Low Temperature (NH$_{3}$CH$_{3}$)Al(SO$_{4}$)$_{2}$ $\cdot$ 12H$_{2}$O Structure: ABC30DE20F2_oP220_29_a_a_30a_a_20a_2a-001
| Prototype | AlCH$_{30}$NO$_{20}$S$_{2}$ |
| AFLOW prototype label | ABC30DE20F2_oP220_29_a_a_30a_a_20a_2a-001 |
| ICSD | 108970 |
| Pearson symbol | oP220 |
| Space group number | 29 |
| Space group symbol | $Pca2_1$ |
| AFLOW prototype command |
aflow --proto=ABC30DE20F2_oP220_29_a_a_30a_a_20a_2a-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak x_{1}, \allowbreak y_{1}, \allowbreak z_{1}, \allowbreak x_{2}, \allowbreak y_{2}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak y_{4}, \allowbreak z_{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}, \allowbreak x_{9}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}, \allowbreak x_{13}, \allowbreak y_{13}, \allowbreak z_{13}, \allowbreak x_{14}, \allowbreak y_{14}, \allowbreak z_{14}, \allowbreak x_{15}, \allowbreak y_{15}, \allowbreak z_{15}, \allowbreak x_{16}, \allowbreak y_{16}, \allowbreak z_{16}, \allowbreak x_{17}, \allowbreak y_{17}, \allowbreak z_{17}, \allowbreak x_{18}, \allowbreak y_{18}, \allowbreak z_{18}, \allowbreak x_{19}, \allowbreak y_{19}, \allowbreak z_{19}, \allowbreak x_{20}, \allowbreak y_{20}, \allowbreak z_{20}, \allowbreak x_{21}, \allowbreak y_{21}, \allowbreak z_{21}, \allowbreak x_{22}, \allowbreak y_{22}, \allowbreak z_{22}, \allowbreak x_{23}, \allowbreak y_{23}, \allowbreak z_{23}, \allowbreak x_{24}, \allowbreak y_{24}, \allowbreak z_{24}, \allowbreak x_{25}, \allowbreak y_{25}, \allowbreak z_{25}, \allowbreak x_{26}, \allowbreak y_{26}, \allowbreak z_{26}, \allowbreak x_{27}, \allowbreak y_{27}, \allowbreak z_{27}, \allowbreak x_{28}, \allowbreak y_{28}, \allowbreak z_{28}, \allowbreak x_{29}, \allowbreak y_{29}, \allowbreak z_{29}, \allowbreak x_{30}, \allowbreak y_{30}, \allowbreak z_{30}, \allowbreak x_{31}, \allowbreak y_{31}, \allowbreak z_{31}, \allowbreak x_{32}, \allowbreak y_{32}, \allowbreak z_{32}, \allowbreak x_{33}, \allowbreak y_{33}, \allowbreak z_{33}, \allowbreak x_{34}, \allowbreak y_{34}, \allowbreak z_{34}, \allowbreak x_{35}, \allowbreak y_{35}, \allowbreak z_{35}, \allowbreak x_{36}, \allowbreak y_{36}, \allowbreak z_{36}, \allowbreak x_{37}, \allowbreak y_{37}, \allowbreak z_{37}, \allowbreak x_{38}, \allowbreak y_{38}, \allowbreak z_{38}, \allowbreak x_{39}, \allowbreak y_{39}, \allowbreak z_{39}, \allowbreak x_{40}, \allowbreak y_{40}, \allowbreak z_{40}, \allowbreak x_{41}, \allowbreak y_{41}, \allowbreak z_{41}, \allowbreak x_{42}, \allowbreak y_{42}, \allowbreak z_{42}, \allowbreak x_{43}, \allowbreak y_{43}, \allowbreak z_{43}, \allowbreak x_{44}, \allowbreak y_{44}, \allowbreak z_{44}, \allowbreak x_{45}, \allowbreak y_{45}, \allowbreak z_{45}, \allowbreak x_{46}, \allowbreak y_{46}, \allowbreak z_{46}, \allowbreak x_{47}, \allowbreak y_{47}, \allowbreak z_{47}, \allowbreak x_{48}, \allowbreak y_{48}, \allowbreak z_{48}, \allowbreak x_{49}, \allowbreak y_{49}, \allowbreak z_{49}, \allowbreak x_{50}, \allowbreak y_{50}, \allowbreak z_{50}, \allowbreak x_{51}, \allowbreak y_{51}, \allowbreak z_{51}, \allowbreak x_{52}, \allowbreak y_{52}, \allowbreak z_{52}, \allowbreak x_{53}, \allowbreak y_{53}, \allowbreak z_{53}, \allowbreak x_{54}, \allowbreak y_{54}, \allowbreak z_{54}, \allowbreak x_{55}, \allowbreak y_{55}, \allowbreak z_{55}$ |
- The alums have the general formula AB(XO$_{4}$)$_{2}$ · 12H$_{2}$O, where A is a monovalent ion, B is a trivalent ion, and X is a chalcogen. In most cases atom B is aluminum and atom X is sulfur, leading to the name alum.
- All alums have their room-temperature form in space group $Pa\overline{3}$ #205, but the bonding between the A and B ions and the XO$_{4}$ complex can be quite different.
- (Lipson, 1935abc) described three general forms of alum based on the sizes of the monovalent ions. Each of these forms was given a Strukturbericht designation by (Gottfried, 1937):
- This classification scheme is not compete, e.g., (Ledsham, 1968) points out that NaCr(SO$_{4}$)$_{2}$ · 12H$_{2}$O does not fit into any of these categories, and that the actual structure depends on the combination of monovalent and trivalent ions.
- As noted above, the $Pa\overline{3}$ structures of alum are the room temperature form. As the temperature decreases the alum structure may transform. For example, in the temperature range 150-170K the $\beta$-alum (NH$_{3}$CH$_{3}$)Al(SO$_{4}$)$_{2}$ · 12H$_{2}$O transforms into this orthorhombic structure with fully ordered NH$_{3}$CH$_{3}$ ions.
