Santite (KB$_{5}$O$_{8}\cdot$4H$_{2}$O, $K3_{5}$) Structure: A5B8CD12_oC104_41_a2b_4b_a_6b-001
| Prototype | B$_{5}$H$_{8}$KO$_{12}$ |
| AFLOW prototype label | A5B8CD12_oC104_41_a2b_4b_a_6b-001 |
| Strukturbericht designation | $K3_{5}$ |
| Mineral name | santite |
| ICSD | 18211 |
| Pearson symbol | oC104 |
| Space group number | 41 |
| Space group symbol | $Aea2$ |
| AFLOW prototype command |
aflow --proto=A5B8CD12_oC104_41_a2b_4b_a_6b-001
--params=$a, \allowbreak b/a, \allowbreak c/a, \allowbreak z_{1}, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak y_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak y_{4}, \allowbreak z_{4}, \allowbreak x_{5}, \allowbreak y_{5}, \allowbreak z_{5}, \allowbreak x_{6}, \allowbreak y_{6}, \allowbreak z_{6}, \allowbreak x_{7}, \allowbreak y_{7}, \allowbreak z_{7}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}, \allowbreak x_{13}, \allowbreak y_{13}, \allowbreak z_{13}, \allowbreak x_{14}, \allowbreak y_{14}, \allowbreak z_{14}$ |
- (Zachariasen, 1938) originally determined the structure of this compound without knowing the positions of the hydrogens. They believed the structure to be correctly written as KH$_{2}$(H$_{3}$O)$_{2}$B$_{5}$O$_{10}$. (Gottfried, 1940) gave this the Strukturbericht designation $K3_{5}$. (Zachariasen, 1963) did a refinement of the structure, including the hydrogen positions, noting that the structure should properly be written K[B$_{5}$O$_{6}$(OH)$_{4}$]·2H$_{2}$O. As neither the positions of the heavy atoms nor the space group have changed, we retain the $K3_{5}$ designation, but we give the prototype as KB$_{5}$O$_{8}$·4H$_{2}$O, which seems to be the standard designation even though it is not structurally correct.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&a \,\mathbf{\hat{x}}\\\mathbf{a_{2}}&=&\frac{1}{2}b \,\mathbf{\hat{y}}- \frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}b \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $- z_{1} \, \mathbf{a}_{2}+z_{1} \, \mathbf{a}_{3}$ | = | $c z_{1} \,\mathbf{\hat{z}}$ | (4a) | B I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}- \left(z_{1} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}+c z_{1} \,\mathbf{\hat{z}}$ | (4a) | B I |
| $\mathbf{B_{3}}$ | = | $- z_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $c z_{2} \,\mathbf{\hat{z}}$ | (4a) | K I |
| $\mathbf{B_{4}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}- \left(z_{2} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}b \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (4a) | K I |
| $\mathbf{B_{5}}$ | = | $x_{3} \, \mathbf{a}_{1}+\left(y_{3} - z_{3}\right) \, \mathbf{a}_{2}+\left(y_{3} + z_{3}\right) \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+b y_{3} \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (8b) | B II |
| $\mathbf{B_{6}}$ | = | $- x_{3} \, \mathbf{a}_{1}- \left(y_{3} + z_{3}\right) \, \mathbf{a}_{2}- \left(y_{3} - z_{3}\right) \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}- b y_{3} \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (8b) | B II |
| $\mathbf{B_{7}}$ | = | $\left(x_{3} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{3} + z_{3} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{3} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (8b) | B II |
| $\mathbf{B_{8}}$ | = | $- \left(x_{3} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{3} - z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{3} + z_{3} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{3} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (8b) | B II |
| $\mathbf{B_{9}}$ | = | $x_{4} \, \mathbf{a}_{1}+\left(y_{4} - z_{4}\right) \, \mathbf{a}_{2}+\left(y_{4} + z_{4}\right) \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+b y_{4} \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (8b) | B III |
| $\mathbf{B_{10}}$ | = | $- x_{4} \, \mathbf{a}_{1}- \left(y_{4} + z_{4}\right) \, \mathbf{a}_{2}- \left(y_{4} - z_{4}\right) \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}- b y_{4} \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (8b) | B III |
