5}$,$y_{5}$,$z_{5}$,$x_{6}$,$y_{6}$,$z_{6}$,$x_{7}$,$y_{7}$,$z_{7}$,$x_{8}$,$y_{8}$,$z_{8}$


  • Analcime is pseudo–cubic, space group $Ia\overline{3}d$ #230, but (Hartwig, 1931) showed that the structure actually is tetragonal, space group $I4_{1}/acd$ #142. (Hermann, 1937) listed both structures in defining Strukturbericht type $S6_{1}$.
  • Analcime can have many crystal structures, as found in e.g., (Mazzi, 1978) and (Pechar, 1988). We take our prototype from (Mazzi, 1978), using the data from their sample ANA 1, which is in space group $I4_{1}/acd$ #142. The major difference between this structure and that found by (Hartwig, 1931) is that sodium atoms were found at the ($8b$) Wyckoff position (Na–I). (Mazzi, 1978) recover the stoichiometry by noting that site Na–I ($8b$) is only occupied 23% of the time, while site Na–II ($16e$) has 82% occupancy. In addition, our Si site ($32e$) is actually 47% aluminum and 53% silicon, and the site which we, following (Hartwig, 1931), label Al ($16f$) is actually 98% silicon and only 2% aluminum. These fractions are highly dependent upon the choice of sample.
  • Removing all sodium from the ($8b$) site results in the original structure found by (Hartwig, 1931) and defined as $S6_{1}$ by (Hermann, 1937).
  • The positions of the hydrogen atoms in the water molecules were not determined, so we only provide the positions of the oxygen atoms (labeled as H2O).