- The data presented here was taken at 113K.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&b \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $x_{1} \, \mathbf{a}_{1}+y_{1} \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $a x_{1} \,\mathbf{\hat{x}}+b y_{1} \,\mathbf{\hat{y}}+c z_{1} \,\mathbf{\hat{z}}$ | (4a) | Al I |
| $\mathbf{B_{2}}$ | = | $- x_{1} \, \mathbf{a}_{1}- y_{1} \, \mathbf{a}_{2}+\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{1} \,\mathbf{\hat{x}}- b y_{1} \,\mathbf{\hat{y}}+c \left(z_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al I |
| $\mathbf{B_{3}}$ | = | $\left(x_{1} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{1} \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $a \left(x_{1} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{1} \,\mathbf{\hat{y}}+c z_{1} \,\mathbf{\hat{z}}$ | (4a) | Al I |
| $\mathbf{B_{4}}$ | = | $- \left(x_{1} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{1} \, \mathbf{a}_{2}+\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{1} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{1} \,\mathbf{\hat{y}}+c \left(z_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | Al I |
| $\mathbf{B_{5}}$ | = | $x_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $a x_{2} \,\mathbf{\hat{x}}+b y_{2} \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (4a) | C I |
| $\mathbf{B_{6}}$ | = | $- x_{2} \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{2} \,\mathbf{\hat{x}}- b y_{2} \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | C I |
| $\mathbf{B_{7}}$ | = | $\left(x_{2} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $a \left(x_{2} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{2} \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (4a) | C I |
| $\mathbf{B_{8}}$ | = | $- \left(x_{2} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{2} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{2} \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | C I |
| $\mathbf{B_{9}}$ | = | $x_{3} \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+b y_{3} \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (4a) | H I |
| $\mathbf{B_{10}}$ | = | $- x_{3} \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}- b y_{3} \,\mathbf{\hat{y}}+c \left(z_{3} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H I |
| $\mathbf{B_{11}}$ | = | $\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{3} \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (4a) | H I |
| $\mathbf{B_{12}}$ | = | $- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{3} \, \mathbf{a}_{2}+\left(z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{3} \,\mathbf{\hat{y}}+c \left(z_{3} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H I |
| $\mathbf{B_{13}}$ | = | $x_{4} \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+b y_{4} \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (4a) | H II |
| $\mathbf{B_{14}}$ | = | $- x_{4} \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}+\left(z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}- b y_{4} \,\mathbf{\hat{y}}+c \left(z_{4} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H II |
| $\mathbf{B_{15}}$ | = | $\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{4} \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (4a) | H II |
| $\mathbf{B_{16}}$ | = | $- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{4} \, \mathbf{a}_{2}+\left(z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{4} \,\mathbf{\hat{y}}+c \left(z_{4} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H II |
| $\mathbf{B_{17}}$ | = | $x_{5} \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+b y_{5} \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (4a) | H III |
| $\mathbf{B_{18}}$ | = | $- x_{5} \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}- b y_{5} \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H III |
| $\mathbf{B_{19}}$ | = | $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{5} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{5} \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (4a) | H III |
| $\mathbf{B_{20}}$ | = | $- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{5} \, \mathbf{a}_{2}+\left(z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{5} \,\mathbf{\hat{y}}+c \left(z_{5} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H III |
| $\mathbf{B_{21}}$ | = | $x_{6} \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (4a) | H IV |
| $\mathbf{B_{22}}$ | = | $- x_{6} \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}- b y_{6} \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H IV |
| $\mathbf{B_{23}}$ | = | $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{6} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{6} \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (4a) | H IV |
| $\mathbf{B_{24}}$ | = | $- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{6} \, \mathbf{a}_{2}+\left(z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}+c \left(z_{6} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H IV |
| $\mathbf{B_{25}}$ | = | $x_{7} \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (4a) | H V |
| $\mathbf{B_{26}}$ | = | $- x_{7} \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- b y_{7} \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H V |
| $\mathbf{B_{27}}$ | = | $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{7} \, \mathbf{a}_{2}+z_{7} \, \mathbf{a}_{3}$ | = | $a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{7} \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (4a) | H V |
| $\mathbf{B_{28}}$ | = | $- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{7} \, \mathbf{a}_{2}+\left(z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}+c \left(z_{7} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H V |
| $\mathbf{B_{29}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+b y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (4a) | H VI |
| $\mathbf{B_{30}}$ | = | $- x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- b y_{8} \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H VI |
| $\mathbf{B_{31}}$ | = | $\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (4a) | H VI |
| $\mathbf{B_{32}}$ | = | $- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+\left(z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{8} \,\mathbf{\hat{y}}+c \left(z_{8} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H VI |
| $\mathbf{B_{33}}$ | = | $x_{9} \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $a x_{9} \,\mathbf{\hat{x}}+b y_{9} \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (4a) | H VII |
| $\mathbf{B_{34}}$ | = | $- x_{9} \, \mathbf{a}_{1}- y_{9} \, \mathbf{a}_{2}+\left(z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{9} \,\mathbf{\hat{x}}- b y_{9} \,\mathbf{\hat{y}}+c \left(z_{9} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H VII |
| $\mathbf{B_{35}}$ | = | $\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $a \left(x_{9} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{9} \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (4a) | H VII |
| $\mathbf{B_{36}}$ | = | $- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}+\left(z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{9} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{9} \,\mathbf{\hat{y}}+c \left(z_{9} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H VII |
| $\mathbf{B_{37}}$ | = | $x_{10} \, \mathbf{a}_{1}+y_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $a x_{10} \,\mathbf{\hat{x}}+b y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (4a) | H VIII |
| $\mathbf{B_{38}}$ | = | $- x_{10} \, \mathbf{a}_{1}- y_{10} \, \mathbf{a}_{2}+\left(z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{10} \,\mathbf{\hat{x}}- b y_{10} \,\mathbf{\hat{y}}+c \left(z_{10} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H VIII |
| $\mathbf{B_{39}}$ | = | $\left(x_{10} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $a \left(x_{10} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (4a) | H VIII |
| $\mathbf{B_{40}}$ | = | $- \left(x_{10} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{10} \, \mathbf{a}_{2}+\left(z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{10} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{10} \,\mathbf{\hat{y}}+c \left(z_{10} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H VIII |
