Rhombohedral Graphite Structure: A_hR2_166_c

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Prototype : C
AFLOW prototype label : A_hR2_166_c
Strukturbericht designation : None
Pearson symbol : hR2
Space group number : 166
Space group symbol : $\mbox{R}\bar{3}\mbox{m}$
AFLOW prototype command : aflow --proto=A_hR2_166_c [--hex]
--params=
$a$,$c/a$,$x_{1}$


  • Graphite also comes in a hexagonal form, which may be either flat (A9) or buckled. When $x_{1} = 1/6$ the graphite sheets are flat. However this does not produce a change in symmetry, as it does in the hexagonal graphite structures. Note that $\alpha$–As (A_hR2_166_c, $\alpha$–As), rhombohedral graphite (A_hR2_166_c, C), and $\beta$–O (A_hR2_166_c, $\beta$–O) have the same AFLOW prototype label. They are generated by the same symmetry operations with different sets of parameters (––params) specified in their corresponding CIF files. Hexagonal settings of this structure can be obtained with the option ––hex.

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:

\[ \begin{array}{ccccccc} & & \mbox{Lattice Coordinates} & & \mbox{Cartesian Coordinates} &\mbox{Wyckoff Position} & \mbox{Atom Type} \\ \mathbf{B}_{1} & =&x_{1} \, \mathbf{a}_{1}+ x_{1} \, \mathbf{a}_{2}+ x_{1} \, \mathbf{a}_{3}& =&x_{1} \, c \, \mathbf{\hat{z}}& \left(2c\right) & \mbox{C} \\ \mathbf{B}_{2} & =&- x_{1} \, \mathbf{a}_{1}- x_{1} \, \mathbf{a}_{2}- x_{1} \, \mathbf{a}_{3}& =&- x_{1} \, c \, \mathbf{\hat{z}}& \left(2c\right) & \mbox{C} \\ \end{array} \]

References

  • H. Lipson and A. R. Stokes, The structure of graphite, Proc. R. Soc. A Math. Phys. Eng. Sci. 181, 101–105 (1942), doi:10.1098/rspa.1942.0063.

Found in

  • J. Donohue, The Structure of the Elements (Robert E. Krieger Publishing Company, Malabar, Florida, 1982)., pp. 258-260.

Geometry files


Prototype Generator

aflow --proto=A_hR2_166_c --params=

Species:

Running:

Output: