B13C2 B4C ($D1_{g}$) Structure : A13B2_hR15_166_b2h_c

Picture of Structure; Click for Big Picture
Prototype : B13C2
AFLOW prototype label : A13B2_hR15_166_b2h_c
Strukturbericht designation : $D1_{g}$
Pearson symbol : hR15
Space group number : 166
Space group symbol : $R\bar{3}m$
AFLOW prototype command : aflow --proto=A13B2_hR15_166_b2h_c
--params=
$a$,$c/a$,$x_{2}$,$x_{3}$,$z_{3}$,$x_{4}$,$z_{4}$


Other compounds with this structure

  • B1–xCx ($0.0813P2, B4Si, B6O

  • This structure has a rather complicated history: \begin{itemize} \item{It is difficult to determine the species of atoms at a given site because of the similar electronic and nuclear cross sections of 11B and 12C (Domnich, 2011).} \item{Early investigations (Clark, 1943) assumed the structure was B4C, 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 than 1 Å. This error is repeated in (Brandes, 1992) and (Wykcoff, 1964).]} \item{In reality, concentrations can range from 8–20% carbon (Domnich, 2011).
  • (Larson, 1986) states that the ($1b$) site in B13C2 is boron, and as the structure becomes more carbon–rich the carbon atoms replace boron in the icosahedra. We follow this and use the structure determined by (Will, 1976) as our reference.} \item{(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).} \end{itemize}

Rhombohedral primitive vectors:

\[ \begin{array}{ccc} \mathbf{a}_1 & = & ~ \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac{1}{\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & - \frac12 \, a \, \mathbf{\hat{x}} - \frac{1}{2\sqrt{3}} \, a \, \mathbf{\hat{y}} + \frac13 \, c \, \mathbf{\hat{z}} \\ \end{array} \]

Basis vectors:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & = & \frac{1}{2} \, \mathbf{a}_{1} + \frac{1}{2} \, \mathbf{a}_{2} + \frac{1}{2} \, \mathbf{a}_{3} & = & \frac{1}{2}c \, \mathbf{\hat{z}} & \left(1b\right) & \mbox{B I} \\ \mathbf{B}_{2} & = & x_{2} \, \mathbf{a}_{1} + x_{2} \, \mathbf{a}_{2} + x_{2} \, \mathbf{a}_{3} & = & x_{2}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{C} \\ \mathbf{B}_{3} & = & -x_{2} \, \mathbf{a}_{1}-x_{2} \, \mathbf{a}_{2}-x_{2} \, \mathbf{a}_{3} & = & -x_{2}c \, \mathbf{\hat{z}} & \left(2c\right) & \mbox{C} \\ \mathbf{B}_{4} & = & x_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + z_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B II} \\ \mathbf{B}_{5} & = & z_{3} \, \mathbf{a}_{1} + x_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B II} \\ \mathbf{B}_{6} & = & x_{3} \, \mathbf{a}_{1} + z_{3} \, \mathbf{a}_{2} + x_{3} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B II} \\ \mathbf{B}_{7} & = & -z_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{y}}-\left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B II} \\ \mathbf{B}_{8} & = & -x_{3} \, \mathbf{a}_{1}-x_{3} \, \mathbf{a}_{2}-z_{3} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{3}+z_{3}\right)a \, \mathbf{\hat{y}}-\left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B II} \\ \mathbf{B}_{9} & = & -x_{3} \, \mathbf{a}_{1}-z_{3} \, \mathbf{a}_{2}-x_{3} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(x_{3}-z_{3}\right)a \, \mathbf{\hat{y}}-\left(\frac{2}{3}x_{3}+\frac{1}{3}z_{3}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B II} \\ \mathbf{B}_{10} & = & x_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + z_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{4}-z_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{4}-z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{4}+\frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{11} & = & z_{4} \, \mathbf{a}_{1} + x_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{4}+z_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(x_{4}-z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{4}+\frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{12} & = & x_{4} \, \mathbf{a}_{1} + z_{4} \, \mathbf{a}_{2} + x_{4} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(-x_{4}+z_{4}\right)a \, \mathbf{\hat{y}} + \left(\frac{2}{3}x_{4}+\frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{13} & = & -z_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(x_{4}-z_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{4}+z_{4}\right)a \, \mathbf{\hat{y}}-\left(\frac{2}{3}x_{4}+\frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{14} & = & -x_{4} \, \mathbf{a}_{1}-x_{4} \, \mathbf{a}_{2}-z_{4} \, \mathbf{a}_{3} & = & \frac{1}{2}\left(-x_{4}+z_{4}\right)a \, \mathbf{\hat{x}} + \frac{1}{2\sqrt{3}}\left(-x_{4}+z_{4}\right)a \, \mathbf{\hat{y}}-\left(\frac{2}{3}x_{4}+\frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B III} \\ \mathbf{B}_{15} & = & -x_{4} \, \mathbf{a}_{1}-z_{4} \, \mathbf{a}_{2}-x_{4} \, \mathbf{a}_{3} & = & \frac{1}{\sqrt{3}}\left(x_{4}-z_{4}\right)a \, \mathbf{\hat{y}}-\left(\frac{2}{3}x_{4}+\frac{1}{3}z_{4}\right)c \, \mathbf{\hat{z}} & \left(6h\right) & \mbox{B III} \\ \end{array} \]

References

  • G. Will and K. H. Kossobutzki, An X–ray structure analysis of boron carbide, B13C2, 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. errata, H. K. Clark and J. L. Hoard, J. Am. Chem. Soc. 67, 2279 (1945).
  • E. A. Brandes and G. B. Brook, eds., Smithells Metals Reference Book (Butterworth Heinemann, Oxford, Auckland, Boston, Johannesburg, Melbourne, New Delhi, 1992), seventh edn. Strukturbericht designations start on page 6–36 (163 in PDF), see table 6.3 on page 6–63 (190).
  • 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 B4C, AIP\ Conf. Proc. 140, 109–113 (1986), doi:10.1063/1.35619.
  • R. Lazzari, N. Vast, J. M. Besson, S. Baroni, and A. Dal Corso, Atomic Structure and Vibrational Properties of Icosahedral B4C Boron Carbide, Phys. Rev. Lett. 83, 3230–3233 (1999), doi:10.1103/PhysRevLett.83.3230. erratum Phys. Rev. Lett. 85, 4194 (2000).

Geometry files


Prototype Generator

aflow --proto=A13B2_hR15_166_b2h_c --params=

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