| $\mathbf{B_{41}}$ | = | $x_{11} \, \mathbf{a}_{1}+y_{11} \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $a x_{11} \,\mathbf{\hat{x}}+b y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (4a) | H IX |
| $\mathbf{B_{42}}$ | = | $- x_{11} \, \mathbf{a}_{1}- y_{11} \, \mathbf{a}_{2}+\left(z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{11} \,\mathbf{\hat{x}}- b y_{11} \,\mathbf{\hat{y}}+c \left(z_{11} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H IX |
| $\mathbf{B_{43}}$ | = | $\left(x_{11} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{11} \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $a \left(x_{11} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (4a) | H IX |
| $\mathbf{B_{44}}$ | = | $- \left(x_{11} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{11} \, \mathbf{a}_{2}+\left(z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{11} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{11} \,\mathbf{\hat{y}}+c \left(z_{11} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H IX |
| $\mathbf{B_{45}}$ | = | $x_{12} \, \mathbf{a}_{1}+y_{12} \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $a x_{12} \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (4a) | H X |
| $\mathbf{B_{46}}$ | = | $- x_{12} \, \mathbf{a}_{1}- y_{12} \, \mathbf{a}_{2}+\left(z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{12} \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}+c \left(z_{12} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H X |
| $\mathbf{B_{47}}$ | = | $\left(x_{12} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{12} \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $a \left(x_{12} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (4a) | H X |
| $\mathbf{B_{48}}$ | = | $- \left(x_{12} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{12} \, \mathbf{a}_{2}+\left(z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{12} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c \left(z_{12} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H X |
| $\mathbf{B_{49}}$ | = | $x_{13} \, \mathbf{a}_{1}+y_{13} \, \mathbf{a}_{2}+z_{13} \, \mathbf{a}_{3}$ | = | $a x_{13} \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (4a) | H XI |
| $\mathbf{B_{50}}$ | = | $- x_{13} \, \mathbf{a}_{1}- y_{13} \, \mathbf{a}_{2}+\left(z_{13} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{13} \,\mathbf{\hat{x}}- b y_{13} \,\mathbf{\hat{y}}+c \left(z_{13} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XI |
| $\mathbf{B_{51}}$ | = | $\left(x_{13} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{13} \, \mathbf{a}_{2}+z_{13} \, \mathbf{a}_{3}$ | = | $a \left(x_{13} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (4a) | H XI |
| $\mathbf{B_{52}}$ | = | $- \left(x_{13} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{13} \, \mathbf{a}_{2}+\left(z_{13} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{13} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}+c \left(z_{13} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XI |
| $\mathbf{B_{53}}$ | = | $x_{14} \, \mathbf{a}_{1}+y_{14} \, \mathbf{a}_{2}+z_{14} \, \mathbf{a}_{3}$ | = | $a x_{14} \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (4a) | H XII |
| $\mathbf{B_{54}}$ | = | $- x_{14} \, \mathbf{a}_{1}- y_{14} \, \mathbf{a}_{2}+\left(z_{14} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{14} \,\mathbf{\hat{x}}- b y_{14} \,\mathbf{\hat{y}}+c \left(z_{14} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XII |
| $\mathbf{B_{55}}$ | = | $\left(x_{14} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{14} \, \mathbf{a}_{2}+z_{14} \, \mathbf{a}_{3}$ | = | $a \left(x_{14} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (4a) | H XII |
| $\mathbf{B_{56}}$ | = | $- \left(x_{14} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{14} \, \mathbf{a}_{2}+\left(z_{14} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{14} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}+c \left(z_{14} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XII |
| $\mathbf{B_{57}}$ | = | $x_{15} \, \mathbf{a}_{1}+y_{15} \, \mathbf{a}_{2}+z_{15} \, \mathbf{a}_{3}$ | = | $a x_{15} \,\mathbf{\hat{x}}+b y_{15} \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (4a) | H XIII |
| $\mathbf{B_{58}}$ | = | $- x_{15} \, \mathbf{a}_{1}- y_{15} \, \mathbf{a}_{2}+\left(z_{15} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{15} \,\mathbf{\hat{x}}- b y_{15} \,\mathbf{\hat{y}}+c \left(z_{15} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XIII |
| $\mathbf{B_{59}}$ | = | $\left(x_{15} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{15} \, \mathbf{a}_{2}+z_{15} \, \mathbf{a}_{3}$ | = | $a \left(x_{15} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{15} \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (4a) | H XIII |
| $\mathbf{B_{60}}$ | = | $- \left(x_{15} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{15} \, \mathbf{a}_{2}+\left(z_{15} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{15} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{15} \,\mathbf{\hat{y}}+c \left(z_{15} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XIII |
| $\mathbf{B_{61}}$ | = | $x_{16} \, \mathbf{a}_{1}+y_{16} \, \mathbf{a}_{2}+z_{16} \, \mathbf{a}_{3}$ | = | $a x_{16} \,\mathbf{\hat{x}}+b y_{16} \,\mathbf{\hat{y}}+c z_{16} \,\mathbf{\hat{z}}$ | (4a) | H XIV |
| $\mathbf{B_{62}}$ | = | $- x_{16} \, \mathbf{a}_{1}- y_{16} \, \mathbf{a}_{2}+\left(z_{16} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{16} \,\mathbf{\hat{x}}- b y_{16} \,\mathbf{\hat{y}}+c \left(z_{16} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XIV |
| $\mathbf{B_{63}}$ | = | $\left(x_{16} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{16} \, \mathbf{a}_{2}+z_{16} \, \mathbf{a}_{3}$ | = | $a \left(x_{16} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{16} \,\mathbf{\hat{y}}+c z_{16} \,\mathbf{\hat{z}}$ | (4a) | H XIV |
| $\mathbf{B_{64}}$ | = | $- \left(x_{16} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{16} \, \mathbf{a}_{2}+\left(z_{16} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{16} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{16} \,\mathbf{\hat{y}}+c \left(z_{16} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XIV |
| $\mathbf{B_{65}}$ | = | $x_{17} \, \mathbf{a}_{1}+y_{17} \, \mathbf{a}_{2}+z_{17} \, \mathbf{a}_{3}$ | = | $a x_{17} \,\mathbf{\hat{x}}+b y_{17} \,\mathbf{\hat{y}}+c z_{17} \,\mathbf{\hat{z}}$ | (4a) | H XV |
| $\mathbf{B_{66}}$ | = | $- x_{17} \, \mathbf{a}_{1}- y_{17} \, \mathbf{a}_{2}+\left(z_{17} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{17} \,\mathbf{\hat{x}}- b y_{17} \,\mathbf{\hat{y}}+c \left(z_{17} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XV |
| $\mathbf{B_{67}}$ | = | $\left(x_{17} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{17} \, \mathbf{a}_{2}+z_{17} \, \mathbf{a}_{3}$ | = | $a \left(x_{17} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{17} \,\mathbf{\hat{y}}+c z_{17} \,\mathbf{\hat{z}}$ | (4a) | H XV |
| $\mathbf{B_{68}}$ | = | $- \left(x_{17} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{17} \, \mathbf{a}_{2}+\left(z_{17} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{17} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{17} \,\mathbf{\hat{y}}+c \left(z_{17} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XV |
| $\mathbf{B_{69}}$ | = | $x_{18} \, \mathbf{a}_{1}+y_{18} \, \mathbf{a}_{2}+z_{18} \, \mathbf{a}_{3}$ | = | $a x_{18} \,\mathbf{\hat{x}}+b y_{18} \,\mathbf{\hat{y}}+c z_{18} \,\mathbf{\hat{z}}$ | (4a) | H XVI |
| $\mathbf{B_{70}}$ | = | $- x_{18} \, \mathbf{a}_{1}- y_{18} \, \mathbf{a}_{2}+\left(z_{18} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{18} \,\mathbf{\hat{x}}- b y_{18} \,\mathbf{\hat{y}}+c \left(z_{18} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XVI |
| $\mathbf{B_{71}}$ | = | $\left(x_{18} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{18} \, \mathbf{a}_{2}+z_{18} \, \mathbf{a}_{3}$ | = | $a \left(x_{18} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{18} \,\mathbf{\hat{y}}+c z_{18} \,\mathbf{\hat{z}}$ | (4a) | H XVI |
| $\mathbf{B_{72}}$ | = | $- \left(x_{18} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{18} \, \mathbf{a}_{2}+\left(z_{18} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{18} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{18} \,\mathbf{\hat{y}}+c \left(z_{18} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XVI |
