** Next:** Notations Used for Representing Close-Packed Structures
**Up:** Close-Packed Structures
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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 6_{3} 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* *c*, = = 90,
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
. 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*,
= = 90. 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 6 shows a rhombohedral lattice in which the primitive cell is defined by the rhombohedral axes

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, and
. If the lattice is rotated through 60
around [001], the hexagonal unit cell will then be centred at
and .
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.

**Copyright © 1981, 1997 International Union of
Crystallography**