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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 shown^2,6 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.

 

Figure 4: The minimum symmetry of a three dimensional close-packing of spheres.

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