5.1 Introduction

5.2 Atom Packing in Crystals

5.3 Close-Packed Structures and Crystal Energies

An example of how we might pack atoms in a plane is shown in Figure 5.1; it is the arrangement in which the reds are set up on a billiard table before starting a game of snooker. The balls are packed in a triangular fashion so as to take up the least possible space on the table. This type of plane is thus called a close-packed plane, and contains three close-packed directions; they are the directions along which the balls touch. The figure shows only a small region of close-packed plane—if we had more reds we could extend the plane sideways and could, if we wished, fill the whole billiard table. The important thing to notice is the way in which the balls are packed in a regularly repeating two-dimensional pattern.

How could we add a second layer of atoms to our close-packed plane? As Figure 5.1 shows, the depressions where the atoms meet are ideal “seats” for the next layer of atoms. By dropping atoms into alternate seats, we can generate a second close-packed plane lying on top of the original one and having an identical packing pattern. Then a third layer can be added, and a fourth, and so on until we have made a sizeable piece of crystal—with, this time, a regularly repeating pattern of atoms in three dimensions. The particular structure we have produced is one in which the atoms take up the least volume and is therefore called a close-packed structure. The atoms in many solid metals are packed in this way.

There is a complication to this apparently simple story. There are two alternative and different sequences in which we can stack the close-packed planes on top of one another. If we follow the stacking sequence in Figure 5.1 rather more closely, we see that, by the time we have reached the fourth atomic plane, we are placing the atoms directly above the original atoms (although, naturally, separated from them by the two interleaving planes of atoms). We then carry on adding atoms as before, generating an ABCABC… sequence.

In Figure 5.2 we show the alternative way of stacking, in which the atoms in the third plane are now directly above those in the first layer. This gives an ABAB … sequence. These two different stacking sequences give two different three-dimensional packing structures—face-centered cubic (f.c.c.) and close–packed hexagonal (c.p.h.) respectively. Many common metals (e.g., Al, Cu and Ni) have the f.c.c. structure and many others (e.g., Mg, Zn, and Ti) have the c.p.h. structure.

Why should Al choose to be f.c.c. while Mg chooses to be c.p.h.? The answer to this is that materials choose the crystal structure that gives minimum energy. This structure may not necessarily be close packed or, indeed, very simple geometrically—although to be a crystal it must still have some sort of three–dimensional repeating pattern.

The difference in energy between alternative structures is often slight. Because of this, the crystal structure that gives the minimum energy at one temperature may not do so at another. Thus tin changes its crystal structure if it is cooled enough; and, incidentally, becomes much more brittle in the process (it is said this caused the tin-alloy coat buttons of Napoleon’s army to fall apart during the harsh Russian winter, and the soldered cans of paraffin on Scott’s South Pole expedition to leak). Cobalt changes its structure at 450ºC, transforming from a c.p.h. structure at lower temperatures to an f.c.c. structure at higher temperatures. More important, pure iron transforms from a b.c.c. structure (defined in the following) to one that is f.c.c. at 911ºC, a process which is important in the heat treatment of steels.

5.4 Crystallography

Let us illustrate the crystallographic approach in the case of f.c.c. Figure 5.3 shows that the atom centers in f.c.c. can be placed at the corners of a cube and in the centers of the cube faces. The cube, of course, has no physical significance but is merely a constructional device. It is called a unit cell. If we look along the cube diagonal, we see the view shown in Figure 5.3 (top center): a triangular pattern which, with a little effort, can be seen to be bits of close-packed planes stacked in an ABCABC sequence.

This unit–cell visualization of the atomic positions is thus exactly equivalent to our earlier approach based on stacking of close-packed planes, but is much more powerful as a descriptive aid. For example, we can see how our complete f.c.c. crystal is built up by attaching further unit cells to the first one (like assembling a set of children’s building cubes) so as to fill space without leaving awkward gaps—something you cannot so easily do with 5-sided shapes (in a plane) or 7-sided shapes (in three dimensions).

Beyond this, inspection of the unit cell reveals planes in which the atoms are packed other than in a close-packed way. On the “cube” faces the atoms are packed in a square array, and on the cube-diagonal planes in separated rows, as shown in Figure 5.3. Obviously, properties like the shear modulus might well be different for close-packed planes and cube planes, because the number of bonds attaching them per unit area is different. This is one of the reasons that it is important to have a method of describing various planar packing arrangements.

Let us now look at the c.p.h. unit cell as shown in Figure 5.4. A view looking down the vertical axis reveals the ABA stacking of close-packed planes. We build up our c.p.h. crystal by adding hexagonal building blocks to one another: hexagonal blocks also stack so that they fill space. Here, again, we can use the unit cell concept to “open up” views of the various types of planes.

5.5 Plane Indices

We could make scale drawings of the many types of planes that we see in unit cells; but the concept of a unit cell also allows us to describe any plane by a set of numbers called Miller indices. The two examples given in Figure 5.5 should enable you to find the Miller index of any plane in a cubic unit cell, although they take a little getting used to. The indices (for a plane) are the reciprocals of the intercepts the plane makes with the three axes, reduced to the smallest integers (reciprocals are used simply to avoid infinities when planes are parallel to axes).