| $\mathbf{B_{73}}$ | = | $x_{19} \, \mathbf{a}_{1}+y_{19} \, \mathbf{a}_{2}+z_{19} \, \mathbf{a}_{3}$ | = | $a x_{19} \,\mathbf{\hat{x}}+b y_{19} \,\mathbf{\hat{y}}+c z_{19} \,\mathbf{\hat{z}}$ | (4a) | H XVII |
| $\mathbf{B_{74}}$ | = | $- x_{19} \, \mathbf{a}_{1}- y_{19} \, \mathbf{a}_{2}+\left(z_{19} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{19} \,\mathbf{\hat{x}}- b y_{19} \,\mathbf{\hat{y}}+c \left(z_{19} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XVII |
| $\mathbf{B_{75}}$ | = | $\left(x_{19} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{19} \, \mathbf{a}_{2}+z_{19} \, \mathbf{a}_{3}$ | = | $a \left(x_{19} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{19} \,\mathbf{\hat{y}}+c z_{19} \,\mathbf{\hat{z}}$ | (4a) | H XVII |
| $\mathbf{B_{76}}$ | = | $- \left(x_{19} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{19} \, \mathbf{a}_{2}+\left(z_{19} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{19} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{19} \,\mathbf{\hat{y}}+c \left(z_{19} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XVII |
| $\mathbf{B_{77}}$ | = | $x_{20} \, \mathbf{a}_{1}+y_{20} \, \mathbf{a}_{2}+z_{20} \, \mathbf{a}_{3}$ | = | $a x_{20} \,\mathbf{\hat{x}}+b y_{20} \,\mathbf{\hat{y}}+c z_{20} \,\mathbf{\hat{z}}$ | (4a) | H XVIII |
| $\mathbf{B_{78}}$ | = | $- x_{20} \, \mathbf{a}_{1}- y_{20} \, \mathbf{a}_{2}+\left(z_{20} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{20} \,\mathbf{\hat{x}}- b y_{20} \,\mathbf{\hat{y}}+c \left(z_{20} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XVIII |
| $\mathbf{B_{79}}$ | = | $\left(x_{20} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{20} \, \mathbf{a}_{2}+z_{20} \, \mathbf{a}_{3}$ | = | $a \left(x_{20} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{20} \,\mathbf{\hat{y}}+c z_{20} \,\mathbf{\hat{z}}$ | (4a) | H XVIII |
| $\mathbf{B_{80}}$ | = | $- \left(x_{20} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{20} \, \mathbf{a}_{2}+\left(z_{20} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{20} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{20} \,\mathbf{\hat{y}}+c \left(z_{20} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XVIII |
| $\mathbf{B_{81}}$ | = | $x_{21} \, \mathbf{a}_{1}+y_{21} \, \mathbf{a}_{2}+z_{21} \, \mathbf{a}_{3}$ | = | $a x_{21} \,\mathbf{\hat{x}}+b y_{21} \,\mathbf{\hat{y}}+c z_{21} \,\mathbf{\hat{z}}$ | (4a) | H XIX |
| $\mathbf{B_{82}}$ | = | $- x_{21} \, \mathbf{a}_{1}- y_{21} \, \mathbf{a}_{2}+\left(z_{21} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{21} \,\mathbf{\hat{x}}- b y_{21} \,\mathbf{\hat{y}}+c \left(z_{21} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XIX |
| $\mathbf{B_{83}}$ | = | $\left(x_{21} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{21} \, \mathbf{a}_{2}+z_{21} \, \mathbf{a}_{3}$ | = | $a \left(x_{21} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{21} \,\mathbf{\hat{y}}+c z_{21} \,\mathbf{\hat{z}}$ | (4a) | H XIX |
| $\mathbf{B_{84}}$ | = | $- \left(x_{21} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{21} \, \mathbf{a}_{2}+\left(z_{21} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{21} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{21} \,\mathbf{\hat{y}}+c \left(z_{21} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XIX |
| $\mathbf{B_{85}}$ | = | $x_{22} \, \mathbf{a}_{1}+y_{22} \, \mathbf{a}_{2}+z_{22} \, \mathbf{a}_{3}$ | = | $a x_{22} \,\mathbf{\hat{x}}+b y_{22} \,\mathbf{\hat{y}}+c z_{22} \,\mathbf{\hat{z}}$ | (4a) | H XX |
| $\mathbf{B_{86}}$ | = | $- x_{22} \, \mathbf{a}_{1}- y_{22} \, \mathbf{a}_{2}+\left(z_{22} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{22} \,\mathbf{\hat{x}}- b y_{22} \,\mathbf{\hat{y}}+c \left(z_{22} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XX |
| $\mathbf{B_{87}}$ | = | $\left(x_{22} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{22} \, \mathbf{a}_{2}+z_{22} \, \mathbf{a}_{3}$ | = | $a \left(x_{22} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{22} \,\mathbf{\hat{y}}+c z_{22} \,\mathbf{\hat{z}}$ | (4a) | H XX |
| $\mathbf{B_{88}}$ | = | $- \left(x_{22} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{22} \, \mathbf{a}_{2}+\left(z_{22} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{22} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{22} \,\mathbf{\hat{y}}+c \left(z_{22} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XX |
| $\mathbf{B_{89}}$ | = | $x_{23} \, \mathbf{a}_{1}+y_{23} \, \mathbf{a}_{2}+z_{23} \, \mathbf{a}_{3}$ | = | $a x_{23} \,\mathbf{\hat{x}}+b y_{23} \,\mathbf{\hat{y}}+c z_{23} \,\mathbf{\hat{z}}$ | (4a) | H XXI |
| $\mathbf{B_{90}}$ | = | $- x_{23} \, \mathbf{a}_{1}- y_{23} \, \mathbf{a}_{2}+\left(z_{23} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{23} \,\mathbf{\hat{x}}- b y_{23} \,\mathbf{\hat{y}}+c \left(z_{23} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXI |
| $\mathbf{B_{91}}$ | = | $\left(x_{23} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{23} \, \mathbf{a}_{2}+z_{23} \, \mathbf{a}_{3}$ | = | $a \left(x_{23} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{23} \,\mathbf{\hat{y}}+c z_{23} \,\mathbf{\hat{z}}$ | (4a) | H XXI |
| $\mathbf{B_{92}}$ | = | $- \left(x_{23} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{23} \, \mathbf{a}_{2}+\left(z_{23} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{23} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{23} \,\mathbf{\hat{y}}+c \left(z_{23} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXI |
| $\mathbf{B_{93}}$ | = | $x_{24} \, \mathbf{a}_{1}+y_{24} \, \mathbf{a}_{2}+z_{24} \, \mathbf{a}_{3}$ | = | $a x_{24} \,\mathbf{\hat{x}}+b y_{24} \,\mathbf{\hat{y}}+c z_{24} \,\mathbf{\hat{z}}$ | (4a) | H XXII |
| $\mathbf{B_{94}}$ | = | $- x_{24} \, \mathbf{a}_{1}- y_{24} \, \mathbf{a}_{2}+\left(z_{24} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{24} \,\mathbf{\hat{x}}- b y_{24} \,\mathbf{\hat{y}}+c \left(z_{24} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXII |
| $\mathbf{B_{95}}$ | = | $\left(x_{24} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{24} \, \mathbf{a}_{2}+z_{24} \, \mathbf{a}_{3}$ | = | $a \left(x_{24} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{24} \,\mathbf{\hat{y}}+c z_{24} \,\mathbf{\hat{z}}$ | (4a) | H XXII |
| $\mathbf{B_{96}}$ | = | $- \left(x_{24} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{24} \, \mathbf{a}_{2}+\left(z_{24} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{24} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{24} \,\mathbf{\hat{y}}+c \left(z_{24} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXII |
| $\mathbf{B_{97}}$ | = | $x_{25} \, \mathbf{a}_{1}+y_{25} \, \mathbf{a}_{2}+z_{25} \, \mathbf{a}_{3}$ | = | $a x_{25} \,\mathbf{\hat{x}}+b y_{25} \,\mathbf{\hat{y}}+c z_{25} \,\mathbf{\hat{z}}$ | (4a) | H XXIII |
| $\mathbf{B_{98}}$ | = | $- x_{25} \, \mathbf{a}_{1}- y_{25} \, \mathbf{a}_{2}+\left(z_{25} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{25} \,\mathbf{\hat{x}}- b y_{25} \,\mathbf{\hat{y}}+c \left(z_{25} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXIII |
| $\mathbf{B_{99}}$ | = | $\left(x_{25} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{25} \, \mathbf{a}_{2}+z_{25} \, \mathbf{a}_{3}$ | = | $a \left(x_{25} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{25} \,\mathbf{\hat{y}}+c z_{25} \,\mathbf{\hat{z}}$ | (4a) | H XXIII |
| $\mathbf{B_{100}}$ | = | $- \left(x_{25} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{25} \, \mathbf{a}_{2}+\left(z_{25} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{25} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{25} \,\mathbf{\hat{y}}+c \left(z_{25} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXIII |
| $\mathbf{B_{101}}$ | = | $x_{26} \, \mathbf{a}_{1}+y_{26} \, \mathbf{a}_{2}+z_{26} \, \mathbf{a}_{3}$ | = | $a x_{26} \,\mathbf{\hat{x}}+b y_{26} \,\mathbf{\hat{y}}+c z_{26} \,\mathbf{\hat{z}}$ | (4a) | H XXIV |
| $\mathbf{B_{102}}$ | = | $- x_{26} \, \mathbf{a}_{1}- y_{26} \, \mathbf{a}_{2}+\left(z_{26} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{26} \,\mathbf{\hat{x}}- b y_{26} \,\mathbf{\hat{y}}+c \left(z_{26} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXIV |
| $\mathbf{B_{103}}$ | = | $\left(x_{26} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{26} \, \mathbf{a}_{2}+z_{26} \, \mathbf{a}_{3}$ | = | $a \left(x_{26} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{26} \,\mathbf{\hat{y}}+c z_{26} \,\mathbf{\hat{z}}$ | (4a) | H XXIV |
