Buckled Graphite Structure: A_hP4_186_ab-001
- According to (Wyckoff, 1963), hexagonal graphite may be either flat, space group $P6_{3}/mmc$ #194 or buckled, space group $P6_{3}mc$ #186 (this structure).
If it is buckled, the buckling parameter is small, less than 1/20 of the ‘c’ parameter of the hexagonal unit cell.
We will assign the $A9$ Strukturbericht designation to the unbuckled structure. - Experimentally, a rhombohedral ($R\overline{3}m$ #166) graphite structure is also observed.
- There is no ICSD entry for (Hull, 1917). Instead we provide the ICSD entry for the somewhat later work of (Hassel, 1924). The two structures have similar volumes and $c/a$ values, but Hull's value of $z_{2}$=0.07143 is substantially larger than Hassel and Mark's value of 0.005. We show the former value to emphasize the buckling.
- When $z_{2}=z_{1}$, this structure is equivalent to unbuckled ($A9$) hexagonal graphite.
- Space group $P6_{3}mc$ #186 is polar and does not specify the origin of the $z$-axis. Here we chose $z_{1} = 0$ for the carbon (2a) site.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{2}a \,\mathbf{\hat{y}}\\\mathbf{a_{3}}&=&c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $z_{1} \, \mathbf{a}_{3}$ | = | $c z_{1} \,\mathbf{\hat{z}}$ | (2a) | C I |
| $\mathbf{B_{2}}$ | = | $\left(z_{1} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $c \left(z_{1} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (2a) | C I |
| $\mathbf{B_{3}}$ | = | $\frac{1}{3} \, \mathbf{a}_{1}+\frac{2}{3} \, \mathbf{a}_{2}+z_{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+c z_{2} \,\mathbf{\hat{z}}$ | (2b) | C II |
| $\mathbf{B_{4}}$ | = | $\frac{2}{3} \, \mathbf{a}_{1}+\frac{1}{3} \, \mathbf{a}_{2}+\left(z_{2} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{\sqrt{3}}{6}a \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{2}\right) \,\mathbf{\hat{z}}$ | (2b) | C II |
References
- A. W. Hull, A New Method of X-Ray Crystal Analysis, Phys. Rev. 10, 661–696 (1917), doi:10.1103/PhysRev.10.661.
- O. Hassel and H. Mark, Über die Kristallstruktur des Graphits, Z. f. Physik 25, 317–337 (1924), doi:10.1007/BF01327534.
Found in
- R. G. W. Wyckoff, Crystal Structure, vol. 1 (Interscience, New York, London, Sydney, 1963).
First cited in
- 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)
Geometry files
Prototype Generator
aflow --proto=A_hP4_186_ab --params=$a,c/a,z_{1},z_{2}$