Packing in three dimensions

Now consider some simple ways of filling space in three dimensions using spheres to represent the atoms or molecules.

Commence by drawing in your notebook the square layer of spheres as shown in the arrangement of Figure 3a. Add a second layer (Figure 3b) followed by a third layer directly above the first (Figure 3c). Note that the spheres of the first and third layers are not in contact.

Confirm by reference to Figure 4, where the spheres (unit radii) are drawn smaller for clarity, that:  x = 2 SQRT(2); % space filling = 74.1%; the sphere coordination number is 12.

This is an unusual representation of cubic close packing. The conventional representation, reflecting the high symmetry of the structure, is given later.

A body centred cubic unit cell can be formed by displacing the corner spheres so that contact is only maintained to the central sphere.

Sketch the cell shown in Figure 5a, following the sequence of Figure 5b (spheres in blue, unit cell and diagonals in black). Calculate from Figure 5c: x; the number and distance of nearest neighbours; % space filling; the radius of the central sphere that would allow complete sphere contact.

This arrangement is called body-centred cubic packing and is quite common in metallic structures, e.g. Potassium (K) and Tungsten (W).

Now fill the smaller triangle with the second arrangement of spheres previously studied (Figure 6a, A). Add a second layer (Figure 6b, AB). There are two ways of adding the third layer. First add the third layer so that it is superirnposed upon the first. The sequence can then continue ABABA ... etc. (Figure 6c).

Sketch Figure 7 (two copies) and check on your model that the unit cell shown is correct. Answer the following questions: What is the basal angle of the unit cell? (The packing is called hexagonal close packing.) What is the sphere coordination number? Mark in the centres of octahedral holes in the first diagram. [NB This and the following exercise are best left until you have built a model from linked polyhedra.] Mark in the centres of tetrahedral holes in the second diagram. How many spheres are there per unit cell?

Examples of hexagonal close packing are metallic Beryllium Be and Molecular H2 (rotating).

Now explore the situation when the third layer does not superimpose the first (ABC). Make a model of the packing sequence of Figure 8 using the larger triangle and four layers of spheres. In order to study this packing more closely sketch (use blue, red, and green) the fragments shown in Figure 9a and Figure 9b. When they are joined together (ABCA) they form the face-centred cube shown in Figure 10 where - the cube diagonals are perpendicular to close packed layers. Confirm that this is the unit cell of the packing sequence (cubic close packing) and expose as much of the unit cell as possible in your model by removing spheres from the top three layers.

Sketch three copies of the unit cell. Complete the following exercises. On the first diagram colour in spheres occupying the same close packed layer. (Take one diagonal as the direction of close packing and use different colours for different layers.) Mark in the centres of octahedral interstitial holes in the 2nd diagrarn. Mark in the centres of tetrahedral interstitial holes in the 3rd diagram. How many spheres are there per unit cell? How many octahedral and tetrahedral sites are there per unit cell? What is the coordination number of a sphere? What is the edge length of the cube? What is the % space filling? What are the radii of the spheres that would just fit into the octahedral and tetrahedral holes?

This type of close packing is common in metals for example: Copper (Cu) , and also in molecular solids for example: H₂S.