Close Packed Structures

In solids the particles pack together as tightly as possible, leaving the least empty space. To understand how, we treat the particles as identical hard spheres and build the structure up in stages.

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(a) Close packing in one dimension

There is only one way to close-pack spheres in one dimension — line them up in a row so that each touches its neighbours (Fig. 1.17). Each sphere then touches two others. The number of nearest neighbours of a particle is its coordination number, so here the coordination number is 2.

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(b) Close packing in two dimensions

A two-dimensional close-packed layer is built by stacking rows. This can be done in two ways.

Square close packing (AAA type). The second row is placed directly above the first, so spheres line up both horizontally and vertically. Every row is identical ("A" type), giving an AAA… pattern (Fig. 1.18a). Each sphere touches four neighbours; their centres form a square, so the 2D coordination number is 4.

Hexagonal close packing (ABAB type). The second row is placed in a staggered way, its spheres nestling in the depressions of the first row ("B" type). The third row lines up with the first (A again), giving an ABAB… pattern (Fig. 1.18b). This packs more efficiently — each sphere now touches six neighbours whose centres form a regular hexagon, so the 2D coordination number is 6. Small triangular voids appear between the spheres, some with apex pointing up and some pointing down.

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(c) Close packing in three dimensions

Real structures are three-dimensional and are made by stacking 2D layers.

From square close-packed layers. Placing each square layer directly above the one below (spheres perfectly aligned) gives an AAA… stack (Fig. 1.19). This is the simple cubic lattice, whose unit cell is the primitive cubic unit cell.

From hexagonal close-packed layers. Take a hexagonal layer A and place a second layer B in its depressions. Not all triangular voids of the first layer get covered, which creates two kinds of voids (Figs. 1.20 and 1.21):

• Where a sphere of the upper layer sits over a void of the lower layer, a tetrahedral void forms (marked T) — the centres of the four surrounding spheres form a tetrahedron.
• Where a triangular void of the second layer lies over a triangular void of the first (the two triangles pointing in opposite directions), an octahedral void forms (marked O), surrounded by six spheres.

If the number of close-packed spheres is $N$, then:

Placing the third layer offers two possibilities: …