β (high) quartz (SiO$_{2}$, $C8$) Structure: A2B_hP9_181_i_d-001
- This is the high-temperature structure of $\alpha$-quartz. It can also be found in the enantiomorphic space group $P6_{2}22$ #180.
- (Wright, 1981) put the oxygen atoms on half-filled (12k) sites, with pairs of oxygen sites separated by only 0.4Å. We approximate the structure here by moving the oxygen atoms to fully-filled (6i) sites.
- The ICSD entry is for the $P6_{2}22$ representation of the structure.
- More information about SiO$_{2}$ structures can be found in our article on SiO$_{2}$ and AlPO$_{4}$.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{4}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (3d) | Si I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}+\frac{5}{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{4}a \,\mathbf{\hat{y}}+\frac{5}{6}c \,\mathbf{\hat{z}}$ | (3d) | Si I |
| $\mathbf{B_{3}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+\frac{1}{6} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{6}c \,\mathbf{\hat{z}}$ | (3d) | Si I |
| $\mathbf{B_{4}}$ | = | $x_{2} \, \mathbf{a}_{1}+2 x_{2} \, \mathbf{a}_{2}$ | = | $\frac{3}{2}a x_{2} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}$ | (6i) | O I |
| $\mathbf{B_{5}}$ | = | $- 2 x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\frac{1}{3} \, \mathbf{a}_{3}$ | = | $- \frac{3}{2}a x_{2} \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}$ | (6i) | O I |
| $\mathbf{B_{6}}$ | = | $x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\frac{2}{3} \, \mathbf{a}_{3}$ | = | $- \sqrt{3}a x_{2} \,\mathbf{\hat{y}}+\frac{2}{3}c \,\mathbf{\hat{z}}$ | (6i) | O I |
| $\mathbf{B_{7}}$ | = | $- x_{2} \, \mathbf{a}_{1}- 2 x_{2} \, \mathbf{a}_{2}$ | = | $- \frac{3}{2}a x_{2} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}$ | (6i) | O I |
| $\mathbf{B_{8}}$ | = | $2 x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+\frac{1}{3} \, \mathbf{a}_{3}$ | = | $\frac{3}{2}a x_{2} \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}$ | (6i) | O I |
| $\mathbf{B_{9}}$ | = | $- x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+\frac{2}{3} \, \mathbf{a}_{3}$ | = | $\sqrt{3}a x_{2} \,\mathbf{\hat{y}}+\frac{2}{3}c \,\mathbf{\hat{z}}$ | (6i) | O I |
References
- A. F. Wright and M. S. Lehmann, The Structure of Quartz at 25 and 590$^\circ$C Determined by Neutron Diffraction, J. Solid State Chem. 36, 371–380 (1981), doi:10.1016/0022-4596(81)90449-7.
Found in
- Mineral Web $\beta$-quartz structure.
First cited in
- D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comp. Mat. Sci. 161, S1-S1011 (2019). (doi=10.1016/j.commatsci.2018.10.043)
Geometry files
Prototype Generator
aflow --proto=A2B_hP9_181_i_d --params=$a,c/a,x_{2}$