| $\mathbf{B_{104}}$ | = | $- \left(x_{26} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{26} \, \mathbf{a}_{2}+\left(z_{26} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{26} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{26} \,\mathbf{\hat{y}}+c \left(z_{26} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXIV |
| $\mathbf{B_{105}}$ | = | $x_{27} \, \mathbf{a}_{1}+y_{27} \, \mathbf{a}_{2}+z_{27} \, \mathbf{a}_{3}$ | = | $a x_{27} \,\mathbf{\hat{x}}+b y_{27} \,\mathbf{\hat{y}}+c z_{27} \,\mathbf{\hat{z}}$ | (4a) | H XXV |
| $\mathbf{B_{106}}$ | = | $- x_{27} \, \mathbf{a}_{1}- y_{27} \, \mathbf{a}_{2}+\left(z_{27} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{27} \,\mathbf{\hat{x}}- b y_{27} \,\mathbf{\hat{y}}+c \left(z_{27} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXV |
| $\mathbf{B_{107}}$ | = | $\left(x_{27} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{27} \, \mathbf{a}_{2}+z_{27} \, \mathbf{a}_{3}$ | = | $a \left(x_{27} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{27} \,\mathbf{\hat{y}}+c z_{27} \,\mathbf{\hat{z}}$ | (4a) | H XXV |
| $\mathbf{B_{108}}$ | = | $- \left(x_{27} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{27} \, \mathbf{a}_{2}+\left(z_{27} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{27} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{27} \,\mathbf{\hat{y}}+c \left(z_{27} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXV |
| $\mathbf{B_{109}}$ | = | $x_{28} \, \mathbf{a}_{1}+y_{28} \, \mathbf{a}_{2}+z_{28} \, \mathbf{a}_{3}$ | = | $a x_{28} \,\mathbf{\hat{x}}+b y_{28} \,\mathbf{\hat{y}}+c z_{28} \,\mathbf{\hat{z}}$ | (4a) | H XXVI |
| $\mathbf{B_{110}}$ | = | $- x_{28} \, \mathbf{a}_{1}- y_{28} \, \mathbf{a}_{2}+\left(z_{28} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{28} \,\mathbf{\hat{x}}- b y_{28} \,\mathbf{\hat{y}}+c \left(z_{28} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXVI |
| $\mathbf{B_{111}}$ | = | $\left(x_{28} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{28} \, \mathbf{a}_{2}+z_{28} \, \mathbf{a}_{3}$ | = | $a \left(x_{28} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{28} \,\mathbf{\hat{y}}+c z_{28} \,\mathbf{\hat{z}}$ | (4a) | H XXVI |
| $\mathbf{B_{112}}$ | = | $- \left(x_{28} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{28} \, \mathbf{a}_{2}+\left(z_{28} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{28} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{28} \,\mathbf{\hat{y}}+c \left(z_{28} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXVI |
| $\mathbf{B_{113}}$ | = | $x_{29} \, \mathbf{a}_{1}+y_{29} \, \mathbf{a}_{2}+z_{29} \, \mathbf{a}_{3}$ | = | $a x_{29} \,\mathbf{\hat{x}}+b y_{29} \,\mathbf{\hat{y}}+c z_{29} \,\mathbf{\hat{z}}$ | (4a) | H XXVII |
| $\mathbf{B_{114}}$ | = | $- x_{29} \, \mathbf{a}_{1}- y_{29} \, \mathbf{a}_{2}+\left(z_{29} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{29} \,\mathbf{\hat{x}}- b y_{29} \,\mathbf{\hat{y}}+c \left(z_{29} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXVII |
| $\mathbf{B_{115}}$ | = | $\left(x_{29} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{29} \, \mathbf{a}_{2}+z_{29} \, \mathbf{a}_{3}$ | = | $a \left(x_{29} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{29} \,\mathbf{\hat{y}}+c z_{29} \,\mathbf{\hat{z}}$ | (4a) | H XXVII |
| $\mathbf{B_{116}}$ | = | $- \left(x_{29} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{29} \, \mathbf{a}_{2}+\left(z_{29} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{29} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{29} \,\mathbf{\hat{y}}+c \left(z_{29} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXVII |
| $\mathbf{B_{117}}$ | = | $x_{30} \, \mathbf{a}_{1}+y_{30} \, \mathbf{a}_{2}+z_{30} \, \mathbf{a}_{3}$ | = | $a x_{30} \,\mathbf{\hat{x}}+b y_{30} \,\mathbf{\hat{y}}+c z_{30} \,\mathbf{\hat{z}}$ | (4a) | H XXVIII |
| $\mathbf{B_{118}}$ | = | $- x_{30} \, \mathbf{a}_{1}- y_{30} \, \mathbf{a}_{2}+\left(z_{30} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{30} \,\mathbf{\hat{x}}- b y_{30} \,\mathbf{\hat{y}}+c \left(z_{30} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXVIII |
| $\mathbf{B_{119}}$ | = | $\left(x_{30} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{30} \, \mathbf{a}_{2}+z_{30} \, \mathbf{a}_{3}$ | = | $a \left(x_{30} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{30} \,\mathbf{\hat{y}}+c z_{30} \,\mathbf{\hat{z}}$ | (4a) | H XXVIII |
| $\mathbf{B_{120}}$ | = | $- \left(x_{30} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{30} \, \mathbf{a}_{2}+\left(z_{30} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{30} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{30} \,\mathbf{\hat{y}}+c \left(z_{30} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXVIII |
| $\mathbf{B_{121}}$ | = | $x_{31} \, \mathbf{a}_{1}+y_{31} \, \mathbf{a}_{2}+z_{31} \, \mathbf{a}_{3}$ | = | $a x_{31} \,\mathbf{\hat{x}}+b y_{31} \,\mathbf{\hat{y}}+c z_{31} \,\mathbf{\hat{z}}$ | (4a) | H XXIX |
| $\mathbf{B_{122}}$ | = | $- x_{31} \, \mathbf{a}_{1}- y_{31} \, \mathbf{a}_{2}+\left(z_{31} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{31} \,\mathbf{\hat{x}}- b y_{31} \,\mathbf{\hat{y}}+c \left(z_{31} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXIX |
| $\mathbf{B_{123}}$ | = | $\left(x_{31} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{31} \, \mathbf{a}_{2}+z_{31} \, \mathbf{a}_{3}$ | = | $a \left(x_{31} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{31} \,\mathbf{\hat{y}}+c z_{31} \,\mathbf{\hat{z}}$ | (4a) | H XXIX |
| $\mathbf{B_{124}}$ | = | $- \left(x_{31} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{31} \, \mathbf{a}_{2}+\left(z_{31} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{31} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{31} \,\mathbf{\hat{y}}+c \left(z_{31} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXIX |
| $\mathbf{B_{125}}$ | = | $x_{32} \, \mathbf{a}_{1}+y_{32} \, \mathbf{a}_{2}+z_{32} \, \mathbf{a}_{3}$ | = | $a x_{32} \,\mathbf{\hat{x}}+b y_{32} \,\mathbf{\hat{y}}+c z_{32} \,\mathbf{\hat{z}}$ | (4a) | H XXX |
| $\mathbf{B_{126}}$ | = | $- x_{32} \, \mathbf{a}_{1}- y_{32} \, \mathbf{a}_{2}+\left(z_{32} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{32} \,\mathbf{\hat{x}}- b y_{32} \,\mathbf{\hat{y}}+c \left(z_{32} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXX |
| $\mathbf{B_{127}}$ | = | $\left(x_{32} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{32} \, \mathbf{a}_{2}+z_{32} \, \mathbf{a}_{3}$ | = | $a \left(x_{32} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{32} \,\mathbf{\hat{y}}+c z_{32} \,\mathbf{\hat{z}}$ | (4a) | H XXX |
| $\mathbf{B_{128}}$ | = | $- \left(x_{32} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{32} \, \mathbf{a}_{2}+\left(z_{32} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{32} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{32} \,\mathbf{\hat{y}}+c \left(z_{32} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | H XXX |
| $\mathbf{B_{129}}$ | = | $x_{33} \, \mathbf{a}_{1}+y_{33} \, \mathbf{a}_{2}+z_{33} \, \mathbf{a}_{3}$ | = | $a x_{33} \,\mathbf{\hat{x}}+b y_{33} \,\mathbf{\hat{y}}+c z_{33} \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{130}}$ | = | $- x_{33} \, \mathbf{a}_{1}- y_{33} \, \mathbf{a}_{2}+\left(z_{33} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{33} \,\mathbf{\hat{x}}- b y_{33} \,\mathbf{\hat{y}}+c \left(z_{33} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{131}}$ | = | $\left(x_{33} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{33} \, \mathbf{a}_{2}+z_{33} \, \mathbf{a}_{3}$ | = | $a \left(x_{33} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{33} \,\mathbf{\hat{y}}+c z_{33} \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{132}}$ | = | $- \left(x_{33} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{33} \, \mathbf{a}_{2}+\left(z_{33} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{33} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{33} \,\mathbf{\hat{y}}+c \left(z_{33} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | N I |
