β-Po ($A_i$) Structure: A_hR1_166_a-001
| Prototype | Po |
| AFLOW prototype label | A_hR1_166_a-001 |
| Strukturbericht designation | $A_{i}$ |
| ICSD | 649161 |
| CCDC | 1763583 |
| Pearson symbol | hR1 |
| Space group number | 166 |
| Space group symbol | $R\overline{3}m$ |
| AFLOW prototype command |
aflow --proto=A_hR1_166_a-001
--params=$a, \allowbreak c/a$ |
- Originally, Po was assigned Strukturbericht designation $A19$, which is now considered to be incorrect. (Donohue, 1982, 390).
- Solid polonium actually exists in two phases: cubic $\alpha$–Po ($A_{h}$) is the ground state and rhombohedral $\beta$–Po ($A_{l}$) (this structure) (Donohue, 1974). There is a large amount of hysteresis observed in the transition, which occurs at 54$^\circ$C for $\alpha \rightarrow \beta$ and 18$^\circ$ for $\beta \rightarrow \alpha$.
- This rhombohedral structure becomes cubic at various values of c/a (or $\alpha$)
$c/a$ $\alpha$ Cubic Lattice $\sqrt{7}$ $60°$ Face-Centered Cubic $\sqrt{\frac32}$ $90°$ Simple Cubic $\sqrt{\frac38}$ $109.47°$ Body-Centered Cubic
- Note that $\beta$–Po and $\alpha$–Hg have the same AFLOW prototype label, A_hR1_166_a. They are generated by the same symmetry operations with different sets of parameters (
--params) specified in their corresponding CIF files. Experimentally, $\beta$–Po ($A_{i}$) has c/a near 1, or $\alpha > 90°$, while $\alpha$–Hg (A10) has c/a near 2, or $\alpha < 90°$. - Hexagonal settings of rhombohedral structures can be obtained with the option
--hex.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{\sqrt{3}}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (1a) | Po I |
References
- W. H. Beamer and C. L. Maxwell, Physical Properties of Polonium. II. X-Ray Studies and Crystal Structure, J. Chem. Phys. 17, 1293–1298 (1949), doi:10.1063/1.1747155.
Found in
- J. Donohue, The Structures of the Elements (Robert E. Krieger Publishing Company, New York, 1974).
First cited in
- M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 1, Comp. Mat. Sci. 136, S1-S828 (2017). (doi=10.1016/j.commatsci.2017.01.017)
Geometry files
Prototype Generator
aflow --proto=A_hR1_166_a --params=$a,c/a$