Chapters / Chapter 1 · Solid state / Topic 6 of 14

To build a crystal in three dimensions, close-packed layers are stacked one on top of the other in a regular way. Consider a sphere S in the first layer: spheres in the next layer nestle in the notches around it. In the second layer there is a notch over S, but there are also notches over other positions. The third layer may repeat the arrangement of the first layer, or it may not.

• If the third layer repeats the first, there are only two kinds of layer, A and B, and the pattern is ABAB… This is the hexagonal close-packed (hcp) structure.
• If the third layer does not repeat the first, it is denoted C, and the pattern is ABCABC… This is the cubic close-packed (ccp) structure, also called face-centred cubic (fcc).

Hexagonal close packing

In ABA stacking, the seven spheres in both the first layer (A) and the third layer occupy simple hexagonal lattice points, while the three spheres of the second layer (B) are placed at the corners of an equilateral triangle midway between the two parallel hexagonal planes, so that the nearest environment of every sphere becomes identical. The height of the unit cell, c, is the distance between the first and third layers; a is the length of the side of the hexagon.

A sphere is hexagonally surrounded by six spheres in its own plane, and touches three spheres below the plane and three above it. Coordination number = 6 + 3 + 3 = 12.

Packing efficiency 74%, coordination number 12, stacking ABAB and so on.

Cubic close packing

In ccp / fcc the stacking sequence is ABCABC… The structure has cubic symmetry. In addition to the eight spheres at the eight corners of the cube, there is one sphere at the centre of each of the six faces, hence the name face-centred cubic. Each sphere is surrounded by 12 other nearest spheres, so the coordination number is 12. Packing efficiency is 74%.

Common features of hcp and ccp

• Coordination number is 12 in both.
• Packing efficiency is 74% in both, 74% of the total volume of the unit cell is occupied by the spheres.

Body-centred cubic, for comparison

Another common arrangement in a few metals is body-centred cubic (bcc). The second layer sits in the notches of the first and the pattern repeats ABAB…, but the coordination number is 8, compared with 12 in a close-packed pattern.