Trigonal Averievite [(CsCl)Cu$_{5}$O$_{2}$(VO$_{4}$)$_{2}$] Structure: A2B7C10D20E4_hP86_147_ae_c2g_d3g_2d6g_dg-001
| Prototype | ClCsCu$_{5}$O$_{2}$(VO$_{4}$)$_{2}$ |
| AFLOW prototype label | A2B7C10D20E4_hP86_147_ae_c2g_d3g_2d6g_dg-001 |
| Mineral name | averievite |
| ICSD | 30939 |
| CCDC | 2048048 |
| Pearson symbol | hP86 |
| Space group number | 147 |
| Space group symbol | $P\overline{3}$ |
| AFLOW prototype command |
aflow --proto=A2B7C10D20E4_hP86_147_ae_c2g_d3g_2d6g_dg-001
--params=$a, \allowbreak c/a, \allowbreak z_{2}, \allowbreak z_{3}, \allowbreak z_{4}, \allowbreak z_{5}, \allowbreak z_{6}, \allowbreak x_{8}, \allowbreak y_{8}, \allowbreak z_{8}, \allowbreak x_{9}, \allowbreak y_{9}, \allowbreak z_{9}, \allowbreak x_{10}, \allowbreak y_{10}, \allowbreak z_{10}, \allowbreak x_{11}, \allowbreak y_{11}, \allowbreak z_{11}, \allowbreak x_{12}, \allowbreak y_{12}, \allowbreak z_{12}, \allowbreak x_{13}, \allowbreak y_{13}, \allowbreak z_{13}, \allowbreak x_{14}, \allowbreak y_{14}, \allowbreak z_{14}, \allowbreak x_{15}, \allowbreak y_{15}, \allowbreak z_{15}, \allowbreak x_{16}, \allowbreak y_{16}, \allowbreak z_{16}, \allowbreak x_{17}, \allowbreak y_{17}, \allowbreak z_{17}, \allowbreak x_{18}, \allowbreak y_{18}, \allowbreak z_{18}, \allowbreak x_{19}, \allowbreak y_{19}, \allowbreak z_{19}$ |
- Averievite is characterized by the presence of a backbone composed of Cu$_{5}$O$_{2}$ clusters and either VO$_{4}$ or PO$_{4}$ tetrahedra. The exact structure depends on atoms added to the backbone. We have included several averievite structures in the Encyclopedia:
- Stoiberite, Cu$_{5}$(VO$_{4}$)$_{2}$,
- (ClCu)$_{2}$Cu$_{5}$O$_{2}$(VO$_{4}$)$_{2}$,
- Cu$_{5}$O$_{2}$(VO$_{4}$)$_{2}$·CuCl$_{2}$,
- (CsCl)Cu$_{5}$O$_{2}$(PO$_{4}$)$_{2}$,
- Monoclinic (CsCl)Cu$_{5}$O$_{2}$(VO$_{4}$)$_{2}$,
- Trigonal (CsCl)Cu$_{5}$O$_{2}$(VO$_{4}$)$_{2}$ (this structure), and
- (KBr)Cu$_{5}$O$_{2}$(VO$_{4}$)$_{2}$.
- See (Starov, 1997), (Botana, 2018), and (Kornyakov, 2021) for more information.
- In the current structure the Cs I (2c) Wyckoff positions are occupied 50% of the time, while the Cs (6g) sites are occupied 28% and 22% of the time. Thus the two blue blobs in the JMol visualization actually only represent two cesium atoms in each unit cell.