| $\mathbf{B_{11}}$ | = | $\left(x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{4} + z_{4} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{4} + z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{4} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (8b) | B III |
| $\mathbf{B_{12}}$ | = | $- \left(x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{4} - z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{4} + z_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{4} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (8b) | B III |
| $\mathbf{B_{13}}$ | = | $x_{5} \, \mathbf{a}_{1}+\left(y_{5} - z_{5}\right) \, \mathbf{a}_{2}+\left(y_{5} + z_{5}\right) \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+b y_{5} \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (8b) | H I |
| $\mathbf{B_{14}}$ | = | $- x_{5} \, \mathbf{a}_{1}- \left(y_{5} + z_{5}\right) \, \mathbf{a}_{2}- \left(y_{5} - z_{5}\right) \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}- b y_{5} \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (8b) | H I |
| $\mathbf{B_{15}}$ | = | $\left(x_{5} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{5} + z_{5} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (8b) | H I |
| $\mathbf{B_{16}}$ | = | $- \left(x_{5} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{5} - z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{5} + z_{5} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (8b) | H I |
| $\mathbf{B_{17}}$ | = | $x_{6} \, \mathbf{a}_{1}+\left(y_{6} - z_{6}\right) \, \mathbf{a}_{2}+\left(y_{6} + z_{6}\right) \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}+b y_{6} \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (8b) | H II |
| $\mathbf{B_{18}}$ | = | $- x_{6} \, \mathbf{a}_{1}- \left(y_{6} + z_{6}\right) \, \mathbf{a}_{2}- \left(y_{6} - z_{6}\right) \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}- b y_{6} \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (8b) | H II |
| $\mathbf{B_{19}}$ | = | $\left(x_{6} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{6} + z_{6} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (8b) | H II |
| $\mathbf{B_{20}}$ | = | $- \left(x_{6} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{6} - z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{6} + z_{6} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (8b) | H II |
| $\mathbf{B_{21}}$ | = | $x_{7} \, \mathbf{a}_{1}+\left(y_{7} - z_{7}\right) \, \mathbf{a}_{2}+\left(y_{7} + z_{7}\right) \, \mathbf{a}_{3}$ | = | $a x_{7} \,\mathbf{\hat{x}}+b y_{7} \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (8b) | H III |
| $\mathbf{B_{22}}$ | = | $- x_{7} \, \mathbf{a}_{1}- \left(y_{7} + z_{7}\right) \, \mathbf{a}_{2}- \left(y_{7} - z_{7}\right) \, \mathbf{a}_{3}$ | = | $- a x_{7} \,\mathbf{\hat{x}}- b y_{7} \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (8b) | H III |
| $\mathbf{B_{23}}$ | = | $\left(x_{7} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{7} + z_{7} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{7} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{7} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (8b) | H III |
| $\mathbf{B_{24}}$ | = | $- \left(x_{7} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{7} - z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{7} + z_{7} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{7} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{7} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{7} \,\mathbf{\hat{z}}$ | (8b) | H III |
| $\mathbf{B_{25}}$ | = | $x_{8} \, \mathbf{a}_{1}+\left(y_{8} - z_{8}\right) \, \mathbf{a}_{2}+\left(y_{8} + z_{8}\right) \, \mathbf{a}_{3}$ | = | $a x_{8} \,\mathbf{\hat{x}}+b y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (8b) | H IV |
| $\mathbf{B_{26}}$ | = | $- x_{8} \, \mathbf{a}_{1}- \left(y_{8} + z_{8}\right) \, \mathbf{a}_{2}- \left(y_{8} - z_{8}\right) \, \mathbf{a}_{3}$ | = | $- a x_{8} \,\mathbf{\hat{x}}- b y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (8b) | H IV |
| $\mathbf{B_{27}}$ | = | $\left(x_{8} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{8} + z_{8} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{8} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{8} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (8b) | H IV |
| $\mathbf{B_{28}}$ | = | $- \left(x_{8} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{8} - z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{8} + z_{8} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{8} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{8} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (8b) | H IV |
| $\mathbf{B_{29}}$ | = | $x_{9} \, \mathbf{a}_{1}+\left(y_{9} - z_{9}\right) \, \mathbf{a}_{2}+\left(y_{9} + z_{9}\right) \, \mathbf{a}_{3}$ | = | $a x_{9} \,\mathbf{\hat{x}}+b y_{9} \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (8b) | O I |
| $\mathbf{B_{30}}$ | = | $- x_{9} \, \mathbf{a}_{1}- \left(y_{9} + z_{9}\right) \, \mathbf{a}_{2}- \left(y_{9} - z_{9}\right) \, \mathbf{a}_{3}$ | = | $- a x_{9} \,\mathbf{\hat{x}}- b y_{9} \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (8b) | O I |
| $\mathbf{B_{31}}$ | = | $\left(x_{9} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{9} + z_{9} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{9} + z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{9} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{9} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (8b) | O I |
| $\mathbf{B_{32}}$ | = | $- \left(x_{9} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{9} - z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{9} + z_{9} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{9} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{9} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (8b) | O I |
| $\mathbf{B_{33}}$ | = | $x_{10} \, \mathbf{a}_{1}+\left(y_{10} - z_{10}\right) \, \mathbf{a}_{2}+\left(y_{10} + z_{10}\right) \, \mathbf{a}_{3}$ | = | $a x_{10} \,\mathbf{\hat{x}}+b y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (8b) | O II |
| $\mathbf{B_{34}}$ | = | $- x_{10} \, \mathbf{a}_{1}- \left(y_{10} + z_{10}\right) \, \mathbf{a}_{2}- \left(y_{10} - z_{10}\right) \, \mathbf{a}_{3}$ | = | $- a x_{10} \,\mathbf{\hat{x}}- b y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (8b) | O II |
| $\mathbf{B_{35}}$ | = | $\left(x_{10} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{10} + z_{10} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{10} + z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{10} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{10} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (8b) | O II |
| $\mathbf{B_{36}}$ | = | $- \left(x_{10} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{10} - z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{10} + z_{10} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{10} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{10} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (8b) | O II |
| $\mathbf{B_{37}}$ | = | $x_{11} \, \mathbf{a}_{1}+\left(y_{11} - z_{11}\right) \, \mathbf{a}_{2}+\left(y_{11} + z_{11}\right) \, \mathbf{a}_{3}$ | = | $a x_{11} \,\mathbf{\hat{x}}+b y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (8b) | O III |
| $\mathbf{B_{38}}$ | = | $- x_{11} \, \mathbf{a}_{1}- \left(y_{11} + z_{11}\right) \, \mathbf{a}_{2}- \left(y_{11} - z_{11}\right) \, \mathbf{a}_{3}$ | = | $- a x_{11} \,\mathbf{\hat{x}}- b y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (8b) | O III |
| $\mathbf{B_{39}}$ | = | $\left(x_{11} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{11} + z_{11} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{11} + z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{11} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{11} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (8b) | O III |
| $\mathbf{B_{40}}$ | = | $- \left(x_{11} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{11} - z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{11} + z_{11} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{11} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{11} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (8b) | O III |
| $\mathbf{B_{41}}$ | = | $x_{12} \, \mathbf{a}_{1}+\left(y_{12} - z_{12}\right) \, \mathbf{a}_{2}+\left(y_{12} + z_{12}\right) \, \mathbf{a}_{3}$ | = | $a x_{12} \,\mathbf{\hat{x}}+b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8b) | O IV |
