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Possible Lattice Types

Close-packing of equal spheres can belong to the trigonal, hexagonal or cubic crystal systems. When the structure has the minimum symmetry discussed earlier it belongs to the trigonal system. When it has a 6₃ axis of symmetry it belongs to the hexagonal system. Structures belonging to the hexagonal system necessarily have a hexagonal lattice, i.e. a lattice in which we can choose a primitive unit cell with a = b $\neq$ c, $\alpha$ = $\beta$ = 90 $^{\circ}$ , in Fig. 5. It should be noted that there are two spheres associated with each lattice point in the hcp structure, one at 000 and the other at $\frac{1}{3}$ $\frac{2}{3}$ $\frac{1}{2}$ . Structures belonging to the trigonal system can have either a hexagonal or a rhombohedral lattice. By a rhombohedral lattice is meant a lattice in which we can choose a primitive unit cell with a = b = c, $\alpha$ = $\beta$ = $\gamma$ $\neq$ 90 $^{\circ}$ . Both types of lattices can be referred to either hexagonal or rhombohedral axes, the unit cell being non-primitive when a hexagonal lattice is referred to rhombohedral axes or vice versa.

**Figure 5:** The primitive unit cell of the hcp structure.
$\begin{figure} \includegraphics {fig5.ps} \end{figure}$

Figure 6 shows a rhombohedral lattice in which the primitive cell is defined by the rhombohedral axes a₁, a₂, a₃; but a non-primitive hexagonal unit cell can be chosen by adopting the axes A₁, A₂, C. The latter has lattice points at 000., $\frac{2}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ and $\frac{1}{3}$ $\frac{2}{3}$ $\frac{2}{3}$ . In the special case of the close-packing ABCABC.... (with the ideal h/a ratio of 0.8165) the primitive rhombohedral lattice has $\alpha$ = $\beta$ = $\gamma$ = 60 $^{\circ}$ , which enhances the symmetry to $F\overline{4}3m$ and enables the choice of a face- centred cubic unit cell. The relationship between the fcc and the primitive rhombohedral unit cell is shown in Fig. 7. The three-fold axis of the rhombohedral unit cell coincides with one of the $\langle$ 111 $\rangle$ directions of the cubic unit cell. The close-packed layers are thus parallel to the $\{$ 111 $\}$ planes in the cubic close-packing.

**Figure 6:** A rhombohedral lattice (a₁, a₂, a₃) referred to hexagonal axes (A₁, A₂, C) (After M. J. Buerger, X-ray crystallography, Wiley: New York 1953).
$\begin{figure} \includegraphics {fig6.ps} \end{figure}$

**Figure 7:** The relationship between the fcc and the primitive rhombohedral unit cell of the ccp structure.
$\begin{figure} \includegraphics {fig7.ps} \end{figure}$

In close-packed structures, it is generally convenient to refer both hexagonal and rhombohedral lattices to hexagonal axes. The projection of the hexagonal lattice on the (001) plane is shown in Fig. 8. The axes, x, y define the smallest hexagonal unit cell, the z axis being normal to the plane of the paper; the hexagonal unit cell is primitive with all the lattice points at 000. Figure 9 depicts the projection of a rhombohedral lattice on the (00.1) plane. The full lines Ox_h, Oy_h represent the hexagonal axes and the three dotted lines represent rhombohedral axes. It is evident from the figure that the hexagonal unit cell of a rhombohedral lattice is non-primitive with lattice points at 000, $\frac{2}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ and $\frac{1}{3}$ $\frac{2}{3}$ $\frac{2}{3}$ . If the lattice is rotated through 60 ${\circ}$ around [001], the hexagonal unit cell will then be centred at $\frac{1}{3}$ $\frac{2}{3}$ $\frac{1}{3}$ and $\frac{2}{3}$ $\frac{1}{3}$ $\frac{2}{3}$ . These two settings of the rhombohedral lattice are called `obverse` and `reverse` settings. They are indistinguishable by X-ray methods since the two are crystallographically equivalent: they represent twin arrangements when both of them occur in the same single crystal.

**Figure 8:** The projection of the hexagonal lattice on the (0001)-plane. Shows different ways of choosing hexagonal axes (after International Tables for Crystallography, Kynoch Press: Birmingham, 1952).
$\begin{figure} \includegraphics {fig8.ps} \end{figure}$

**Figure 9:** Projection of a rhombohedral lattice (obverse setting). Shows the choice of hexagonal ( $\rightarrow$ ) and rhombohedral (- - - -) axes (after International Tables for Crystallography Vol. I, Kynoch Press: Birmingham, 1952).
$\begin{figure} \includegraphics {fig9.ps} \end{figure}$

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