$D2_{a}$ (approximate TiBe$_{12}$) Structure: A12B_hP13_191_cdei_a-001
| Prototype | Be$_{12}$Ti |
| AFLOW prototype label | A12B_hP13_191_cdei_a-001 |
| Strukturbericht designation | $D2_{a}$ |
| ICSD | 58745 |
| Pearson symbol | hP13 |
| Space group number | 191 |
| Space group symbol | $P6/mmm$ |
| AFLOW prototype command |
aflow --proto=A12B_hP13_191_cdei_a-001
--params=$a, \allowbreak c/a, \allowbreak z_{4}, \allowbreak z_{5}$ |
- The structure of TiBe$_{12}$ is not settled. (Raeuchle, 1952) described the structure as a somewhat disordered supercell containing 48 atoms with lattice constants $a = 29.44$Å and $c = 7.33$Å, but they stated that a ‘pseudo-cell’ existed with dimensions $a = 4.23$Å and $c = 7.33$Å. This pseudo-cell is described here, and was designated Strukturbericht $D2_{a}$ by Smithells (Brandes, 1992). Rauchle and Rundle suggested that the larger primitive cell was constructed from the multiple pseudo-cells, with the titanium atom alternating between the $(1a)$ and $(1b)$ Wyckoff positions.
- Other experimenters have suggested that the actual structure of TiBe$_{12}$ is tetragonal. (Jackson, 2016) presents first-principles calculations which suggest that the actual structure is the tetragonal ThMn$_{12}$ ($D2_{b}$) type.
- HfFe$_{6}$Ge$_{6}$ is the ternary form of this structure.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (1a) | Ti I |
| $\mathbf{B_{2}}$ | = | $\frac{1}{3} \, \mathbf{a}_{1}+\frac{2}{3} \, \mathbf{a}_{2}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}$ | (2c) | Be I |
| $\mathbf{B_{3}}$ | = | $\frac{2}{3} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{2}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}$ | (2c) | Be I |
| $\mathbf{B_{4}}$ | = | $\frac{1}{3} \, \mathbf{a}_{1}+\frac{2}{3} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (2d) | Be II |
| $\mathbf{B_{5}}$ | = | $\frac{2}{3} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}$ | (2d) | Be II |
| $\mathbf{B_{6}}$ | = | $z_{4} \, \mathbf{a}_{3}$ | = | $c z_{4} \,\mathbf{\hat{z}}$ | (2e) | Be III |
| $\mathbf{B_{7}}$ | = | $- z_{4} \, \mathbf{a}_{3}$ | = | $- c z_{4} \,\mathbf{\hat{z}}$ | (2e) | Be III |
| $\mathbf{B_{8}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{4}a \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (6i) | Be IV |
| $\mathbf{B_{9}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{4}a \,\mathbf{\hat{y}}+c z_{5} \,\mathbf{\hat{z}}$ | (6i) | Be IV |
| $\mathbf{B_{10}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}+z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+c z_{5} \,\mathbf{\hat{z}}$ | (6i) | Be IV |
| $\mathbf{B_{11}}$ | = | $\frac{1}{2} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{4}a \,\mathbf{\hat{y}}- c z_{5} \,\mathbf{\hat{z}}$ | (6i) | Be IV |
| $\mathbf{B_{12}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}- z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{4}a \,\mathbf{\hat{y}}- c z_{5} \,\mathbf{\hat{z}}$ | (6i) | Be IV |
| $\mathbf{B_{13}}$ | = | $\frac{1}{2} \, \mathbf{a}_{1}+\frac{1}{2} \, \mathbf{a}_{2}- z_{5} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- c z_{5} \,\mathbf{\hat{z}}$ | (6i) | Be IV |
References
- R. F. Raeuchle and R. E. Rundle, The Structure of TiBe$_{12}$, Acta Cryst. 5, 85–93 (1952), doi:10.1107/S0365110X52000186.
- E. A. Brandes and G. B. Brook, eds., Smithells Metals Reference Book (Butterworth Heinemann, Oxford, Auckland, Boston, Johannesburg, Melbourne, New Delhi, 1992), seventh edn.
Found in
- M. L. Jackson, P. A. Burr, and R. W. Grimes, Resolving the structure of TiBe$_{12}$, Acta Crystallogr. Sect. B 72, 277–280 (2016), doi:10.1107/S205252061600322X.
Prototype Generator
aflow --proto=A12B_hP13_191_cdei_a --params=$a,c/a,z_{4},z_{5}$