- AFLOW shifts the origin by (0 0 c/2), transforming the published Cl (1b) and (3f) positions to (1a) and (3e), respectively.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (1a) | Cl I |
| $\mathbf{B_{2}}$ | = | $z_{2} \, \mathbf{a}_{3}$ | = | $c z_{2} \,\mathbf{\hat{z}}$ | (2c) | Cs I |
| $\mathbf{B_{3}}$ | = | $- z_{2} \, \mathbf{a}_{3}$ | = | $- c z_{2} \,\mathbf{\hat{z}}$ | (2c) | Cs I |
| $\mathbf{B_{4}}$ | = | $\frac{1}{3} \, \mathbf{a}_{1}+\frac{2}{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+c z_{3} \,\mathbf{\hat{z}}$ | (2d) | Cu I |
| $\mathbf{B_{5}}$ | = | $\frac{2}{3} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}- c z_{3} \,\mathbf{\hat{z}}$ | (2d) | Cu I |
| $\mathbf{B_{6}}$ | = | $\frac{1}{3} \, \mathbf{a}_{1}+\frac{2}{3} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+c z_{4} \,\mathbf{\hat{z}}$ | (2d) | O I |
| $\mathbf{B_{7}}$ | = | $\frac{2}{3} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}- c z_{4} \,\mathbf{\hat{z}}$ | (2d) | O I |
| $\mathbf{B_{8}}$ | = | $\frac{1}{3} \, \mathbf{a}_{1}+\frac{2}{3} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (2d) | O II |
| $\mathbf{B_{9}}$ | = | $\frac{2}{3} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}- c z_{5} \,\mathbf{\hat{z}}$ | (2d) | O II |
| $\mathbf{B_{10}}$ | = | $\frac{1}{3} \, \mathbf{a}_{1}+\frac{2}{3} \, \mathbf{a}_{2}+z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+c z_{6} \,\mathbf{\hat{z}}$ | (2d) | V I |
| $\mathbf{B_{11}}$ | = | $\frac{2}{3} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{2}- z_{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}- c z_{6} \,\mathbf{\hat{z}}$ | (2d) | V I |
| $\mathbf{B_{12}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{4}a \,\mathbf{\hat{y}}$ | (3e) | Cl II |
| $\mathbf{B_{13}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{4}a \,\mathbf{\hat{y}}$ | (3e) | Cl II |
| $\mathbf{B_{14}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}$ | (3e) | Cl II |
| $\mathbf{B_{15}}$ | = | $x_{8} \, \mathbf{a}_{1}+y_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} + y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (6g) | Cs II |
| $\mathbf{B_{16}}$ | = | $- y_{8} \, \mathbf{a}_{1}+\left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{8} - 2 y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (6g) | Cs II |
| $\mathbf{B_{17}}$ | = | $- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}- x_{8} \, \mathbf{a}_{2}+z_{8} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{8} - y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{8} \,\mathbf{\hat{y}}+c z_{8} \,\mathbf{\hat{z}}$ | (6g) | Cs II |
| $\mathbf{B_{18}}$ | = | $- x_{8} \, \mathbf{a}_{1}- y_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{8} + y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{8} - y_{8}\right) \,\mathbf{\hat{y}}- c z_{8} \,\mathbf{\hat{z}}$ | (6g) | Cs II |
| $\mathbf{B_{19}}$ | = | $y_{8} \, \mathbf{a}_{1}- \left(x_{8} - y_{8}\right) \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{8} + 2 y_{8}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{8} \,\mathbf{\hat{y}}- c z_{8} \,\mathbf{\hat{z}}$ | (6g) | Cs II |
| $\mathbf{B_{20}}$ | = | $\left(x_{8} - y_{8}\right) \, \mathbf{a}_{1}+x_{8} \, \mathbf{a}_{2}- z_{8} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{8} - y_{8}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{8} \,\mathbf{\hat{y}}- c z_{8} \,\mathbf{\hat{z}}$ | (6g) | Cs II |
| $\mathbf{B_{21}}$ | = | $x_{9} \, \mathbf{a}_{1}+y_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} + y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (6g) | Cs III |
| $\mathbf{B_{22}}$ | = | $- y_{9} \, \mathbf{a}_{1}+\left(x_{9} - y_{9}\right) \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{9} - 2 y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{9} \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (6g) | Cs III |