Body-centered Tetragonal primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & - \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{y}} + \frac12 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac12 \, a \, \mathbf{\hat{y}} + \frac12 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} + \frac12 \, a \, \mathbf{\hat{y}} - \frac12 \, c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & \frac{3}{8} \, \mathbf{a}_{1} + \frac{1}{8} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{1}{8}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Na I} \\ \mathbf{B}_{2} & = & \frac{1}{8} \, \mathbf{a}_{1} + \frac{3}{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}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Na I} \\ \mathbf{B}_{3} & = & \frac{5}{8} \, \mathbf{a}_{1} + \frac{7}{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{3}{8}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Na I} \\ \mathbf{B}_{4} & = & \frac{7}{8} \, \mathbf{a}_{1} + \frac{5}{8} \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{y}} + \frac{5}{8}c \, \mathbf{\hat{z}} & \left(8b\right) & \mbox{Na I} \\ \mathbf{B}_{5} & = & \frac{1}{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} +x_{2}\right) \, \mathbf{a}_{2} + x_{2} \, \mathbf{a}_{3} & = & x_{2}a \, \mathbf{\hat{x}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(16e\right) & \mbox{Na II} \\ \mathbf{B}_{6} & = & \frac{3}{4} \, \mathbf{a}_{1} + \left(\frac{1}{4} - x_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}} + \frac{1}{2}a \, \mathbf{\hat{y}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(16e\right) & \mbox{Na II} \\ \mathbf{B}_{7} & = & \left(\frac{1}{4} +x_{2}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + x_{2} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(16e\right) & \mbox{Na II} \\ \mathbf{B}_{8} & = & \left(\frac{1}{4} - x_{2}\right) \, \mathbf{a}_{1} + \frac{1}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{2}\right)a \, \mathbf{\hat{y}} & \left(16e\right) & \mbox{Na II} \\ \mathbf{B}_{9} & = & \frac{3}{4} \, \mathbf{a}_{1} + \left(\frac{3}{4} - x_{2}\right) \, \mathbf{a}_{2}-x_{2} \, \mathbf{a}_{3} & = & -x_{2}a \, \mathbf{\hat{x}} + \frac{3}{4}c \, \mathbf{\hat{z}} & \left(16e\right) & \mbox{Na II} \\ \mathbf{B}_{10} & = & \frac{1}{4} \, \mathbf{a}_{1} + \left(\frac{3}{4} +x_{2}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{2}\right)a \, \mathbf{\hat{x}} + \frac{1}{4}c \, \mathbf{\hat{z}} & \left(16e\right) & \mbox{Na II} \\ \mathbf{B}_{11} & = & \left(\frac{1}{4} - x_{2} + y_{2}\right) \, \mathbf{a}_{1} + \left(\frac{3}{4} +y_{2}\right) \, \mathbf{a}_{2}-x_{2} \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}}-a\left(x_{2}+\frac{1}{4}\right) \, \mathbf{\hat{y}} + \left(\frac{1}{2} +y_{2}\right)c \, \mathbf{\hat{z}} & \left(16e\right) & \mbox{Na II} \\ \mathbf{B}_{12} & = & \left(\frac{3}{4} +x_{2}\right) \, \mathbf{a}_{1} + \frac{3}{4} \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{2}\right) \, \mathbf{a}_{3} & = & \frac{1}{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{2}\right)a \, \mathbf{\hat{y}} + \frac{1}{2}c \, \mathbf{\hat{z}} & \left(16e\right) & \mbox{Na II} \\ \mathbf{B}_{13} & = & \left(\frac{3}{8} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} +x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +2x_{3}\right) \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Al} \\ \mathbf{B}_{14} & = & \left(\frac{3}{8} - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} - x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - 2x_{3}\right) \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Al} \\ \mathbf{B}_{15} & = & \left(\frac{1}{8} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} - x_{3}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}}- \frac{1}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Al} \\ \mathbf{B}_{16} & = & \left(\frac{1}{8} - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} +x_{3}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \frac{7}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Al} \\ \mathbf{B}_{17} & = & \left(\frac{5}{8} - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{7}{8} - x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} - 2x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \frac{3}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Al} \\ \mathbf{B}_{18} & = & \left(\frac{5}{8} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{7}{8} +x_{3}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +2x_{3}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{3}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \frac{3}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Al} \\ \mathbf{B}_{19} & = & \left(\frac{7}{8} - x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{5}{8} +x_{3}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{3}\right)a \, \mathbf{\hat{y}} + \frac{5}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Al} \\ \mathbf{B}_{20} & = & \left(\frac{7}{8} +x_{3}\right) \, \mathbf{a}_{1} + \left(\frac{5}{8} - x_{3}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & -x_{3}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{3}\right)a \, \mathbf{\hat{y}} + \frac{5}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{Al} \\ \mathbf{B}_{21} & = & \left(\frac{3}{8} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} +x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} +2x_{4}\right) \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{H$_{2}$O} \\ \mathbf{B}_{22} & = & \left(\frac{3}{8} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{1}{8} - x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{1}{4} - 2x_{4}\right) \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \frac{1}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{H$_{2}$O} \\ \mathbf{B}_{23} & = & \left(\frac{1}{8} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} - x_{4}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}}- \frac{1}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{H$_{2}$O} \\ \mathbf{B}_{24} & = & \left(\frac{1}{8} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{3}{8} +x_{4}\right) \, \mathbf{a}_{2} + \frac{3}{4} \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \frac{7}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{H$_{2}$O} \\ \mathbf{B}_{25} & = & \left(\frac{5}{8} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{7}{8} - x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} - 2x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \frac{3}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{H$_{2}$O} \\ \mathbf{B}_{26} & = & \left(\frac{5}{8} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{7}{8} +x_{4}\right) \, \mathbf{a}_{2} + \left(\frac{3}{4} +2x_{4}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{4}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \frac{3}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{H$_{2}$O} \\ \mathbf{B}_{27} & = & \left(\frac{7}{8} - x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{5}{8} +x_{4}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{4}\right)a \, \mathbf{\hat{y}} + \frac{5}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{H$_{2}$O} \\ \mathbf{B}_{28} & = & \left(\frac{7}{8} +x_{4}\right) \, \mathbf{a}_{1} + \left(\frac{5}{8} - x_{4}\right) \, \mathbf{a}_{2} + \frac{1}{4} \, \mathbf{a}_{3} & = & -x_{4}a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{4}\right)a \, \mathbf{\hat{y}} + \frac{5}{8}c \, \mathbf{\hat{z}} & \left(16f\right) & \mbox{H$_{2}$O} \\ \mathbf{B}_{29} & = & \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{O I} \\ \mathbf{B}_{30} & = & \left(\frac{1}{2} - y_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(-x_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5} - y_{5}\right) \, \mathbf{a}_{3} & = & -x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{5}\right)a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{31} & = & \left(x_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(x_{5}-y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{32} & = & \left(-x_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(y_{5}+z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5} + y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{33} & = & \left(y_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5} + y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{5}\right)a \, \mathbf{\hat{x}} + y_{5}a \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{34} & = & \left(\frac{1}{2} - y_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +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(\frac{1}{2}-z_{5}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{35} & = & \left(\frac{1}{2} +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(- \frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{5}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{36} & = & \left(\frac{1}{2} - x_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{5} - y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{5}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{37} & = & \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{O I} \\ \mathbf{B}_{38} & = & \left(\frac{1}{2} +y_{5} - z_{5}\right) \, \mathbf{a}_{1} + \left(x_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5} + y_{5}\right) \, \mathbf{a}_{3} & = & x_{5}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{5}\right)a \, \mathbf{\hat{y}}-z_{5}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{39} & = & \left(-x_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5} - z_{5}\right) \, \mathbf{a}_{2} + \left(-x_{5}+y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{x}}-a\left(x_{5}+\frac{1}{4}\right) \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{5}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{40} & = & \left(x_{5}-z_{5}\right) \, \mathbf{a}_{1} + \left(-y_{5}-z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5} - y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}}-c\left(z_{5}+\frac{1}{4}\right) \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{41} & = & \left(-y_{5}+z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5} - y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{5}\right)a \, \mathbf{\hat{x}}-y_{5}a \, \mathbf{\hat{y}} + z_{5}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{42} & = & \left(\frac{1}{2} +y_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - 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(\frac{1}{2} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{43} & = & \left(\frac{1}{2} - 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(\frac{3}{4}-y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{44} & = & \left(\frac{1}{2} +x_{5} + z_{5}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{5} + z_{5}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{5} + y_{5}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{5}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{5}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{5}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O I} \\ \mathbf{B}_{45} & = & \left(y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}+y_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{46} & = & \left(\frac{1}{2} - y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6} - y_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{6}\right)a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{47} & = & \left(x_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}-y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{48} & = & \left(-x_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6} + y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{49} & = & \left(y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6} + y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{6}\right)a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{50} & = & \left(\frac{1}{2} - y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}-y_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{51} & = & \left(\frac{1}{2} +x_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(x_{6}+y_{6}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{52} & = & \left(\frac{1}{2} - x_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{6} - y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{53} & = & \left(-y_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(-x_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}-y_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{54} & = & \left(\frac{1}{2} +y_{6} - z_{6}\right) \, \mathbf{a}_{1} + \left(x_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6} + y_{6}\right) \, \mathbf{a}_{3} & = & x_{6}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{6}\right)a \, \mathbf{\hat{y}}-z_{6}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{55} & = & \left(-x_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6} - z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}+y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{x}}-a\left(x_{6}+\frac{1}{4}\right) \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{6}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{56} & = & \left(x_{6}-z_{6}\right) \, \mathbf{a}_{1} + \left(-y_{6}-z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6} - y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{y}}-c\left(z_{6}+\frac{1}{4}\right) \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{57} & = & \left(-y_{6}+z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6} - y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{6}\right)a \, \mathbf{\hat{x}}-y_{6}a \, \mathbf{\hat{y}} + z_{6}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{58} & = & \left(\frac{1}{2} +y_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}+y_{6}\right) \, \mathbf{a}_{3} & = & -x_{6}a \, \mathbf{\hat{x}} + y_{6}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{59} & = & \left(\frac{1}{2} - x_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(-y_{6}+z_{6}\right) \, \mathbf{a}_{2} + \left(-x_{6}-y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{60} & = & \left(\frac{1}{2} +x_{6} + z_{6}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{6} + z_{6}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{6} + y_{6}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{6}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{6}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{6}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O II} \\ \mathbf{B}_{61} & = & \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}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{62} & = & \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}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{63} & = & \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(\frac{3}{4} +x_{7}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{64} & = & \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(\frac{3}{4} +z_{7}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{65} & = & \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}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{66} & = & \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)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{67} & = & \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(- \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)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{68} & = & \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)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{69} & = & \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}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{70} & = & \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}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{71} & = & \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}}-a\left(x_{7}+\frac{1}{4}\right) \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{7}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{72} & = & \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}}-c\left(z_{7}+\frac{1}{4}\right) \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{73} & = & \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}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{74} & = & \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)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{75} & = & \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(\frac{3}{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)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{76} & = & \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)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{O III} \\ \mathbf{B}_{77} & = & \left(y_{8}+z_{8}\right) \, \mathbf{a}_{1} + \left(x_{8}+z_{8}\right) \, \mathbf{a}_{2} + \left(x_{8}+y_{8}\right) \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + y_{8}a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{78} & = & \left(\frac{1}{2} - y_{8} + z_{8}\right) \, \mathbf{a}_{1} + \left(-x_{8}+z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{8} - y_{8}\right) \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2}-y_{8}\right)a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{79} & = & \left(x_{8}+z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{8} + z_{8}\right) \, \mathbf{a}_{2} + \left(x_{8}-y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{3}{4} +x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{80} & = & \left(-x_{8}+z_{8}\right) \, \mathbf{a}_{1} + \left(y_{8}+z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{8} + y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{3}{4} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{81} & = & \left(y_{8}-z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{8} - z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{8} + y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2}-x_{8}\right)a \, \mathbf{\hat{x}} + y_{8}a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{82} & = & \left(\frac{1}{2} - y_{8} - z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{8} - z_{8}\right) \, \mathbf{a}_{2} + \left(x_{8}-y_{8}\right) \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}}-y_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{83} & = & \left(\frac{1}{2} +x_{8} - z_{8}\right) \, \mathbf{a}_{1} + \left(y_{8}-z_{8}\right) \, \mathbf{a}_{2} + \left(x_{8}+y_{8}\right) \, \mathbf{a}_{3} & = & \left(- \frac{1}{4} +y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{84} & = & \left(\frac{1}{2} - x_{8} - z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - y_{8} - z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} - x_{8} - y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{85} & = & \left(-y_{8}-z_{8}\right) \, \mathbf{a}_{1} + \left(-x_{8}-z_{8}\right) \, \mathbf{a}_{2} + \left(-x_{8}-y_{8}\right) \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}}-y_{8}a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{86} & = & \left(\frac{1}{2} +y_{8} - z_{8}\right) \, \mathbf{a}_{1} + \left(x_{8}-z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{8} + y_{8}\right) \, \mathbf{a}_{3} & = & x_{8}a \, \mathbf{\hat{x}} + \left(\frac{1}{2} +y_{8}\right)a \, \mathbf{\hat{y}}-z_{8}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{87} & = & \left(-x_{8}-z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{8} - z_{8}\right) \, \mathbf{a}_{2} + \left(-x_{8}+y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{8}\right)a \, \mathbf{\hat{x}}-a\left(x_{8}+\frac{1}{4}\right) \, \mathbf{\hat{y}} + \left(\frac{1}{4}-z_{8}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{88} & = & \left(x_{8}-z_{8}\right) \, \mathbf{a}_{1} + \left(-y_{8}-z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{8} - y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4}-y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{8}\right)a \, \mathbf{\hat{y}}-c\left(z_{8}+\frac{1}{4}\right) \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{89} & = & \left(-y_{8}+z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +x_{8} + z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{8} - y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{2} +x_{8}\right)a \, \mathbf{\hat{x}}-y_{8}a \, \mathbf{\hat{y}} + z_{8}c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{90} & = & \left(\frac{1}{2} +y_{8} + z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} - x_{8} + z_{8}\right) \, \mathbf{a}_{2} + \left(-x_{8}+y_{8}\right) \, \mathbf{a}_{3} & = & -x_{8}a \, \mathbf{\hat{x}} + y_{8}a \, \mathbf{\hat{y}} + \left(\frac{1}{2} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{91} & = & \left(\frac{1}{2} - x_{8} + z_{8}\right) \, \mathbf{a}_{1} + \left(-y_{8}+z_{8}\right) \, \mathbf{a}_{2} + \left(-x_{8}-y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{3}{4}-y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4}-x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \mathbf{B}_{92} & = & \left(\frac{1}{2} +x_{8} + z_{8}\right) \, \mathbf{a}_{1} + \left(\frac{1}{2} +y_{8} + z_{8}\right) \, \mathbf{a}_{2} + \left(\frac{1}{2} +x_{8} + y_{8}\right) \, \mathbf{a}_{3} & = & \left(\frac{1}{4} +y_{8}\right)a \, \mathbf{\hat{x}} + \left(\frac{1}{4} +x_{8}\right)a \, \mathbf{\hat{y}} + \left(\frac{1}{4} +z_{8}\right)c \, \mathbf{\hat{z}} & \left(32g\right) & \mbox{Si} \\ \end{array} \]

References

  • F. Mazzi and E. Galli, Is each analcime different?, Am. Mineral. 63, 448–460 (1978).
  • W. Hartwig, Zur Strukturbestimmung des Analcims, Zeitschrift für Kristallographie – Crystalline Materials 78, 173–207 (1931), doi:10.1524/zkri.1931.78.1.173.
  • C. Hermann, O. Lohrmann, and H. Philipp, eds., Strukturbericht Band II 1928–1932 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).

Found in

  • F. Pechar, The crystal structure of natural monoclinic analcime (NaAlSi2O6·H2O), Zeitschrift für Kristallographie – Crystalline Materials 184, 63–70 (1988), doi:10.1524/zkri.1988.184.14.63.

Geometry files


Prototype Generator

aflow --proto=A2B2C3D12E4_tI184_142_f_f_be_3g_g --params=

Species:

Running:

Output: