# Close-Packed Structures

A brief but comprehensive reference on close-packing in materials science, which goes beyond the scope of this learning unit, is the International Union of Crystallography’s pamphlet “Close-Packed Structures” by P. Krishna and D. Pandey. This can be freely downloaded in HTML or PDF format.

## Introduction

Fig. 1: Arrangement of billiard balls before a game

Johannes Kepler (1571–1630) was intrigued about the structure and symmetrical nature of snowflakes, and this prompted him to speculate about the structure of matter. In his essay “On the Six-Cornered Snowflake”, he discussed the packing of spheres and proposed what has come to be known as the Kepler conjecture, that the most efficient ways of packing spheres in three dimensions are the hexagonal close-packed and cubic close-packed structures that we are about to consider. The same close-packing arrangements were considered by Thomas Harriot relating to the packing of cannonballs and we can also see them in the packing of approximately spherical fruit such as oranges in the shops.

An attempt to arrange spherical or circular objects in two dimensions leads to the conclusion that the most efficient way of filling space has hexagonal symmetry. A good example of this is the arrangement of billiard balls before a game (Fig. 1).

## Hexagonal close-packing (HCP)

Fig. 2a: A single close-packed layer

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We now consider close-packed structures in three dimensions.

We first consider a single layer of close-packed spheres similar to the billiard-ball arrangement that we saw above (Fig. 2a). What happens if we try to put another layer on top? If we put the next layer directly above the first, this will not be a particularly stable arrangement – a small nudge and the spheres will fall into the approximately triangular gaps between the spheres of the first layer. We therefore expect that the spheres of the second layer will sit in these interstices. However, if we look closely, we see that there are two types of interstices – those with the point of the triangle pointing to the right (marked in blue here) and those in which the point is to the left (marked in pink here). It is not possible for the second layer of spheres to occupy both types of interstices simultaneously since there simply isn’t sufficient space, so they must “choose” one set or the other.

Fig. 2b: Second (B) layer on top of original (A) layer

Here, in Fig. 2b, the second layer (B) is occupying the “blue” interstices in the first layer (A).

Fig. 2c: Hexagonal close-packed structure consisting of two alternating layers

In the arrangement known as hexagonal close-packing, or HCP for short, the third layer forms directly above the first layer, and the stacking proceeds ABABABA... The arrangement has hexagonal symmetry about an axis perpendicular to the ABABAB planes (Fig. 2c).

Fig. 2d: The unit cell of the HCP structure

Fig. 2d shows the conventional unit cell, or basic repeating unit, of the HCP structure – if translated and reproduced through space, this will give the overall structure. (Note that the “B” atom, although it appears in a different colour to aid understanding of the stacking sequence, is chemically identical to the “A” atoms.) In the top right of Fig. 2d, the atoms have been shown a little smaller so that the structure of the unit cell can be more easily understood, and in the bottom right diagram, we see the cell as it would appear when viewed down an axis perpendicular to the close-packed planes. In this diagram it is no longer obvious that the B atom is at a different height from the A atoms, and when a two-dimensional plan of a crystal structure is drawn in this way, the height is indicated as a fraction of the height of the cell, as shown at bottom right. Where no co-ordinate is present, this indicates that the atoms in question are at a height of 0 (and also at 1 cell height, since the structure repeats).

## Introduction to the unit cell concept

Fig. 3a illustrates the concept of the “unit cell”. If we have a periodic (repeating) structure, we can represent the whole structure by a small “building-block” that repeats through space. The building shown in the picture is made up of periodic units, horizontally and vertically. We can identify one of these units (shown in yellow).

Fig. 3a: The concept of a unit cell as a repeating unit making up a structure

Then, if we repeat the unit through space, we obtain something resembling the original building (although without the edges!), Fig. 3b.

Fig. 3b: A repeating pattern built up from multiple unit cells translated through space
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## Cubic close-packing (CCP)

Fig. 4a: A third layer (C) placed on top of the layer B in Fig. 3b

There is an alternative way of stacking the spheres that also produces a close-packed structure (Fig. 4a). The A and B layers are formed in the same way as in the HCP arrangement (Fig. 3b), but the third layer forms above the pink-coloured interstices of the A layer (see Fig. 2a), instead of above the atoms of the A layer. The third layer is therefore no longer identical to the A layer, and is given a different label, C.

Fig. 4b: Upper A-layer

The fourth layer forms above the A layer, and the sequence repeats, ABCABCA (Fig. 4b.)

Note that in this ABCABCA arrangement, each layer occupies the same type of interstice – here the right-pointing type of interstice, marked in blue in the top left diagram – in the layer below. In contrast, in the ABABABA stacking in the HCP structure, the B layer occupies the right-pointing interstices of the A layer, and the A layer occupies the left-pointing interstices of the B layer.

Fig. 4c: Stacking sequence to give the cubic close-packed structure

The ABCABCA arrangement (Fig. 4c) is known as cubic close-packed (CCP).