| $\mathbf{B_{23}}$ | = | $- \left(x_{9} - y_{9}\right) \, \mathbf{a}_{1}- x_{9} \, \mathbf{a}_{2}+z_{9} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{9} - y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{9} \,\mathbf{\hat{y}}+c z_{9} \,\mathbf{\hat{z}}$ | (6g) | Cs III |
| $\mathbf{B_{24}}$ | = | $- x_{9} \, \mathbf{a}_{1}- y_{9} \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{9} + y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{9} - y_{9}\right) \,\mathbf{\hat{y}}- c z_{9} \,\mathbf{\hat{z}}$ | (6g) | Cs III |
| $\mathbf{B_{25}}$ | = | $y_{9} \, \mathbf{a}_{1}- \left(x_{9} - y_{9}\right) \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{9} + 2 y_{9}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{9} \,\mathbf{\hat{y}}- c z_{9} \,\mathbf{\hat{z}}$ | (6g) | Cs III |
| $\mathbf{B_{26}}$ | = | $\left(x_{9} - y_{9}\right) \, \mathbf{a}_{1}+x_{9} \, \mathbf{a}_{2}- z_{9} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{9} - y_{9}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{9} \,\mathbf{\hat{y}}- c z_{9} \,\mathbf{\hat{z}}$ | (6g) | Cs III |
| $\mathbf{B_{27}}$ | = | $x_{10} \, \mathbf{a}_{1}+y_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} + y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (6g) | Cu II |
| $\mathbf{B_{28}}$ | = | $- y_{10} \, \mathbf{a}_{1}+\left(x_{10} - y_{10}\right) \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{10} - 2 y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (6g) | Cu II |
| $\mathbf{B_{29}}$ | = | $- \left(x_{10} - y_{10}\right) \, \mathbf{a}_{1}- x_{10} \, \mathbf{a}_{2}+z_{10} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{10} - y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{10} \,\mathbf{\hat{y}}+c z_{10} \,\mathbf{\hat{z}}$ | (6g) | Cu II |
| $\mathbf{B_{30}}$ | = | $- x_{10} \, \mathbf{a}_{1}- y_{10} \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{10} + y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{10} - y_{10}\right) \,\mathbf{\hat{y}}- c z_{10} \,\mathbf{\hat{z}}$ | (6g) | Cu II |
| $\mathbf{B_{31}}$ | = | $y_{10} \, \mathbf{a}_{1}- \left(x_{10} - y_{10}\right) \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{10} + 2 y_{10}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{10} \,\mathbf{\hat{y}}- c z_{10} \,\mathbf{\hat{z}}$ | (6g) | Cu II |
| $\mathbf{B_{32}}$ | = | $\left(x_{10} - y_{10}\right) \, \mathbf{a}_{1}+x_{10} \, \mathbf{a}_{2}- z_{10} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{10} - y_{10}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{10} \,\mathbf{\hat{y}}- c z_{10} \,\mathbf{\hat{z}}$ | (6g) | Cu II |
| $\mathbf{B_{33}}$ | = | $x_{11} \, \mathbf{a}_{1}+y_{11} \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} + y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (6g) | Cu III |
| $\mathbf{B_{34}}$ | = | $- y_{11} \, \mathbf{a}_{1}+\left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{11} - 2 y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (6g) | Cu III |
| $\mathbf{B_{35}}$ | = | $- \left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}- x_{11} \, \mathbf{a}_{2}+z_{11} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{11} - y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{11} \,\mathbf{\hat{y}}+c z_{11} \,\mathbf{\hat{z}}$ | (6g) | Cu III |
| $\mathbf{B_{36}}$ | = | $- x_{11} \, \mathbf{a}_{1}- y_{11} \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{11} + y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{11} - y_{11}\right) \,\mathbf{\hat{y}}- c z_{11} \,\mathbf{\hat{z}}$ | (6g) | Cu III |
| $\mathbf{B_{37}}$ | = | $y_{11} \, \mathbf{a}_{1}- \left(x_{11} - y_{11}\right) \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{11} + 2 y_{11}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{11} \,\mathbf{\hat{y}}- c z_{11} \,\mathbf{\hat{z}}$ | (6g) | Cu III |
| $\mathbf{B_{38}}$ | = | $\left(x_{11} - y_{11}\right) \, \mathbf{a}_{1}+x_{11} \, \mathbf{a}_{2}- z_{11} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{11} - y_{11}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{11} \,\mathbf{\hat{y}}- c z_{11} \,\mathbf{\hat{z}}$ | (6g) | Cu III |
| $\mathbf{B_{39}}$ | = | $x_{12} \, \mathbf{a}_{1}+y_{12} \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{12} + y_{12}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{12} - y_{12}\right) \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (6g) | Cu IV |
| $\mathbf{B_{40}}$ | = | $- y_{12} \, \mathbf{a}_{1}+\left(x_{12} - y_{12}\right) \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{12} - 2 y_{12}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (6g) | Cu IV |
| $\mathbf{B_{41}}$ | = | $- \left(x_{12} - y_{12}\right) \, \mathbf{a}_{1}- x_{12} \, \mathbf{a}_{2}+z_{12} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{12} - y_{12}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{12} \,\mathbf{\hat{y}}+c z_{12} \,\mathbf{\hat{z}}$ | (6g) | Cu IV |
| $\mathbf{B_{42}}$ | = | $- x_{12} \, \mathbf{a}_{1}- y_{12} \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{12} + y_{12}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{12} - y_{12}\right) \,\mathbf{\hat{y}}- c z_{12} \,\mathbf{\hat{z}}$ | (6g) | Cu IV |
| $\mathbf{B_{43}}$ | = | $y_{12} \, \mathbf{a}_{1}- \left(x_{12} - y_{12}\right) \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{12} + 2 y_{12}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{12} \,\mathbf{\hat{y}}- c z_{12} \,\mathbf{\hat{z}}$ | (6g) | Cu IV |
| $\mathbf{B_{44}}$ | = | $\left(x_{12} - y_{12}\right) \, \mathbf{a}_{1}+x_{12} \, \mathbf{a}_{2}- z_{12} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{12} - y_{12}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{12} \,\mathbf{\hat{y}}- c z_{12} \,\mathbf{\hat{z}}$ | (6g) | Cu IV |
| $\mathbf{B_{45}}$ | = | $x_{13} \, \mathbf{a}_{1}+y_{13} \, \mathbf{a}_{2}+z_{13} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{13} + y_{13}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{13} - y_{13}\right) \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (6g) | O III |
| $\mathbf{B_{46}}$ | = | $- y_{13} \, \mathbf{a}_{1}+\left(x_{13} - y_{13}\right) \, \mathbf{a}_{2}+z_{13} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{13} - 2 y_{13}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (6g) | O III |
| $\mathbf{B_{47}}$ | = | $- \left(x_{13} - y_{13}\right) \, \mathbf{a}_{1}- x_{13} \, \mathbf{a}_{2}+z_{13} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{13} - y_{13}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{13} \,\mathbf{\hat{y}}+c z_{13} \,\mathbf{\hat{z}}$ | (6g) | O III |
| $\mathbf{B_{48}}$ | = | $- x_{13} \, \mathbf{a}_{1}- y_{13} \, \mathbf{a}_{2}- z_{13} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{13} + y_{13}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{13} - y_{13}\right) \,\mathbf{\hat{y}}- c z_{13} \,\mathbf{\hat{z}}$ | (6g) | O III |
| $\mathbf{B_{49}}$ | = | $y_{13} \, \mathbf{a}_{1}- \left(x_{13} - y_{13}\right) \, \mathbf{a}_{2}- z_{13} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{13} + 2 y_{13}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{13} \,\mathbf{\hat{y}}- c z_{13} \,\mathbf{\hat{z}}$ | (6g) | O III |
| $\mathbf{B_{50}}$ | = | $\left(x_{13} - y_{13}\right) \, \mathbf{a}_{1}+x_{13} \, \mathbf{a}_{2}- z_{13} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{13} - y_{13}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{13} \,\mathbf{\hat{y}}- c z_{13} \,\mathbf{\hat{z}}$ | (6g) | O III |
| $\mathbf{B_{51}}$ | = | $x_{14} \, \mathbf{a}_{1}+y_{14} \, \mathbf{a}_{2}+z_{14} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{14} + y_{14}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{14} - y_{14}\right) \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (6g) | O IV |