| $\mathbf{B_{133}}$ | = | $x_{34} \, \mathbf{a}_{1}+y_{34} \, \mathbf{a}_{2}+z_{34} \, \mathbf{a}_{3}$ | = | $a x_{34} \,\mathbf{\hat{x}}+b y_{34} \,\mathbf{\hat{y}}+c z_{34} \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{134}}$ | = | $- x_{34} \, \mathbf{a}_{1}- y_{34} \, \mathbf{a}_{2}+\left(z_{34} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{34} \,\mathbf{\hat{x}}- b y_{34} \,\mathbf{\hat{y}}+c \left(z_{34} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{135}}$ | = | $\left(x_{34} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{34} \, \mathbf{a}_{2}+z_{34} \, \mathbf{a}_{3}$ | = | $a \left(x_{34} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{34} \,\mathbf{\hat{y}}+c z_{34} \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{136}}$ | = | $- \left(x_{34} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{34} \, \mathbf{a}_{2}+\left(z_{34} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{34} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{34} \,\mathbf{\hat{y}}+c \left(z_{34} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{137}}$ | = | $x_{35} \, \mathbf{a}_{1}+y_{35} \, \mathbf{a}_{2}+z_{35} \, \mathbf{a}_{3}$ | = | $a x_{35} \,\mathbf{\hat{x}}+b y_{35} \,\mathbf{\hat{y}}+c z_{35} \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{138}}$ | = | $- x_{35} \, \mathbf{a}_{1}- y_{35} \, \mathbf{a}_{2}+\left(z_{35} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{35} \,\mathbf{\hat{x}}- b y_{35} \,\mathbf{\hat{y}}+c \left(z_{35} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{139}}$ | = | $\left(x_{35} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{35} \, \mathbf{a}_{2}+z_{35} \, \mathbf{a}_{3}$ | = | $a \left(x_{35} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{35} \,\mathbf{\hat{y}}+c z_{35} \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{140}}$ | = | $- \left(x_{35} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{35} \, \mathbf{a}_{2}+\left(z_{35} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{35} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{35} \,\mathbf{\hat{y}}+c \left(z_{35} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O II |
| $\mathbf{B_{141}}$ | = | $x_{36} \, \mathbf{a}_{1}+y_{36} \, \mathbf{a}_{2}+z_{36} \, \mathbf{a}_{3}$ | = | $a x_{36} \,\mathbf{\hat{x}}+b y_{36} \,\mathbf{\hat{y}}+c z_{36} \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{142}}$ | = | $- x_{36} \, \mathbf{a}_{1}- y_{36} \, \mathbf{a}_{2}+\left(z_{36} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{36} \,\mathbf{\hat{x}}- b y_{36} \,\mathbf{\hat{y}}+c \left(z_{36} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{143}}$ | = | $\left(x_{36} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{36} \, \mathbf{a}_{2}+z_{36} \, \mathbf{a}_{3}$ | = | $a \left(x_{36} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{36} \,\mathbf{\hat{y}}+c z_{36} \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{144}}$ | = | $- \left(x_{36} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{36} \, \mathbf{a}_{2}+\left(z_{36} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{36} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{36} \,\mathbf{\hat{y}}+c \left(z_{36} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O III |
| $\mathbf{B_{145}}$ | = | $x_{37} \, \mathbf{a}_{1}+y_{37} \, \mathbf{a}_{2}+z_{37} \, \mathbf{a}_{3}$ | = | $a x_{37} \,\mathbf{\hat{x}}+b y_{37} \,\mathbf{\hat{y}}+c z_{37} \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{146}}$ | = | $- x_{37} \, \mathbf{a}_{1}- y_{37} \, \mathbf{a}_{2}+\left(z_{37} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{37} \,\mathbf{\hat{x}}- b y_{37} \,\mathbf{\hat{y}}+c \left(z_{37} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{147}}$ | = | $\left(x_{37} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{37} \, \mathbf{a}_{2}+z_{37} \, \mathbf{a}_{3}$ | = | $a \left(x_{37} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{37} \,\mathbf{\hat{y}}+c z_{37} \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{148}}$ | = | $- \left(x_{37} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{37} \, \mathbf{a}_{2}+\left(z_{37} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{37} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{37} \,\mathbf{\hat{y}}+c \left(z_{37} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O IV |
| $\mathbf{B_{149}}$ | = | $x_{38} \, \mathbf{a}_{1}+y_{38} \, \mathbf{a}_{2}+z_{38} \, \mathbf{a}_{3}$ | = | $a x_{38} \,\mathbf{\hat{x}}+b y_{38} \,\mathbf{\hat{y}}+c z_{38} \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{150}}$ | = | $- x_{38} \, \mathbf{a}_{1}- y_{38} \, \mathbf{a}_{2}+\left(z_{38} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{38} \,\mathbf{\hat{x}}- b y_{38} \,\mathbf{\hat{y}}+c \left(z_{38} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{151}}$ | = | $\left(x_{38} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{38} \, \mathbf{a}_{2}+z_{38} \, \mathbf{a}_{3}$ | = | $a \left(x_{38} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{38} \,\mathbf{\hat{y}}+c z_{38} \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{152}}$ | = | $- \left(x_{38} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{38} \, \mathbf{a}_{2}+\left(z_{38} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{38} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{38} \,\mathbf{\hat{y}}+c \left(z_{38} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O V |
| $\mathbf{B_{153}}$ | = | $x_{39} \, \mathbf{a}_{1}+y_{39} \, \mathbf{a}_{2}+z_{39} \, \mathbf{a}_{3}$ | = | $a x_{39} \,\mathbf{\hat{x}}+b y_{39} \,\mathbf{\hat{y}}+c z_{39} \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{154}}$ | = | $- x_{39} \, \mathbf{a}_{1}- y_{39} \, \mathbf{a}_{2}+\left(z_{39} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{39} \,\mathbf{\hat{x}}- b y_{39} \,\mathbf{\hat{y}}+c \left(z_{39} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{155}}$ | = | $\left(x_{39} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{39} \, \mathbf{a}_{2}+z_{39} \, \mathbf{a}_{3}$ | = | $a \left(x_{39} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{39} \,\mathbf{\hat{y}}+c z_{39} \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{156}}$ | = | $- \left(x_{39} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{39} \, \mathbf{a}_{2}+\left(z_{39} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{39} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{39} \,\mathbf{\hat{y}}+c \left(z_{39} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VI |
| $\mathbf{B_{157}}$ | = | $x_{40} \, \mathbf{a}_{1}+y_{40} \, \mathbf{a}_{2}+z_{40} \, \mathbf{a}_{3}$ | = | $a x_{40} \,\mathbf{\hat{x}}+b y_{40} \,\mathbf{\hat{y}}+c z_{40} \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{158}}$ | = | $- x_{40} \, \mathbf{a}_{1}- y_{40} \, \mathbf{a}_{2}+\left(z_{40} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{40} \,\mathbf{\hat{x}}- b y_{40} \,\mathbf{\hat{y}}+c \left(z_{40} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{159}}$ | = | $\left(x_{40} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{40} \, \mathbf{a}_{2}+z_{40} \, \mathbf{a}_{3}$ | = | $a \left(x_{40} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{40} \,\mathbf{\hat{y}}+c z_{40} \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{160}}$ | = | $- \left(x_{40} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{40} \, \mathbf{a}_{2}+\left(z_{40} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{40} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{40} \,\mathbf{\hat{y}}+c \left(z_{40} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VII |
| $\mathbf{B_{161}}$ | = | $x_{41} \, \mathbf{a}_{1}+y_{41} \, \mathbf{a}_{2}+z_{41} \, \mathbf{a}_{3}$ | = | $a x_{41} \,\mathbf{\hat{x}}+b y_{41} \,\mathbf{\hat{y}}+c z_{41} \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{162}}$ | = | $- x_{41} \, \mathbf{a}_{1}- y_{41} \, \mathbf{a}_{2}+\left(z_{41} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{41} \,\mathbf{\hat{x}}- b y_{41} \,\mathbf{\hat{y}}+c \left(z_{41} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{163}}$ | = | $\left(x_{41} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{41} \, \mathbf{a}_{2}+z_{41} \, \mathbf{a}_{3}$ | = | $a \left(x_{41} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{41} \,\mathbf{\hat{y}}+c z_{41} \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{164}}$ | = | $- \left(x_{41} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{41} \, \mathbf{a}_{2}+\left(z_{41} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{41} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{41} \,\mathbf{\hat{y}}+c \left(z_{41} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O VIII |
