β-RuCl$_{3}$ Structure: A3B_hP8_158_d_a-001
| Prototype | Cl$_{3}$Ru |
| AFLOW prototype label | A3B_hP8_158_d_a-001 |
| ICSD | 22093 |
| Pearson symbol | hP8 |
| Space group number | 158 |
| Space group symbol | $P3c1$ |
| AFLOW prototype command |
aflow --proto=A3B_hP8_158_d_a-001
--params=$a, \allowbreak c/a, \allowbreak z_{1}, \allowbreak x_{2}, \allowbreak y_{2}, \allowbreak z_{2}$ |
- The crystal structure of both $\alpha$- and $\beta$–RuCl$_{3}$ is somewhat uncertain:
- $\alpha$–RuCl$_{3}$ has been reported in the trigonal CrCl$_{3}$ ($D0_{4}$) structure (Fletcher, 1967) and in the monoclinic AlCl$_{3}$ structure (Johnson, 2015).
- (Fletcher, 1967) states that the structure of $\beta$–RuCl$_{3}$ is consistent with the hexagonal space groups $P6_{3}cm$ #185, $P6c2$ #188, and $P6_{3}/mcm$ #193, in addition to the trigonal space group $P3c1$ #158, which we present here. We also provide the structure with space group #185: hexagonal $\beta$–RuCl$_{3}$ (A3B_hP8_185_c_a).
- Space group $P3c1$ #158 allows an arbitary choice of the origin of the $z$-axis. Here we set $z_{1} = 0$ for the ruthenium (2a) atoms.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $z_{1} \, \mathbf{a}_{3}$ | = | $c z_{1} \,\mathbf{\hat{z}}$ | (2a) | Ru I |
| $\mathbf{B_{2}}$ | = | $\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $c \left(z_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (2a) | Ru I |
| $\mathbf{B_{3}}$ | = | $x_{2} \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} + y_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{2} - y_{2}\right) \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (6d) | Cl I |
| $\mathbf{B_{4}}$ | = | $- y_{2} \, \mathbf{a}_{1}+\left(x_{2} - y_{2}\right) \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{2} - 2 y_{2}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (6d) | Cl I |
| $\mathbf{B_{5}}$ | = | $- \left(x_{2} - y_{2}\right) \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(2 x_{2} - y_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{2} \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (6d) | Cl I |
| $\mathbf{B_{6}}$ | = | $- y_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{2} + y_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \left(x_{2} - y_{2}\right) \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (6d) | Cl I |
| $\mathbf{B_{7}}$ | = | $- \left(x_{2} - y_{2}\right) \, \mathbf{a}_{1}+y_{2} \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(- x_{2} + 2 y_{2}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a x_{2} \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (6d) | Cl I |
| $\mathbf{B_{8}}$ | = | $x_{2} \, \mathbf{a}_{1}+\left(x_{2} - y_{2}\right) \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(2 x_{2} - y_{2}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a y_{2} \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (6d) | Cl I |
References
- J. M. Fletcher, W. E. Gardner, A. C. Fox, and G. Topping, X-Ray, infrared, and magnetic studies of α- and β-ruthenium trichloride, J. Chem. Soc. A pp. 1038–1045 (1967), doi:10.1039/J19670001038.
- R. D. Johnson, S. C. Williams, A. A. Haghighirad, J. Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O. Jeschke, R. Valentí, and R. Coldea, Monoclinic crystal structure of α-RuCl$_{3}$ and the zigzag antiferromagnetic ground state, Phys. Rev. B 92, 235119 (2015), doi:10.1103/PhysRevB.92.235119.
Prototype Generator
aflow --proto=A3B_hP8_158_d_a --params=$a,c/a,z_{1},x_{2},y_{2},z_{2}$