| $\mathbf{B_{52}}$ | = | $- y_{14} \, \mathbf{a}_{1}+\left(x_{14} - y_{14}\right) \, \mathbf{a}_{2}+z_{14} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{14} - 2 y_{14}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (6g) | O IV |
| $\mathbf{B_{53}}$ | = | $- \left(x_{14} - y_{14}\right) \, \mathbf{a}_{1}- x_{14} \, \mathbf{a}_{2}+z_{14} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{14} - y_{14}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{14} \,\mathbf{\hat{y}}+c z_{14} \,\mathbf{\hat{z}}$ | (6g) | O IV |
| $\mathbf{B_{54}}$ | = | $- x_{14} \, \mathbf{a}_{1}- y_{14} \, \mathbf{a}_{2}- z_{14} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{14} + y_{14}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{14} - y_{14}\right) \,\mathbf{\hat{y}}- c z_{14} \,\mathbf{\hat{z}}$ | (6g) | O IV |
| $\mathbf{B_{55}}$ | = | $y_{14} \, \mathbf{a}_{1}- \left(x_{14} - y_{14}\right) \, \mathbf{a}_{2}- z_{14} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{14} + 2 y_{14}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{14} \,\mathbf{\hat{y}}- c z_{14} \,\mathbf{\hat{z}}$ | (6g) | O IV |
| $\mathbf{B_{56}}$ | = | $\left(x_{14} - y_{14}\right) \, \mathbf{a}_{1}+x_{14} \, \mathbf{a}_{2}- z_{14} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{14} - y_{14}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{14} \,\mathbf{\hat{y}}- c z_{14} \,\mathbf{\hat{z}}$ | (6g) | O IV |
| $\mathbf{B_{57}}$ | = | $x_{15} \, \mathbf{a}_{1}+y_{15} \, \mathbf{a}_{2}+z_{15} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{15} + y_{15}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{15} - y_{15}\right) \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (6g) | O V |
| $\mathbf{B_{58}}$ | = | $- y_{15} \, \mathbf{a}_{1}+\left(x_{15} - y_{15}\right) \, \mathbf{a}_{2}+z_{15} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{15} - 2 y_{15}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{15} \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (6g) | O V |
| $\mathbf{B_{59}}$ | = | $- \left(x_{15} - y_{15}\right) \, \mathbf{a}_{1}- x_{15} \, \mathbf{a}_{2}+z_{15} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{15} - y_{15}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{15} \,\mathbf{\hat{y}}+c z_{15} \,\mathbf{\hat{z}}$ | (6g) | O V |
| $\mathbf{B_{60}}$ | = | $- x_{15} \, \mathbf{a}_{1}- y_{15} \, \mathbf{a}_{2}- z_{15} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{15} + y_{15}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{15} - y_{15}\right) \,\mathbf{\hat{y}}- c z_{15} \,\mathbf{\hat{z}}$ | (6g) | O V |
| $\mathbf{B_{61}}$ | = | $y_{15} \, \mathbf{a}_{1}- \left(x_{15} - y_{15}\right) \, \mathbf{a}_{2}- z_{15} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{15} + 2 y_{15}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{15} \,\mathbf{\hat{y}}- c z_{15} \,\mathbf{\hat{z}}$ | (6g) | O V |
| $\mathbf{B_{62}}$ | = | $\left(x_{15} - y_{15}\right) \, \mathbf{a}_{1}+x_{15} \, \mathbf{a}_{2}- z_{15} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{15} - y_{15}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{15} \,\mathbf{\hat{y}}- c z_{15} \,\mathbf{\hat{z}}$ | (6g) | O V |
| $\mathbf{B_{63}}$ | = | $x_{16} \, \mathbf{a}_{1}+y_{16} \, \mathbf{a}_{2}+z_{16} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{16} + y_{16}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{16} - y_{16}\right) \,\mathbf{\hat{y}}+c z_{16} \,\mathbf{\hat{z}}$ | (6g) | O VI |
| $\mathbf{B_{64}}$ | = | $- y_{16} \, \mathbf{a}_{1}+\left(x_{16} - y_{16}\right) \, \mathbf{a}_{2}+z_{16} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{16} - 2 y_{16}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{16} \,\mathbf{\hat{y}}+c z_{16} \,\mathbf{\hat{z}}$ | (6g) | O VI |
| $\mathbf{B_{65}}$ | = | $- \left(x_{16} - y_{16}\right) \, \mathbf{a}_{1}- x_{16} \, \mathbf{a}_{2}+z_{16} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{16} - y_{16}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{16} \,\mathbf{\hat{y}}+c z_{16} \,\mathbf{\hat{z}}$ | (6g) | O VI |
| $\mathbf{B_{66}}$ | = | $- x_{16} \, \mathbf{a}_{1}- y_{16} \, \mathbf{a}_{2}- z_{16} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{16} + y_{16}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{16} - y_{16}\right) \,\mathbf{\hat{y}}- c z_{16} \,\mathbf{\hat{z}}$ | (6g) | O VI |
| $\mathbf{B_{67}}$ | = | $y_{16} \, \mathbf{a}_{1}- \left(x_{16} - y_{16}\right) \, \mathbf{a}_{2}- z_{16} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{16} + 2 y_{16}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{16} \,\mathbf{\hat{y}}- c z_{16} \,\mathbf{\hat{z}}$ | (6g) | O VI |
| $\mathbf{B_{68}}$ | = | $\left(x_{16} - y_{16}\right) \, \mathbf{a}_{1}+x_{16} \, \mathbf{a}_{2}- z_{16} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{16} - y_{16}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{16} \,\mathbf{\hat{y}}- c z_{16} \,\mathbf{\hat{z}}$ | (6g) | O VI |
| $\mathbf{B_{69}}$ | = | $x_{17} \, \mathbf{a}_{1}+y_{17} \, \mathbf{a}_{2}+z_{17} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{17} + y_{17}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{17} - y_{17}\right) \,\mathbf{\hat{y}}+c z_{17} \,\mathbf{\hat{z}}$ | (6g) | O VII |
| $\mathbf{B_{70}}$ | = | $- y_{17} \, \mathbf{a}_{1}+\left(x_{17} - y_{17}\right) \, \mathbf{a}_{2}+z_{17} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{17} - 2 y_{17}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{17} \,\mathbf{\hat{y}}+c z_{17} \,\mathbf{\hat{z}}$ | (6g) | O VII |
| $\mathbf{B_{71}}$ | = | $- \left(x_{17} - y_{17}\right) \, \mathbf{a}_{1}- x_{17} \, \mathbf{a}_{2}+z_{17} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{17} - y_{17}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{17} \,\mathbf{\hat{y}}+c z_{17} \,\mathbf{\hat{z}}$ | (6g) | O VII |
| $\mathbf{B_{72}}$ | = | $- x_{17} \, \mathbf{a}_{1}- y_{17} \, \mathbf{a}_{2}- z_{17} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{17} + y_{17}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{17} - y_{17}\right) \,\mathbf{\hat{y}}- c z_{17} \,\mathbf{\hat{z}}$ | (6g) | O VII |
| $\mathbf{B_{73}}$ | = | $y_{17} \, \mathbf{a}_{1}- \left(x_{17} - y_{17}\right) \, \mathbf{a}_{2}- z_{17} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{17} + 2 y_{17}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{17} \,\mathbf{\hat{y}}- c z_{17} \,\mathbf{\hat{z}}$ | (6g) | O VII |
| $\mathbf{B_{74}}$ | = | $\left(x_{17} - y_{17}\right) \, \mathbf{a}_{1}+x_{17} \, \mathbf{a}_{2}- z_{17} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{17} - y_{17}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{17} \,\mathbf{\hat{y}}- c z_{17} \,\mathbf{\hat{z}}$ | (6g) | O VII |
| $\mathbf{B_{75}}$ | = | $x_{18} \, \mathbf{a}_{1}+y_{18} \, \mathbf{a}_{2}+z_{18} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{18} + y_{18}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{18} - y_{18}\right) \,\mathbf{\hat{y}}+c z_{18} \,\mathbf{\hat{z}}$ | (6g) | O VIII |
| $\mathbf{B_{76}}$ | = | $- y_{18} \, \mathbf{a}_{1}+\left(x_{18} - y_{18}\right) \, \mathbf{a}_{2}+z_{18} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{18} - 2 y_{18}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{18} \,\mathbf{\hat{y}}+c z_{18} \,\mathbf{\hat{z}}$ | (6g) | O VIII |
| $\mathbf{B_{77}}$ | = | $- \left(x_{18} - y_{18}\right) \, \mathbf{a}_{1}- x_{18} \, \mathbf{a}_{2}+z_{18} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{18} - y_{18}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{18} \,\mathbf{\hat{y}}+c z_{18} \,\mathbf{\hat{z}}$ | (6g) | O VIII |
