Analcime (NaAlSi$_{2}$O$_{6}\cdot$H$_{2}$O, $S6_{1}$) Structure: A2B2C3D12E4_tI184_142_f_f_be_3g_g-001
| Prototype | AlH$_{2}$NaO$_{7}$Si$_{2}$ |
| AFLOW prototype label | A2B2C3D12E4_tI184_142_f_f_be_3g_g-001 |
| Strukturbericht designation | $S6_{1}$ |
| Mineral name | analcime |
| ICSD | 34874 |
| 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-001
--params=$a, \allowbreak c/a, \allowbreak x_{2}, \allowbreak x_{3}, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}$ |
- 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 there 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 H$_{2}$O).
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}- \frac{1}{2}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $\frac{3}{8} \, \mathbf{a}_{1}+\frac{1}{8} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (8b) | Na I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{8} \, \mathbf{a}_{1}+\frac{3}{8} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}- \frac{1}{8}c \,\mathbf{\hat{z}}$ | (8b) | Na I |
| $\mathbf{B_{3}}$ | = | $\frac{5}{8} \, \mathbf{a}_{1}+\frac{7}{8} \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (8b) | Na I |
| $\mathbf{B_{4}}$ | = | $\frac{7}{8} \, \mathbf{a}_{1}+\frac{5}{8} \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{5}{8}c \,\mathbf{\hat{z}}$ | (8b) | Na I |
| $\mathbf{B_{5}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+\left(x_{2} + \frac{1}{4}\right) \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $a x_{2} \,\mathbf{\hat{x}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16e) | Na II |
| $\mathbf{B_{6}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}- \left(x_{2} - \frac{1}{4}\right) \, \mathbf{a}_{2}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{2} \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16e) | Na II |
| $\mathbf{B_{7}}$ | = | $\left(x_{2} + \frac{1}{4}\right) \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+a \left(x_{2} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (16e) | Na II |
| $\mathbf{B_{8}}$ | = | $- \left(x_{2} - \frac{1}{4}\right) \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}- a \left(x_{2} - \frac{1}{4}\right) \,\mathbf{\hat{y}}$ | (16e) | Na II |
| $\mathbf{B_{9}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}- \left(x_{2} - \frac{3}{4}\right) \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $- a x_{2} \,\mathbf{\hat{x}}+\frac{3}{4}c \,\mathbf{\hat{z}}$ | (16e) | Na II |
| $\mathbf{B_{10}}$ | = | $\frac{1}{4} \, \mathbf{a}_{1}+\left(x_{2} + \frac{3}{4}\right) \, \mathbf{a}_{2}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{2} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (16e) | Na II |
| $\mathbf{B_{11}}$ | = | $- \left(x_{2} - \frac{3}{4}\right) \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $- \frac{1}{4}a \,\mathbf{\hat{x}}- a \left(x_{2} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (16e) | Na II |
| $\mathbf{B_{12}}$ | = | $\left(x_{2} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\frac{3}{4} \, \mathbf{a}_{2}+\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+a \left(x_{2} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (16e) | Na II |
| $\mathbf{B_{13}}$ | = | $\left(x_{3} + \frac{3}{8}\right) \, \mathbf{a}_{1}+\left(x_{3} + \frac{1}{8}\right) \, \mathbf{a}_{2}+\left(2 x_{3} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Al I |
| $\mathbf{B_{14}}$ | = | $- \left(x_{3} - \frac{3}{8}\right) \, \mathbf{a}_{1}- \left(x_{3} - \frac{1}{8}\right) \, \mathbf{a}_{2}- \left(2 x_{3} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Al I |
| $\mathbf{B_{15}}$ | = | $\left(x_{3} + \frac{1}{8}\right) \, \mathbf{a}_{1}- \left(x_{3} - \frac{3}{8}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- \frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Al I |
| $\mathbf{B_{16}}$ | = | $- \left(x_{3} - \frac{1}{8}\right) \, \mathbf{a}_{1}+\left(x_{3} + \frac{3}{8}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- \frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | Al I |
| $\mathbf{B_{17}}$ | = | $- \left(x_{3} - \frac{5}{8}\right) \, \mathbf{a}_{1}- \left(x_{3} - \frac{7}{8}\right) \, \mathbf{a}_{2}- \left(2 x_{3} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16f) | Al I |
| $\mathbf{B_{18}}$ | = | $\left(x_{3} + \frac{5}{8}\right) \, \mathbf{a}_{1}+\left(x_{3} + \frac{7}{8}\right) \, \mathbf{a}_{2}+\left(2 x_{3} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16f) | Al I |
| $\mathbf{B_{19}}$ | = | $- \left(x_{3} - \frac{7}{8}\right) \, \mathbf{a}_{1}+\left(x_{3} + \frac{5}{8}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{5}{8}c \,\mathbf{\hat{z}}$ | (16f) | Al I |
| $\mathbf{B_{20}}$ | = | $\left(x_{3} + \frac{7}{8}\right) \, \mathbf{a}_{1}- \left(x_{3} - \frac{5}{8}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{5}{8}c \,\mathbf{\hat{z}}$ | (16f) | Al I |
| $\mathbf{B_{21}}$ | = | $\left(x_{4} + \frac{3}{8}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{1}{8}\right) \, \mathbf{a}_{2}+\left(2 x_{4} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | H I |
| $\mathbf{B_{22}}$ | = | $- \left(x_{4} - \frac{3}{8}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{8}\right) \, \mathbf{a}_{2}- \left(2 x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | H I |
| $\mathbf{B_{23}}$ | = | $\left(x_{4} + \frac{1}{8}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{3}{8}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- \frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | H I |
| $\mathbf{B_{24}}$ | = | $- \left(x_{4} - \frac{1}{8}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{3}{8}\right) \, \mathbf{a}_{2}+\frac{3}{4} \, \mathbf{a}_{3}$ | = | $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- \frac{1}{8}c \,\mathbf{\hat{z}}$ | (16f) | H I |
| $\mathbf{B_{25}}$ | = | $- \left(x_{4} - \frac{5}{8}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{7}{8}\right) \, \mathbf{a}_{2}- \left(2 x_{4} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16f) | H I |
