# Crystal Structure of Solids

Many construction materials, particularly metals and some ceramics, consist of small crystals or grains within which the atoms are packed in regular, repeating, three-dimensional patterns giving a longrange order. The grains are ‘glued’ together at the grain boundaries; we will consider the importance of these later, but first we will discuss the possible arrangements of atoms within the grains.

For this, we will assume that atoms are hard spheres – a considerable but convenient simplification. It is also convenient to start with the atomic structure of elements (which of course consist of single-sized atoms) that have non-directional bonding (e.g. pure metals with metallic bonds).

## Crystal Structure of Solids

The simplest structure is one in which the atoms adopt a cubic pattern i.e. each atom is held at the corner of a cube. For obvious reasons, this is called the simple cubic (SC) structure. The atoms touch at the centre of each edge of the cube (Fig. 1a). The structure is sometimes more conveniently shown as in Fig. 1b.

We can use this figure to define some properties of crystalline structures:

• the unit cell: the smallest repeating unit within the structure, in this case a cube (Fig. 1)
• the coordination number: the number of atoms touching a particular atom or the numbers of its nearest neighbours, in this case 6 (Fig. 2)
• the closed-packed direction: the direction along which atoms are in continuous contact, in this case any of the sides on the unit cell
• the atomic packing factor (APF): the volume of atoms in the unit cell/volume of the unit cell, which therefore represents the efficiency of packing of the atoms.

The APF can be calculated from simple geometry. In this case:

• there are eight corner atoms, and each is shared between eight adjoining cells

∴ each unit cell contains 8 × 1/8 = 1 atom

• the atoms are touching along the sides of the cube (the close-packed direction)

∴ radius of each atom, r = 0.5a (Fig. 3) when a = length of the side of the unit cube

the volume of each atom = 4/3 πr3 = 4/3 π(0.5a)3

∴ APF = [atoms/cell].[volume each atom]/volume of unit cell

= [1] × [(4/3 π(0.5a)3]/[a3] = 0.52.

There are two other crystal structures that have cubic structures with atoms located at the eight corners but which have additional atoms:

• the body-centred cubic (BCC) structure, which also has an atom at the centre of the cube (Fig. 4)
• the face-centred cubic (FCC) structure, which also an atom at the centre of each of the six faces (Fig. 5).

With the body-centred cubic structure, the coordination number is 8 (the atom in the cell centre touches the eight corner atoms) and the close-packed direction is the cell diagonal. It should be apparent from Fig. 4 that:

• each unit cell contains 8/8 + 1 = 2 atoms
• considering the close-packed direction gives: 4r = √3a or r = √3a/4.

∴ APF = [2] × [(4/3 π(√3a/4)3 ]/[a3 ] = 0.68.

With the face-centred cubic structure, a little thought should convince you that the coordination number is 12 and the close-packed direction is the face diagonal. From Fig. 5:

• each unit cell contains 8/8 + 6/2 = 4 atoms
• considering the close-packed direction gives: 4r = √2a or r = √2a/4.

∴ APF = [4] × [(4/3 π(√2a/4)3]/[a3] = 0.74.

Moving from the SC to the BCC to the FCC structure therefore gives an increase in the coordination number (from 6 to 8 to 12) and in the efficiency of packing (from an APF of 0.52 to 0.68 to 0.74).

One further structure that might be expected to have efficient packing needs consideration – the hexagonal close-packed (HCP) structure. If we start with a single plane, then the most efficient packing is a hexagonal layout, i.e. as the atoms labelled A in Fig. 6.

In adding a second layer the atoms (labelled B) place themselves in the hollows in the first layer. There are then two possible positions for the atoms in the third layer, either directly above the A atoms or in the positions labelled C. The first of these options gives the structure and unit cell shown in Fig. 7.

Two dimensions, a and c, are required to define the unit cell, with c/a = 1.633. The coordination number is 12 and atomic packing factor 0.74, i.e. the same as for the face-centred cubic structure.

If we know the crystal structure and the atomic weight and size of an element then we can make an estimate of its density. For example, take copper, which has a face-centred cubic structure, an atomic weight of 63.5, and an atomic radius of 0.128 nm (atomic weights and sizes are readily available from tables of properties of elements).

• atomic weight = 63.5, therefore 63.5 g of copper contain 6.023 × 1023 atoms

∴ mass of one atom = 10.5 × 10−23 g

• In the FCC structure there are 4 atoms/unit cell

∴ mass of unit cell = 4.22 × 10−22 g

• As above, in the FCC structure, length of side of unit cell (a) = 4r/√2

∴ a = 4 × 0.128/√2 = 0.362 nm

∴ unit cell volume = 4.74 × 10−2 nm3

∴ density = weight/volume = 4.22 × 10−22 g/4.74 × 10−2 nm3 = 8900 kg/m3

A typical measured value of the density of copper is 8940 kg/m3 , so our estimate is close.

We generally expect that elements that adopt one of the crystal structures described above will prefer to adopt the one that has the lowest internal energy. The efficiency of packing (i.e. the APF) is an important, but not the sole, factor in this.

In practice, no common metals adopt the simple cubic structure, but the energy difference between the other three structures is often small, and the structures adopted by some common metals are:

• FCC – aluminium, copper, nickel, iron (above 910°C), lead, silver, gold
• HCP – magnesium, zinc, cadmium, cobalt, titanium
• BCC – iron (below 910°C), chromium, molybdenum, niobium, vanadium.

The two structures for iron show that the crystal structure can have different minimum energies at different temperatures. These various forms are called allotropes.

Changes from one structure to another brought about by changes of temperature are of fundamental importance to metallurgical practice. For example, the change from FCC to BCC as the temperature of iron is reduced through 910°C forms the foundation of the metallurgy of steel.

## Imperfections and Impurities

In practice it is impossible for a perfect and uniform atomic structure to be formed throughout the material and there will always be a number of imperfections. Point defects occur at discrete sites in the atomic lattice and can be either missing or extra atoms, called vacancies and interstitial atoms respectively, as shown in Fig. 8.

A linear dislocation is a onedimensional defect; an example is when part of a plane of atoms is missing and causes an edge dislocation, as shown in Fig. 3.9. The result of all such defects is that the surrounding atomic structure is distorted and so is not in its preferred lowest energy state.

This has important consequences during loading; when the internal strain energy is sufficient to locally rearrange the structure a dislocation is, in effect, moved. The dislocation does not move back to its original position on unloading, and so the resulting deformation is irreversible i.e. it is plastic. If the required internal energy needed to trigger the dislocation movement is sharply defined then this gives rise to a distinct yield point.

It is also impossible to produce a completely pure material, and some foreign atoms will also be present, thus producing a solid solution. A substitutional impurity occurs when the foreign atoms take the place of the parent atoms, resulting in a substitutional solid solution (Fig. 10a). If the atoms of the two materials are of a similar size then there will be little distortion to the atomic lattice, but if their size differs significantly then some distortion will occur.

An interstitial impurity, as the name implies, occurs when the foreign atoms are forced between the parent atoms, resulting in an interstitial solid solution; again the degree of distortion depends on the relative sizes of the atoms involved (Fig. 10b).

The impurities may occur by chance during manufacture, but nearly all metals used in construction are in fact alloys, in which controlled quantities of carefully selected impurities have been deliberately added to enhance one or more properties.