| $\mathbf{B_{78}}$ | = | $- x_{18} \, \mathbf{a}_{1}- y_{18} \, \mathbf{a}_{2}- z_{18} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{18} + y_{18}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{18} - y_{18}\right) \,\mathbf{\hat{y}}- c z_{18} \,\mathbf{\hat{z}}$ | (6g) | O VIII |
| $\mathbf{B_{79}}$ | = | $y_{18} \, \mathbf{a}_{1}- \left(x_{18} - y_{18}\right) \, \mathbf{a}_{2}- z_{18} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{18} + 2 y_{18}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{18} \,\mathbf{\hat{y}}- c z_{18} \,\mathbf{\hat{z}}$ | (6g) | O VIII |
| $\mathbf{B_{80}}$ | = | $\left(x_{18} - y_{18}\right) \, \mathbf{a}_{1}+x_{18} \, \mathbf{a}_{2}- z_{18} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{18} - y_{18}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{18} \,\mathbf{\hat{y}}- c z_{18} \,\mathbf{\hat{z}}$ | (6g) | O VIII |
| $\mathbf{B_{81}}$ | = | $x_{19} \, \mathbf{a}_{1}+y_{19} \, \mathbf{a}_{2}+z_{19} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{19} + y_{19}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{19} - y_{19}\right) \,\mathbf{\hat{y}}+c z_{19} \,\mathbf{\hat{z}}$ | (6g) | V II |
| $\mathbf{B_{82}}$ | = | $- y_{19} \, \mathbf{a}_{1}+\left(x_{19} - y_{19}\right) \, \mathbf{a}_{2}+z_{19} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{19} - 2 y_{19}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{19} \,\mathbf{\hat{y}}+c z_{19} \,\mathbf{\hat{z}}$ | (6g) | V II |
| $\mathbf{B_{83}}$ | = | $- \left(x_{19} - y_{19}\right) \, \mathbf{a}_{1}- x_{19} \, \mathbf{a}_{2}+z_{19} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{19} - y_{19}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{19} \,\mathbf{\hat{y}}+c z_{19} \,\mathbf{\hat{z}}$ | (6g) | V II |
| $\mathbf{B_{84}}$ | = | $- x_{19} \, \mathbf{a}_{1}- y_{19} \, \mathbf{a}_{2}- z_{19} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{19} + y_{19}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \left(x_{19} - y_{19}\right) \,\mathbf{\hat{y}}- c z_{19} \,\mathbf{\hat{z}}$ | (6g) | V II |
| $\mathbf{B_{85}}$ | = | $y_{19} \, \mathbf{a}_{1}- \left(x_{19} - y_{19}\right) \, \mathbf{a}_{2}- z_{19} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{19} + 2 y_{19}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{19} \,\mathbf{\hat{y}}- c z_{19} \,\mathbf{\hat{z}}$ | (6g) | V II |
| $\mathbf{B_{86}}$ | = | $\left(x_{19} - y_{19}\right) \, \mathbf{a}_{1}+x_{19} \, \mathbf{a}_{2}- z_{19} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{19} - y_{19}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a y_{19} \,\mathbf{\hat{y}}- c z_{19} \,\mathbf{\hat{z}}$ | (6g) | V II |
References
- I. V. Kornyakov, V. A. Vladimirova, O. I. Siidra, and S. V. Krivovichev, Expanding the Averievite Family, (MX)Cu$_{5}$O$_{2}$(T$^{5+}$O$_{4}$)$_{2}$ (T$^{5+}$ = P, V; M = K, Rb, Cs, Cu; X = Cl, Br): Synthesis and Single-Crystal X-ray Diffraction Study, Molecules 26, 1833 (2021), doi:10.3390/molecules26071833.
- G. L. Starova, S. V. Krivovichev, V. S. Fundamensky, and S. K. Filatov, The crystal structure of averievite, Cu$_{5}$O$_{2}$(VO$_{4}$)$_{2}\cdot$nMX; comparison with related compounds, Mineral. Mag. 61, 441–446 (1997).
- A. S. Botana, H. Zheng, S. H. Lapidus, J. F. Mitchell, and M. R. Norman, Averievite: A copper oxide kagome antiferromagnet, Phys. Rev. B 98, August (2018), doi:10.1103/PhysRevB.98.054421.
First cited in
- N. Anderson, M. J. Mehl, H. Eckert, S. Divilov, X. Campilongo, S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 5. Submitted to Computational Materials Science (2026).
Geometry files
Prototype Generator
aflow --proto=A2B7C10D20E4_hP86_147_ae_c2g_d3g_2d6g_dg --params=$a,c/a,z_{2},z_{3},z_{4},z_{5},z_{6},x_{8},y_{8},z_{8},x_{9},y_{9},z_{9},x_{10},y_{10},z_{10},x_{11},y_{11},z_{11},x_{12},y_{12},z_{12},x_{13},y_{13},z_{13},x_{14},y_{14},z_{14},x_{15},y_{15},z_{15},x_{16},y_{16},z_{16},x_{17},y_{17},z_{17},x_{18},y_{18},z_{18},x_{19},y_{19},z_{19}$