Fig. 4d: The unit cell of the CCP structure

It is not immediately obvious from examining the layers, with their hexagonal symmetry, where this name comes from. However, if we remove all but one of the A atoms in the top layer, the A atom immediately below it and triangles of 6 atoms in each of the B and C layers, we are left with a cubic arrangement (Fig. 4d).

Fig. 4e: The unit cell of the CCP structure (rotated)
Fig. 4f: The unit cell of the CCP structure (small atoms)

The cubic shape can be seen more easily if the arrangement is rotated (Fig. 4e) and the atoms shown smaller (Fig. 4f). The structure consists of atoms at each corner of a cube, together with atoms at the centres of each of the cube faces, giving rise to an alternative name, face-centred cubic (FCC).

Fig. 4g: The unit cell of the CCP structure (projection)

A plan of the structure, with heights marked as fractional co-ordinates, is shown in Fig. 4g.

Rotate the model in the window below, or try the preset views using the buttons.

## Comparison of HCP and CCP

Fig. 5: Overview of close-packed structures

To recap, the top part of Fig. 5 shows the first layer (A) with the two different types of interstices. At bottom left we see the B layer over the top, occupying one of the sets of interstices (here the right-pointing ones). The HCP structure is made up of further repeats of this A-B stacking. At bottom right, on the other hand, we see the C layer of the CCP structure; this sits on the right-pointing interstices of the B layer. The A layer comes above this and occupies the right-pointing interstices of the C layer.

The HCP structure has hexagonal symmetry but the CCP structure does not: try to convince yourself of this by looking at the diagrams!

## More on the CCP structure

Fig. 6a: Structure made of several unit cells of CCP

In the CCP structure, the unit cell has cubic symmetry, so the larger-scale structure, made up of many unit cells, also has cubic symmetry (Fig. 6a).

Fig. 6b: The unit cell of CCP, drawn with the atoms unique to a single unit cell

So far, unit cells have been drawn with full spheres at each corner and on the faces in CCP. However, these corner and face atoms are actually shared between neighbouring cells, and this can make the determination of the number of atoms per unit cell rather difficult. Fig. 6b attempts to show the number of atoms uniquely associated with one cell by means of fractional spheres. Each corner atom is shared by 8 unit cells, so there is 1/8 of an atom per unit cell at each corner. There are 8 corners per unit cell, giving a total of 1 corner atom per cell. Each face-centring atom is shared between two neighbouring unit cells, giving 1/2 atom per face in each cell. There are in total 6 faces, giving 6 x 1/2 = 3 face-centring atoms per unit cell. The total number of atoms per unit cell is thus 4.

The packing density (fraction of space filled by atoms) for both HCP and CCP is:

$\frac{\pi }{3\sqrt 2}\approx 0.74$

## The body-centred cubic (BCC) structure

Fig. 7a: First layer of the body-centred cubic structure

Another important crystal structure in metallic materials, in addition to HCP and CCP, is the body-centred cubic (BCC) structure. The first layer consists of a square arrangement of atoms (Fig. 7a).

Fig. 7b: Second layer of the body-centred cubic structure

The second layer lies in the square interstices (Fig. 7b) and the third layer lies directly above the first layer.

Fig. 7c: Unit cell of the body-centred cubic structure

BCC is not a close-packed structure – it has a lower packing density than FCC and HCP - but it does have planes on which the atoms are close-packed along specific directions. The BCC structure arises when there is some directionality to the bonding. The unit cell is cubic with an atom at the centre of the cube (hence the name, body-centred cubic) as well as the ones on the corners.

Fig. 7d: Unit cell of the body-centred cubic structure, showing only the atoms that fully belong to the cell

Applying the same approach as with the CCP cell, we can easily see that there are 2 atoms per unit cell in this case (Fig. 7d). The packing density for BCC is $\frac{\pi \sqrt 3 }{8}\approx 0.68$.

## Closest-packed planes

Fig. 8: Closest-packed planes in each of the metallic structures studied in this chapter

Fig. 8 compares the close-packed plane in CCP and HCP and the near-close-packed plane in BCC. Note that all three structures have close-packed directions (directions along which the atoms are touching) but only CCP and HCP have close-packed planes (planes on which all atoms are touching). We also need to remember, of course, that this billiard-ball picture is a simplification and atoms are not really solid spheres touching one another!

## Exercise

We saw earlier that the packing density for CCP was $$\frac{\pi }{3\sqrt 2}\approx 0.74$$, and that for BCC was $$\frac{\pi \sqrt 3 }{8}\approx 0.68$$.

Show that these expressions are correct, assuming that the closest-packed atoms touch one another.

## For further thought...

In some materials, mistakes in the stacking sequence can occur quite easily. What might the resulting structures look like?

So far, we have seen some common arrangements of atoms in pure metals. What might happen if there was a little of another element mixed in? Or a lot?

## Learning goals

• Understand the concept of close-packing in metals (approximation of atoms as hard spheres; most efficient way of packing these spheres).

• Know the difference between HCP and CCP packing sequences.

• Be able to compare and explain the features of the HCP, CCP and BCC crystal structures (appearance of unit cell, close-packed planes, packing density etc.)