| $\mathbf{B_{26}}$ | = | $\left(x_{4} + \frac{5}{8}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{7}{8}\right) \, \mathbf{a}_{2}+\left(2 x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (16f) | H I |
| $\mathbf{B_{27}}$ | = | $- \left(x_{4} - \frac{7}{8}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{5}{8}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{5}{8}c \,\mathbf{\hat{z}}$ | (16f) | H I |
| $\mathbf{B_{28}}$ | = | $\left(x_{4} + \frac{7}{8}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{5}{8}\right) \, \mathbf{a}_{2}+\frac{1}{4} \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+\frac{5}{8}c \,\mathbf{\hat{z}}$ | (16f) | H I |
| $\mathbf{B_{29}}$ | = | $\left(y_{5} + z_{5}\right) \, \mathbf{a}_{1}+\left(x_{5} + z_{5}\right) \, \mathbf{a}_{2}+\left(x_{5} + y_{5}\right) \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+a y_{5} \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{30}}$ | = | $\left(- y_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{5} - z_{5}\right) \, \mathbf{a}_{2}- \left(x_{5} + y_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}- a \left(y_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{31}}$ | = | $\left(x_{5} + z_{5}\right) \, \mathbf{a}_{1}+\left(- y_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{5} - y_{5}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{32}}$ | = | $- \left(x_{5} - z_{5}\right) \, \mathbf{a}_{1}+\left(y_{5} + z_{5}\right) \, \mathbf{a}_{2}+\left(- x_{5} + y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{33}}$ | = | $\left(y_{5} - z_{5}\right) \, \mathbf{a}_{1}- \left(x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- x_{5} + y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a y_{5} \,\mathbf{\hat{y}}- c z_{5} \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{34}}$ | = | $- \left(y_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{5} - z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{5} - y_{5}\right) \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}- a y_{5} \,\mathbf{\hat{y}}- c \left(z_{5} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{35}}$ | = | $\left(x_{5} - z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{5} - z_{5}\right) \, \mathbf{a}_{2}+\left(x_{5} + y_{5}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{36}}$ | = | $- \left(x_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{5} + y_{5} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{37}}$ | = | $- \left(y_{5} + z_{5}\right) \, \mathbf{a}_{1}- \left(x_{5} + z_{5}\right) \, \mathbf{a}_{2}- \left(x_{5} + y_{5}\right) \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}- a y_{5} \,\mathbf{\hat{y}}- c z_{5} \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{38}}$ | = | $\left(y_{5} - z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{5} - z_{5}\right) \, \mathbf{a}_{2}+\left(x_{5} + y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+a \left(y_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{5} \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{39}}$ | = | $- \left(x_{5} + z_{5}\right) \, \mathbf{a}_{1}+\left(y_{5} - z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{5} - y_{5}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{5} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{40}}$ | = | $\left(x_{5} - z_{5}\right) \, \mathbf{a}_{1}- \left(y_{5} + z_{5}\right) \, \mathbf{a}_{2}+\left(x_{5} - y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{5} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{41}}$ | = | $- \left(y_{5} - z_{5}\right) \, \mathbf{a}_{1}+\left(x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{5} - y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a y_{5} \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{42}}$ | = | $\left(y_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(- x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{5} - y_{5}\right) \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}+a y_{5} \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{43}}$ | = | $\left(- x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{5} - z_{5}\right) \, \mathbf{a}_{2}- \left(x_{5} + y_{5}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{44}}$ | = | $\left(x_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{5} + y_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{5} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O I |
| $\mathbf{B_{45}}$ | = | $\left(y_{6} + z_{6}\right) \, \mathbf{a}_{1}+\left(x_{6} + z_{6}\right) \, \mathbf{a}_{2}+\left(x_{6} + y_{6}\right) \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{46}}$ | = | $\left(- y_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{6} - z_{6}\right) \, \mathbf{a}_{2}- \left(x_{6} + y_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}- a \left(y_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{47}}$ | = | $\left(x_{6} + z_{6}\right) \, \mathbf{a}_{1}+\left(- y_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{6} - y_{6}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{6} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{6} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{48}}$ | = | $- \left(x_{6} - z_{6}\right) \, \mathbf{a}_{1}+\left(y_{6} + z_{6}\right) \, \mathbf{a}_{2}+\left(- x_{6} + y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{6} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{6} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{6} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{49}}$ | = | $\left(y_{6} - z_{6}\right) \, \mathbf{a}_{1}- \left(x_{6} + z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- x_{6} + y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{y}}- c z_{6} \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{50}}$ | = | $- \left(y_{6} + z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{6} - z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{6} - y_{6}\right) \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{51}}$ | = | $\left(x_{6} - z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{6} - z_{6}\right) \, \mathbf{a}_{2}+\left(x_{6} + y_{6}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{6} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{6} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{52}}$ | = | $- \left(x_{6} + z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{6} + z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{6} + y_{6} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{6} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{6} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{53}}$ | = | $- \left(y_{6} + z_{6}\right) \, \mathbf{a}_{1}- \left(x_{6} + z_{6}\right) \, \mathbf{a}_{2}- \left(x_{6} + y_{6}\right) \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{y}}- c z_{6} \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{54}}$ | = | $\left(y_{6} - z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{6} - z_{6}\right) \, \mathbf{a}_{2}+\left(x_{6} + y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}+a \left(y_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{6} \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{55}}$ | = | $- \left(x_{6} + z_{6}\right) \, \mathbf{a}_{1}+\left(y_{6} - z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{6} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{6} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{6} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{56}}$ | = | $\left(x_{6} - z_{6}\right) \, \mathbf{a}_{1}- \left(y_{6} + z_{6}\right) \, \mathbf{a}_{2}+\left(x_{6} - y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{6} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{6} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{6} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{57}}$ | = | $- \left(y_{6} - z_{6}\right) \, \mathbf{a}_{1}+\left(x_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{6} - y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a y_{6} \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{58}}$ | = | $\left(y_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(- x_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{6} - y_{6}\right) \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}+a y_{6} \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{59}}$ | = | $\left(- x_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{6} - z_{6}\right) \, \mathbf{a}_{2}- \left(x_{6} + y_{6}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{6} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{6} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{60}}$ | = | $\left(x_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{6} + y_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{6} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{6} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O II |
| $\mathbf{B_{61}}$ | = | $\left(y_{7} + z_{7}\right) \, \mathbf{a}_{1}+\left(x_{7} + z_{7}\right) \, \mathbf{a}_{2}+\left(x_{7} + y_{7}\right) \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{62}}$ | = | $\left(- y_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{7} - z_{7}\right) \, \mathbf{a}_{2}- \left(x_{7} + y_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- a \left(y_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{63}}$ | = | $\left(x_{7} + z_{7}\right) \, \mathbf{a}_{1}+\left(- y_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{7} - y_{7}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{7} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{7} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{64}}$ | = | $- \left(x_{7} - z_{7}\right) \, \mathbf{a}_{1}+\left(y_{7} + z_{7}\right) \, \mathbf{a}_{2}+\left(- x_{7} + y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{7} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{7} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{7} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{65}}$ | = | $\left(y_{7} - z_{7}\right) \, \mathbf{a}_{1}- \left(x_{7} + z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- x_{7} + y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}- c z_{7} \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{66}}$ | = | $- \left(y_{7} + z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{7} - z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{7} - y_{7}\right) \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{67}}$ | = | $\left(x_{7} - z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{7} - z_{7}\right) \, \mathbf{a}_{2}+\left(x_{7} + y_{7}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{7} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{7} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{68}}$ | = | $- \left(x_{7} + z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{7} + z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{7} + y_{7} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{7} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{7} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{69}}$ | = | $- \left(y_{7} + z_{7}\right) \, \mathbf{a}_{1}- \left(x_{7} + z_{7}\right) \, \mathbf{a}_{2}- \left(x_{7} + y_{7}\right) \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}- c z_{7} \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{70}}$ | = | $\left(y_{7} - z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{7} - z_{7}\right) \, \mathbf{a}_{2}+\left(x_{7} + y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+a \left(y_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{7} \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{71}}$ | = | $- \left(x_{7} + z_{7}\right) \, \mathbf{a}_{1}+\left(y_{7} - z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{7} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{7} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{7} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{72}}$ | = | $\left(x_{7} - z_{7}\right) \, \mathbf{a}_{1}- \left(y_{7} + z_{7}\right) \, \mathbf{a}_{2}+\left(x_{7} - y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{7} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{7} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{7} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{73}}$ | = | $- \left(y_{7} - z_{7}\right) \, \mathbf{a}_{1}+\left(x_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{7} - y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a y_{7} \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{74}}$ | = | $\left(y_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(- x_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{7} - y_{7}\right) \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}+a y_{7} \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{75}}$ | = | $\left(- x_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{7} - z_{7}\right) \, \mathbf{a}_{2}- \left(x_{7} + y_{7}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{7} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{7} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{76}}$ | = | $\left(x_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{7} + y_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{7} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{7} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | O III |
| $\mathbf{B_{77}}$ | = | $\left(y_{8} + z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} + z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} + y_{8}\right) \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{78}}$ | = | $\left(- y_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(x_{8} - z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} + y_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a \left(y_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{79}}$ | = | $\left(x_{8} + z_{8}\right) \, \mathbf{a}_{1}+\left(- y_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{8} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{8} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{80}}$ | = | $- \left(x_{8} - z_{8}\right) \, \mathbf{a}_{1}+\left(y_{8} + z_{8}\right) \, \mathbf{a}_{2}+\left(- x_{8} + y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{8} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{8} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{8} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{81}}$ | = | $\left(y_{8} - z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} + z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- x_{8} + y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}- c z_{8} \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{82}}$ | = | $- \left(y_{8} + z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{8} - z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{83}}$ | = | $\left(x_{8} - z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{8} - z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} + y_{8}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{8} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{8} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{84}}$ | = | $- \left(x_{8} + z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{8} + z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{8} + y_{8} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{8} - \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{8} - \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{85}}$ | = | $- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{1}- \left(x_{8} + z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}- c z_{8} \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{86}}$ | = | $\left(y_{8} - z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(x_{8} - z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} + y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+a \left(y_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}- c z_{8} \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{87}}$ | = | $- \left(x_{8} + z_{8}\right) \, \mathbf{a}_{1}+\left(y_{8} - z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{8} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{8} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{8} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{88}}$ | = | $\left(x_{8} - z_{8}\right) \, \mathbf{a}_{1}- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{2}+\left(x_{8} - y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{8} - \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{8} + \frac{1}{4}\right) \,\mathbf{\hat{y}}- c \left(z_{8} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{89}}$ | = | $- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{1}+\left(x_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{8} - y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- a y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{90}}$ | = | $\left(y_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(- x_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}+a y_{8} \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{91}}$ | = | $\left(- x_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{2}- \left(x_{8} + y_{8}\right) \, \mathbf{a}_{3}$ | = | $- a \left(y_{8} + \frac{1}{4}\right) \,\mathbf{\hat{x}}- a \left(x_{8} - \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
| $\mathbf{B_{92}}$ | = | $\left(x_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(x_{8} + y_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(y_{8} + \frac{1}{4}\right) \,\mathbf{\hat{x}}+a \left(x_{8} + \frac{1}{4}\right) \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (32g) | Si I |
References
- F. Mazzi and E. Galli, Is each analcime different?, Am. Mineral. 63, 448–460 (1988).
- W. Hartwig, Zur Strukturbestimmung des Analcims, Zeitschrift für Kristallographie 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 (NaAlSi$_{2}$O$_{6}$$\cdot$H$_{2}$O), Z. Kristallogr. 184, 63–69 (1988), doi:10.1524/zkri.1988.184.14.63.
Prototype Generator
aflow --proto=A2B2C3D12E4_tI184_142_f_f_be_3g_g --params=$a,c/a,x_{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}$