B$_{13}$C$_{2}$ B$_{4}$C
($D1_{g}$) Structure: A13B2_hR15_166_a2h_c-001
| Prototype | B$_{13}$C$_{2}$ |
| AFLOW prototype label | A13B2_hR15_166_a2h_c-001 |
| Strukturbericht designation | $D1_{g}$ |
| ICSD | 612566 |
| Pearson symbol | hR15 |
| Space group number | 166 |
| Space group symbol | $R\overline{3}m$ |
| AFLOW prototype command |
aflow --proto=A13B2_hR15_166_a2h_c-001
--params=$a, \allowbreak c/a, \allowbreak x_{2}, \allowbreak x_{3}, \allowbreak z_{3}, \allowbreak x_{4}, \allowbreak z_{4}$ |
Other compounds with this structure
B$_{1-x}$C$_{x}$, B$_{13}$P$_{2}$, B$_{4}$Si, B$_{6}$O
- This structure has a rather complicated history:
- It is difficult to determine the species of atoms at a given site because of the similar electronic and nuclear cross sections of $^{11}$B and $^{12}$C (Domnich, 2011).
- Early investigations (Clark, 1943) assumed the structure was B$_{4}$C, with the extra carbon atom replacing the boron on the (1b) site. [Note that Clark has an error in the coordinates of one set of boron atoms, giving a boron-boron distance of less that 1Å. This error is repeated in (Brandes, 1992) and (Wykcoff, 1964).]
- In reality, concentrations can range from 8-20% carbon (Domnich, 2011).
- (Larson, 1986) states that in B$_{13}$C$_{2}$ the (1b) site is boron and as the structure becomes more carbon rich the carbon atoms replace borons in the iscosahedra. We follow this and use the structure determined by (Will, 1976) as our reference.
- (Lazzari, 1999) states that excess electrons go on the
polar
sites of the icosahedron, i.e. the sites closest to the carbon atoms on the chains (the B III atoms in our notation). - The original version of this page (Hicks, 2021) listed $a = 6.617$\AA{} rather than $a = 5.617$\AA. This has now been corrected.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{\sqrt{3}}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+\frac{1}{3}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (1a) | B I |
| $\mathbf{B_{2}}$ | = | $x_{2} \, \mathbf{a}_{1}+x_{2} \, \mathbf{a}_{2}+x_{2} \, \mathbf{a}_{3}$ | = | $c x_{2} \,\mathbf{\hat{z}}$ | (2c) | C I |
| $\mathbf{B_{3}}$ | = | $- x_{2} \, \mathbf{a}_{1}- x_{2} \, \mathbf{a}_{2}- x_{2} \, \mathbf{a}_{3}$ | = | $- c x_{2} \,\mathbf{\hat{z}}$ | (2c) | C I |
| $\mathbf{B_{4}}$ | = | $x_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+z_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{5}}$ | = | $z_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{6}}$ | = | $x_{3} \, \mathbf{a}_{1}+z_{3} \, \mathbf{a}_{2}+x_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{\sqrt{3}}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{7}}$ | = | $- z_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{8}}$ | = | $- x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- z_{3} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{9}}$ | = | $- x_{3} \, \mathbf{a}_{1}- z_{3} \, \mathbf{a}_{2}- x_{3} \, \mathbf{a}_{3}$ | = | $\frac{1}{\sqrt{3}}a \left(x_{3} - z_{3}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{3} + z_{3}\right) \,\mathbf{\hat{z}}$ | (6h) | B II |
| $\mathbf{B_{10}}$ | = | $x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+z_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6h) | B III |
| $\mathbf{B_{11}}$ | = | $z_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6h) | B III |
| $\mathbf{B_{12}}$ | = | $x_{4} \, \mathbf{a}_{1}+z_{4} \, \mathbf{a}_{2}+x_{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{\sqrt{3}}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{y}}+\frac{1}{3}c \left(2 x_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6h) | B III |
| $\mathbf{B_{13}}$ | = | $- z_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6h) | B III |
| $\mathbf{B_{14}}$ | = | $- x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}- z_{4} \, \mathbf{a}_{3}$ | = | $- \frac{1}{2}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6h) | B III |
| $\mathbf{B_{15}}$ | = | $- x_{4} \, \mathbf{a}_{1}- z_{4} \, \mathbf{a}_{2}- x_{4} \, \mathbf{a}_{3}$ | = | $\frac{1}{\sqrt{3}}a \left(x_{4} - z_{4}\right) \,\mathbf{\hat{y}}- \frac{1}{3}c \left(2 x_{4} + z_{4}\right) \,\mathbf{\hat{z}}$ | (6h) | B III |
References
- G. Will and K. H. Kossobutzki, An X-ray structure analysis of boron carbide, B$_{13}$C$_{2}$, J. Less-Common Met. 44, 87–97 (1976), doi:10.1016/0022-5088(76)90120-X.
- V. Domnich, S. Reynaud, R. A. Haber, and M. Chhowalla, Boron Carbide: Structure, Properties, and Stability under Stress, J. Am. Ceram. Soc. 94, 3605–3628 (2011), doi:10.1111/j.1551-2916.2011.04865.x.
- H. K. Clark and J. L. Hoard, The Crystal Structure of Boron Carbide}, J. Am. Chem. Soc. 65, 2115–2119 (1943), doi:10.1021/ja01251a026. {\em errata}, H. K. Clark and J. L. Hoard, J. Am. Chem. Soc. 67, 2279 (1945). \bibitem{brandes92:SmithellsE. A. Brandes and G. B. Brook, eds., Smithells Metals Reference Book} (Butterworth Heinemann, Oxford, Auckland, Boston, Johannesburg, Melbourne, New Delhi, 1992), seventh edn.\bibAnnoteFile{brandes92:Smithells
- R. W. G. Wyckoff, Crystal Structures, vol. 2 (Interscience (John Wiley & Sons), New York, London, Sydney, 1964), second edn.
- A. C. Larson, Comments concerning the crystal structure of B$_{4}$C, AIP Conf. Proc. 140, 109–113 (1986), doi:10.1063/1.35619.
- R. Lazzari, N. Vast, J. M. Besson, S. Baroni, and A. D. Corso, Atomic Structure and Vibrational Properties of Icosahedral B$_{4}$C Boron Carbide}, Phys. Rev. Lett. 83, 3230–3233 (1999), doi:10.1103/PhysRevLett.83.3230. {\em erratum Phys. Rev. Lett. 85, 4194 (2000).
- D. Hicks, M. J.Mehl, M. Esters, C. Oses, O. Levy, G. L. W. Hart, C. Toher, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 3, Comput. Mater. Sci. 199, 110450 (2021), doi:10.1016/j.commatsci.2021.110450.
Prototype Generator
aflow --proto=A13B2_hR15_166_a2h_c --params=$a,c/a,x_{2},x_{3},z_{3},x_{4},z_{4}$