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Symmetry and Space Group of Close-Packed Structures

The symmetry of a single close-packed layer of spheres is 6mm . It has 2- , 3- and 6- fold axes of rotation normal to its plane as shown in Fig. 3. In addition it has three symmetry planes--one perpendicular to the x -axis, one perpendicular to the y axis and the third equally inclined to x and y . When two or more layers are stacked over each other in a close-packing the resulting structure retains all the three symmetry planes and has at least 3-fold axes parallel to [00.1] through the points 000, $\frac{1}{3}$ $\frac{2}{3}$ 0 and $\frac{2}{3}$ $\frac{1}{3}$ 0 as shown in Fig. 4. Such a structure belongs to the trigonal system and has a space group P3m1 or R3m1, according as the lattice is hexagonal or rhombohedral. This represents the lowest symmetry of a close-packing of spheres comprised of a completely arbitrary periodic stacking sequence of close-packed layers. If the arbitrariness in stacking successive layers in the unit cell is limited then higher symmetries can also result. It can be shown2,6 that it is possible to have three additional symmetry elements, namely, a centre of symmetry ($\overline{1}$, a mirror plane perpendicular to [00.1], and a screw axis 63. It was shown by Belov7 that consistent combinations of these symmetry elements can give rise to only eight possible space groups:

P3m1, $P\overline{3}m1$, $P\overline{6}m2$, P63mc
P63/mmc, R3m, $R\overline{3}m$ and $F\overline{4}3m$

Of these eight space groups, $F\overline{4}3m$ is the only one that is cubic and corresponds to the cubic close-packed structure ABCABC... In compounds, the presence of the other atoms occupying the voids further restricts the possible space groups.


 
Figure 3: Symmetry axes of a single close-packed layer of spheres.
\begin{figure}
\includegraphics {fig3.ps}
\end{figure}


 
Figure 4: The minimum symmetry of a three dimensional close-packing of spheres.
\begin{figure}
\includegraphics {fig4.ps}
\end{figure}


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Next: Possible Lattice Types Up: Close-Packed Structures Previous: Voids in a Close-Packing

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