| $\mathbf{B_{165}}$ | = | $x_{42} \, \mathbf{a}_{1}+y_{42} \, \mathbf{a}_{2}+z_{42} \, \mathbf{a}_{3}$ | = | $a x_{42} \,\mathbf{\hat{x}}+b y_{42} \,\mathbf{\hat{y}}+c z_{42} \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{166}}$ | = | $- x_{42} \, \mathbf{a}_{1}- y_{42} \, \mathbf{a}_{2}+\left(z_{42} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{42} \,\mathbf{\hat{x}}- b y_{42} \,\mathbf{\hat{y}}+c \left(z_{42} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{167}}$ | = | $\left(x_{42} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{42} \, \mathbf{a}_{2}+z_{42} \, \mathbf{a}_{3}$ | = | $a \left(x_{42} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{42} \,\mathbf{\hat{y}}+c z_{42} \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{168}}$ | = | $- \left(x_{42} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{42} \, \mathbf{a}_{2}+\left(z_{42} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{42} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{42} \,\mathbf{\hat{y}}+c \left(z_{42} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O IX |
| $\mathbf{B_{169}}$ | = | $x_{43} \, \mathbf{a}_{1}+y_{43} \, \mathbf{a}_{2}+z_{43} \, \mathbf{a}_{3}$ | = | $a x_{43} \,\mathbf{\hat{x}}+b y_{43} \,\mathbf{\hat{y}}+c z_{43} \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{170}}$ | = | $- x_{43} \, \mathbf{a}_{1}- y_{43} \, \mathbf{a}_{2}+\left(z_{43} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{43} \,\mathbf{\hat{x}}- b y_{43} \,\mathbf{\hat{y}}+c \left(z_{43} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{171}}$ | = | $\left(x_{43} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{43} \, \mathbf{a}_{2}+z_{43} \, \mathbf{a}_{3}$ | = | $a \left(x_{43} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{43} \,\mathbf{\hat{y}}+c z_{43} \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{172}}$ | = | $- \left(x_{43} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{43} \, \mathbf{a}_{2}+\left(z_{43} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{43} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{43} \,\mathbf{\hat{y}}+c \left(z_{43} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O X |
| $\mathbf{B_{173}}$ | = | $x_{44} \, \mathbf{a}_{1}+y_{44} \, \mathbf{a}_{2}+z_{44} \, \mathbf{a}_{3}$ | = | $a x_{44} \,\mathbf{\hat{x}}+b y_{44} \,\mathbf{\hat{y}}+c z_{44} \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{174}}$ | = | $- x_{44} \, \mathbf{a}_{1}- y_{44} \, \mathbf{a}_{2}+\left(z_{44} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{44} \,\mathbf{\hat{x}}- b y_{44} \,\mathbf{\hat{y}}+c \left(z_{44} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{175}}$ | = | $\left(x_{44} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{44} \, \mathbf{a}_{2}+z_{44} \, \mathbf{a}_{3}$ | = | $a \left(x_{44} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{44} \,\mathbf{\hat{y}}+c z_{44} \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{176}}$ | = | $- \left(x_{44} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{44} \, \mathbf{a}_{2}+\left(z_{44} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{44} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{44} \,\mathbf{\hat{y}}+c \left(z_{44} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XI |
| $\mathbf{B_{177}}$ | = | $x_{45} \, \mathbf{a}_{1}+y_{45} \, \mathbf{a}_{2}+z_{45} \, \mathbf{a}_{3}$ | = | $a x_{45} \,\mathbf{\hat{x}}+b y_{45} \,\mathbf{\hat{y}}+c z_{45} \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{178}}$ | = | $- x_{45} \, \mathbf{a}_{1}- y_{45} \, \mathbf{a}_{2}+\left(z_{45} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{45} \,\mathbf{\hat{x}}- b y_{45} \,\mathbf{\hat{y}}+c \left(z_{45} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{179}}$ | = | $\left(x_{45} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{45} \, \mathbf{a}_{2}+z_{45} \, \mathbf{a}_{3}$ | = | $a \left(x_{45} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{45} \,\mathbf{\hat{y}}+c z_{45} \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{180}}$ | = | $- \left(x_{45} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{45} \, \mathbf{a}_{2}+\left(z_{45} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{45} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{45} \,\mathbf{\hat{y}}+c \left(z_{45} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XII |
| $\mathbf{B_{181}}$ | = | $x_{46} \, \mathbf{a}_{1}+y_{46} \, \mathbf{a}_{2}+z_{46} \, \mathbf{a}_{3}$ | = | $a x_{46} \,\mathbf{\hat{x}}+b y_{46} \,\mathbf{\hat{y}}+c z_{46} \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{182}}$ | = | $- x_{46} \, \mathbf{a}_{1}- y_{46} \, \mathbf{a}_{2}+\left(z_{46} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{46} \,\mathbf{\hat{x}}- b y_{46} \,\mathbf{\hat{y}}+c \left(z_{46} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{183}}$ | = | $\left(x_{46} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{46} \, \mathbf{a}_{2}+z_{46} \, \mathbf{a}_{3}$ | = | $a \left(x_{46} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{46} \,\mathbf{\hat{y}}+c z_{46} \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{184}}$ | = | $- \left(x_{46} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{46} \, \mathbf{a}_{2}+\left(z_{46} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{46} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{46} \,\mathbf{\hat{y}}+c \left(z_{46} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIII |
| $\mathbf{B_{185}}$ | = | $x_{47} \, \mathbf{a}_{1}+y_{47} \, \mathbf{a}_{2}+z_{47} \, \mathbf{a}_{3}$ | = | $a x_{47} \,\mathbf{\hat{x}}+b y_{47} \,\mathbf{\hat{y}}+c z_{47} \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{186}}$ | = | $- x_{47} \, \mathbf{a}_{1}- y_{47} \, \mathbf{a}_{2}+\left(z_{47} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{47} \,\mathbf{\hat{x}}- b y_{47} \,\mathbf{\hat{y}}+c \left(z_{47} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{187}}$ | = | $\left(x_{47} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{47} \, \mathbf{a}_{2}+z_{47} \, \mathbf{a}_{3}$ | = | $a \left(x_{47} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{47} \,\mathbf{\hat{y}}+c z_{47} \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{188}}$ | = | $- \left(x_{47} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{47} \, \mathbf{a}_{2}+\left(z_{47} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{47} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{47} \,\mathbf{\hat{y}}+c \left(z_{47} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIV |
| $\mathbf{B_{189}}$ | = | $x_{48} \, \mathbf{a}_{1}+y_{48} \, \mathbf{a}_{2}+z_{48} \, \mathbf{a}_{3}$ | = | $a x_{48} \,\mathbf{\hat{x}}+b y_{48} \,\mathbf{\hat{y}}+c z_{48} \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{190}}$ | = | $- x_{48} \, \mathbf{a}_{1}- y_{48} \, \mathbf{a}_{2}+\left(z_{48} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{48} \,\mathbf{\hat{x}}- b y_{48} \,\mathbf{\hat{y}}+c \left(z_{48} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{191}}$ | = | $\left(x_{48} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{48} \, \mathbf{a}_{2}+z_{48} \, \mathbf{a}_{3}$ | = | $a \left(x_{48} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{48} \,\mathbf{\hat{y}}+c z_{48} \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{192}}$ | = | $- \left(x_{48} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{48} \, \mathbf{a}_{2}+\left(z_{48} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{48} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{48} \,\mathbf{\hat{y}}+c \left(z_{48} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XV |
| $\mathbf{B_{193}}$ | = | $x_{49} \, \mathbf{a}_{1}+y_{49} \, \mathbf{a}_{2}+z_{49} \, \mathbf{a}_{3}$ | = | $a x_{49} \,\mathbf{\hat{x}}+b y_{49} \,\mathbf{\hat{y}}+c z_{49} \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{194}}$ | = | $- x_{49} \, \mathbf{a}_{1}- y_{49} \, \mathbf{a}_{2}+\left(z_{49} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{49} \,\mathbf{\hat{x}}- b y_{49} \,\mathbf{\hat{y}}+c \left(z_{49} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{195}}$ | = | $\left(x_{49} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{49} \, \mathbf{a}_{2}+z_{49} \, \mathbf{a}_{3}$ | = | $a \left(x_{49} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{49} \,\mathbf{\hat{y}}+c z_{49} \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{196}}$ | = | $- \left(x_{49} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{49} \, \mathbf{a}_{2}+\left(z_{49} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{49} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{49} \,\mathbf{\hat{y}}+c \left(z_{49} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVI |
| $\mathbf{B_{197}}$ | = | $x_{50} \, \mathbf{a}_{1}+y_{50} \, \mathbf{a}_{2}+z_{50} \, \mathbf{a}_{3}$ | = | $a x_{50} \,\mathbf{\hat{x}}+b y_{50} \,\mathbf{\hat{y}}+c z_{50} \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{198}}$ | = | $- x_{50} \, \mathbf{a}_{1}- y_{50} \, \mathbf{a}_{2}+\left(z_{50} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{50} \,\mathbf{\hat{x}}- b y_{50} \,\mathbf{\hat{y}}+c \left(z_{50} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{199}}$ | = | $\left(x_{50} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{50} \, \mathbf{a}_{2}+z_{50} \, \mathbf{a}_{3}$ | = | $a \left(x_{50} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{50} \,\mathbf{\hat{y}}+c z_{50} \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{200}}$ | = | $- \left(x_{50} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{50} \, \mathbf{a}_{2}+\left(z_{50} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{50} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{50} \,\mathbf{\hat{y}}+c \left(z_{50} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVII |
| $\mathbf{B_{201}}$ | = | $x_{51} \, \mathbf{a}_{1}+y_{51} \, \mathbf{a}_{2}+z_{51} \, \mathbf{a}_{3}$ | = | $a x_{51} \,\mathbf{\hat{x}}+b y_{51} \,\mathbf{\hat{y}}+c z_{51} \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{202}}$ | = | $- x_{51} \, \mathbf{a}_{1}- y_{51} \, \mathbf{a}_{2}+\left(z_{51} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{51} \,\mathbf{\hat{x}}- b y_{51} \,\mathbf{\hat{y}}+c \left(z_{51} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{203}}$ | = | $\left(x_{51} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{51} \, \mathbf{a}_{2}+z_{51} \, \mathbf{a}_{3}$ | = | $a \left(x_{51} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{51} \,\mathbf{\hat{y}}+c z_{51} \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{204}}$ | = | $- \left(x_{51} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{51} \, \mathbf{a}_{2}+\left(z_{51} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{51} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{51} \,\mathbf{\hat{y}}+c \left(z_{51} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XVIII |
| $\mathbf{B_{205}}$ | = | $x_{52} \, \mathbf{a}_{1}+y_{52} \, \mathbf{a}_{2}+z_{52} \, \mathbf{a}_{3}$ | = | $a x_{52} \,\mathbf{\hat{x}}+b y_{52} \,\mathbf{\hat{y}}+c z_{52} \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{206}}$ | = | $- x_{52} \, \mathbf{a}_{1}- y_{52} \, \mathbf{a}_{2}+\left(z_{52} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{52} \,\mathbf{\hat{x}}- b y_{52} \,\mathbf{\hat{y}}+c \left(z_{52} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{207}}$ | = | $\left(x_{52} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{52} \, \mathbf{a}_{2}+z_{52} \, \mathbf{a}_{3}$ | = | $a \left(x_{52} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{52} \,\mathbf{\hat{y}}+c z_{52} \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{208}}$ | = | $- \left(x_{52} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{52} \, \mathbf{a}_{2}+\left(z_{52} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{52} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{52} \,\mathbf{\hat{y}}+c \left(z_{52} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XIX |
| $\mathbf{B_{209}}$ | = | $x_{53} \, \mathbf{a}_{1}+y_{53} \, \mathbf{a}_{2}+z_{53} \, \mathbf{a}_{3}$ | = | $a x_{53} \,\mathbf{\hat{x}}+b y_{53} \,\mathbf{\hat{y}}+c z_{53} \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{210}}$ | = | $- x_{53} \, \mathbf{a}_{1}- y_{53} \, \mathbf{a}_{2}+\left(z_{53} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{53} \,\mathbf{\hat{x}}- b y_{53} \,\mathbf{\hat{y}}+c \left(z_{53} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{211}}$ | = | $\left(x_{53} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{53} \, \mathbf{a}_{2}+z_{53} \, \mathbf{a}_{3}$ | = | $a \left(x_{53} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{53} \,\mathbf{\hat{y}}+c z_{53} \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{212}}$ | = | $- \left(x_{53} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{53} \, \mathbf{a}_{2}+\left(z_{53} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{53} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{53} \,\mathbf{\hat{y}}+c \left(z_{53} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | O XX |
| $\mathbf{B_{213}}$ | = | $x_{54} \, \mathbf{a}_{1}+y_{54} \, \mathbf{a}_{2}+z_{54} \, \mathbf{a}_{3}$ | = | $a x_{54} \,\mathbf{\hat{x}}+b y_{54} \,\mathbf{\hat{y}}+c z_{54} \,\mathbf{\hat{z}}$ | (4a) | S I |
| $\mathbf{B_{214}}$ | = | $- x_{54} \, \mathbf{a}_{1}- y_{54} \, \mathbf{a}_{2}+\left(z_{54} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{54} \,\mathbf{\hat{x}}- b y_{54} \,\mathbf{\hat{y}}+c \left(z_{54} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | S I |
| $\mathbf{B_{215}}$ | = | $\left(x_{54} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{54} \, \mathbf{a}_{2}+z_{54} \, \mathbf{a}_{3}$ | = | $a \left(x_{54} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{54} \,\mathbf{\hat{y}}+c z_{54} \,\mathbf{\hat{z}}$ | (4a) | S I |
| $\mathbf{B_{216}}$ | = | $- \left(x_{54} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{54} \, \mathbf{a}_{2}+\left(z_{54} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{54} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{54} \,\mathbf{\hat{y}}+c \left(z_{54} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | S I |
| $\mathbf{B_{217}}$ | = | $x_{55} \, \mathbf{a}_{1}+y_{55} \, \mathbf{a}_{2}+z_{55} \, \mathbf{a}_{3}$ | = | $a x_{55} \,\mathbf{\hat{x}}+b y_{55} \,\mathbf{\hat{y}}+c z_{55} \,\mathbf{\hat{z}}$ | (4a) | S II |
| $\mathbf{B_{218}}$ | = | $- x_{55} \, \mathbf{a}_{1}- y_{55} \, \mathbf{a}_{2}+\left(z_{55} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{55} \,\mathbf{\hat{x}}- b y_{55} \,\mathbf{\hat{y}}+c \left(z_{55} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | S II |
| $\mathbf{B_{219}}$ | = | $\left(x_{55} + \frac{1}{2}\right) \, \mathbf{a}_{1}- y_{55} \, \mathbf{a}_{2}+z_{55} \, \mathbf{a}_{3}$ | = | $a \left(x_{55} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b y_{55} \,\mathbf{\hat{y}}+c z_{55} \,\mathbf{\hat{z}}$ | (4a) | S II |
| $\mathbf{B_{220}}$ | = | $- \left(x_{55} - \frac{1}{2}\right) \, \mathbf{a}_{1}+y_{55} \, \mathbf{a}_{2}+\left(z_{55} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{55} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b y_{55} \,\mathbf{\hat{y}}+c \left(z_{55} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (4a) | S II |
References
- H. Lipson, Existence of Three Alum Structures, Nature 135, 912 (1935), doi:10.1038/135912b0.
- H. Lipson, The Relation between the Alum Structures, Proc. Roy. Soc. London 151, 347–356 (1935), doi:10.1098/rspa.1935.0154.
- A. H. C. Ledsham and H. Steeple, The crystal structure of sodium chromium alum and caesium chromium alum, Acta Crystallogr. Sect. B 24, 1287–1289 (1968), doi:10.1107/S0567740868004188.
- C. Gottfried and F. Schossberger, eds., Strukturbericht Band III 1933-1935 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1937).
Prototype Generator
aflow --proto=ABC30DE20F2_oP220_29_a_a_30a_a_20a_2a --params=$a,b/a,c/a,x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},x_{3},y_{3},z_{3},x_{4},y_{4},z_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12},x_{13},y_{13},z_{13},x_{14},y_{14},z_{14},x_{15},y_{15},z_{15},x_{16},y_{16},z_{16},x_{17},y_{17},z_{17},x_{18},y_{18},z_{18},x_{19},y_{19},z_{19},x_{20},y_{20},z_{20},x_{21},y_{21},z_{21},x_{22},y_{22},z_{22},x_{23},y_{23},z_{23},x_{24},y_{24},z_{24},x_{25},y_{25},z_{25},x_{26},y_{26},z_{26},x_{27},y_{27},z_{27},x_{28},y_{28},z_{28},x_{29},y_{29},z_{29},x_{30},y_{30},z_{30},x_{31},y_{31},z_{31},x_{32},y_{32},z_{32},x_{33},y_{33},z_{33},x_{34},y_{34},z_{34},x_{35},y_{35},z_{35},x_{36},y_{36},z_{36},x_{37},y_{37},z_{37},x_{38},y_{38},z_{38},x_{39},y_{39},z_{39},x_{40},y_{40},z_{40},x_{41},y_{41},z_{41},x_{42},y_{42},z_{42},x_{43},y_{43},z_{43},x_{44},y_{44},z_{44},x_{45},y_{45},z_{45},x_{46},y_{46},z_{46},x_{47},y_{47},z_{47},x_{48},y_{48},z_{48},x_{49},y_{49},z_{49},x_{50},y_{50},z_{50},x_{51},y_{51},z_{51},x_{52},y_{52},z_{52},x_{53},y_{53},z_{53},x_{54},y_{54},z_{54},x_{55},y_{55},z_{55}$