| $\mathbf{B_{42}}$ | = | $- x_{12} \, \mathbf{a}_{1}- \left(y_{12} + z_{12}\right) \, \mathbf{a}_{2}- \left(y_{12} - z_{12}\right) \, \mathbf{a}_{3}$ | = | $- a x_{12} \,\mathbf{\hat{x}}- b y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8b) | O IV |
| $\mathbf{B_{43}}$ | = | $\left(x_{12} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{12} + z_{12} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{12} + z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{12} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{12} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8b) | O IV |
| $\mathbf{B_{44}}$ | = | $- \left(x_{12} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{12} - z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{12} + z_{12} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{12} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{12} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (8b) | O IV |
| $\mathbf{B_{45}}$ | = | $x_{13} \, \mathbf{a}_{1}+\left(y_{13} - z_{13}\right) \, \mathbf{a}_{2}+\left(y_{13} + z_{13}\right) \, \mathbf{a}_{3}$ | = | $a x_{13} \,\mathbf{\hat{x}}+b y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (8b) | O V |
| $\mathbf{B_{46}}$ | = | $- x_{13} \, \mathbf{a}_{1}- \left(y_{13} + z_{13}\right) \, \mathbf{a}_{2}- \left(y_{13} - z_{13}\right) \, \mathbf{a}_{3}$ | = | $- a x_{13} \,\mathbf{\hat{x}}- b y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (8b) | O V |
| $\mathbf{B_{47}}$ | = | $\left(x_{13} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{13} + z_{13} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{13} + z_{13} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{13} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{13} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (8b) | O V |
| $\mathbf{B_{48}}$ | = | $- \left(x_{13} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{13} - z_{13} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{13} + z_{13} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{13} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{13} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (8b) | O V |
| $\mathbf{B_{49}}$ | = | $x_{14} \, \mathbf{a}_{1}+\left(y_{14} - z_{14}\right) \, \mathbf{a}_{2}+\left(y_{14} + z_{14}\right) \, \mathbf{a}_{3}$ | = | $a x_{14} \,\mathbf{\hat{x}}+b y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (8b) | O VI |
| $\mathbf{B_{50}}$ | = | $- x_{14} \, \mathbf{a}_{1}- \left(y_{14} + z_{14}\right) \, \mathbf{a}_{2}- \left(y_{14} - z_{14}\right) \, \mathbf{a}_{3}$ | = | $- a x_{14} \,\mathbf{\hat{x}}- b y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (8b) | O VI |
| $\mathbf{B_{51}}$ | = | $\left(x_{14} + \frac{1}{2}\right) \, \mathbf{a}_{1}- \left(y_{14} + z_{14} - \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(- y_{14} + z_{14} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a \left(x_{14} + \frac{1}{2}\right) \,\mathbf{\hat{x}}- b \left(y_{14} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (8b) | O VI |
| $\mathbf{B_{52}}$ | = | $- \left(x_{14} - \frac{1}{2}\right) \, \mathbf{a}_{1}+\left(y_{14} - z_{14} + \frac{1}{2}\right) \, \mathbf{a}_{2}+\left(y_{14} + z_{14} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a \left(x_{14} - \frac{1}{2}\right) \,\mathbf{\hat{x}}+b \left(y_{14} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (8b) | O VI |
References
- W. H. Zachariasen and H. A. Plettinger, Refinement of the structure of potassium pentaborate tetrahydrate, Acta Cryst. 16, 376–379 (1963), doi:10.1107/S0365110X63001006.
- W. H. Zachariasen, The Crystal Structure of Potassium Acid Dihydronium Pentaborate KH$_{2}$(H$_{3}$Ο)$_{2}$B$_{5}$O$_{10}$, (Potassium Pentaborate Tetrahydrate), Z. Kristallogr. 98, 266–274 (1938), doi:10.1524/zkri.1938.98.1.266.
- C. Gottfried, ed., Strukturbericht Band V 1937 (Akademische Verlagsgesellschaft M. B. H., Leipzig, 1940).
Prototype Generator
aflow --proto=A5B8CD12_oC104_41_a2b_4b_a_6b --params=$a,b/a,c/a,z_{1},z_{2},x_{3},y_{3},z_{3},x_{4},y_{4},z_{4},x_{5},y_{5},z_{5},x_{6},y_{6},z_{6},x_{7},y_{7},z_{7},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12},x_{13},y_{13},z_{13},x_{14},y_{14},z_{14}$