Encyclopedia of Crystallographic Prototypes

AFLOW Prototype: AB_hP12_186_a2b_a2b

  • 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)
  • D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comp. Mat. Sci. 161, S1-S1011 (2019). (doi=10.1016/j.commatsci.2018.10.043)
  • 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, Comp. Mat. Sci. 199, 110450 (2021). (doi=10.1016/j.commatsci.2021.110450)

Moissanite–6H SiC ($B6$) Structure: AB_hP12_186_a2b_a2b

Picture of Structure; Click for Big Picture
Prototype : SiC
AFLOW prototype label : AB_hP12_186_a2b_a2b
Strukturbericht designation : $B6$
Pearson symbol : hP12
Space group number : 186
Space group symbol : $\text{P6}_{3}\text{mc}$
AFLOW prototype command : aflow --proto=AB_hP12_186_a2b_a2b
--params=
$a$,$c/a$,$z_{1}$,$z_{2}$,$z_{3}$,$z_{4}$,$z_{5}$,$z_{6}$


  • This is an alternate stacking (ABCACB) for tetrahedral structures. Compare this to zincblende (ABCABC), moissanite–4H (ABAC), and wurtzite (ABABAB). The 6H refers to the fact that there are 6 CSi dimers in a hexagonal unit cell. Zincblende is denoted 3C, and wurtzite is 2H. Without loss of generality, we can take any of the $z_{i}$ to be zero. In the pictures here we take $z_{1} = 0$.

Hexagonal primitive vectors:

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

Basis vectors:

\[ \begin{array}{ccccccc} & & \text{Lattice Coordinates} & & \text{Cartesian Coordinates} &\text{Wyckoff Position} & \text{Atom Type} \\ \mathbf{B}_{1}& = &z_{1} \, \mathbf{a}_{3}& = &z_{1} \, c \, \mathbf{\hat{z}}& \left(2a\right) & \text{C I} \\ \mathbf{B}_{2}& = &\left(\frac12 + z_{1}\right) \, \mathbf{a}_{3}& = &\left(\frac12 + z_{1}\right) \, c \, \mathbf{\hat{z}}& \left(2a\right) & \text{C I} \\ \mathbf{B}_{3}& = &z_{2} \, \mathbf{a}_{3}& = &z_{2} \, c \, \mathbf{\hat{z}}& \left(2a\right) & \text{Si I} \\ \mathbf{B}_{4}& = &\left(\frac12 + z_{2}\right) \, \mathbf{a}_{3}& = &\left(\frac12 + z_{2}\right) \, c \, \mathbf{\hat{z}}& \left(2a\right) & \text{Si I} \\ \mathbf{B}_{5}& = &\frac13 \, \mathbf{a}_{1}+ \frac23 \, \mathbf{a}_{2}+ z_{3} \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ z_{3} \, c \, \mathbf{\hat{z}}& \left(2b\right) & \text{C II} \\ \mathbf{B}_{6}& = &\frac23 \, \mathbf{a}_{1}+ \frac13 \, \mathbf{a}_{2}+ \left(\frac12 + z_{3}\right) \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{3}\right) \, c \, \mathbf{\hat{z}}& \left(2b\right) & \text{C II} \\ \mathbf{B}_{7}& = &\frac13 \, \mathbf{a}_{1}+ \frac23 \, \mathbf{a}_{2}+ z_{4} \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ z_{4} \, c \, \mathbf{\hat{z}}& \left(2b\right) & \text{C III} \\ \mathbf{B}_{8}& = &\frac23 \, \mathbf{a}_{1}+ \frac13 \, \mathbf{a}_{2}+ \left(\frac12 + z_{4}\right) \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{4}\right) \, c \, \mathbf{\hat{z}}& \left(2b\right) & \text{C III} \\ \mathbf{B}_{9}& = &\frac13 \, \mathbf{a}_{1}+ \frac23 \, \mathbf{a}_{2}+ z_{5} \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ z_{5} \, c \, \mathbf{\hat{z}}& \left(2b\right) & \text{Si II} \\ \mathbf{B}_{10}& = &\frac23 \, \mathbf{a}_{1}+ \frac13 \, \mathbf{a}_{2}+ \left(\frac12 + z_{5}\right) \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{5}\right) \, c \, \mathbf{\hat{z}}& \left(2b\right) & \text{Si II} \\ \mathbf{B}_{11}& = &\frac13 \, \mathbf{a}_{1}+ \frac23 \, \mathbf{a}_{2}+ z_{6} \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}+ \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ z_{6} \, c \, \mathbf{\hat{z}}& \left(2b\right) & \text{Si III} \\ \mathbf{B}_{12}& = &\frac23 \, \mathbf{a}_{1}+ \frac13 \, \mathbf{a}_{2}+ \left(\frac12 + z_{6}\right) \, \mathbf{a}_{3}& = &\frac12 \, a \, \mathbf{\hat{x}}- \frac1{2\sqrt3} \, a \, \mathbf{\hat{y}}+ \left(\frac12 + z_{6}\right) \, c \, \mathbf{\hat{z}}& \left(2b\right) & \text{Si III} \\ \end{array} \]

References

  • A. Bauer, P. Reischauer, J. Kräusslich, N. Schell, W. Matz, and K. Goetz, Structure refinement of the silicon carbide polytypes 4H and 6H: unambiguous determination of the refinement parameters, Acta Crystallogr. Sect. A 57, 60–67 (2001), doi:10.1107/S0108767300012915.

Geometry files


Prototype Generator

aflow --proto=AB_hP12_186_a2b_a2b --params=

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