A very high percentage of the elements are metals. However, only a few types
of structures occur for most elemental metals. This leads to the use of
sphere packings to models metal structures.
In the above diagram, the ccp (fcc) structures are represented by a red circle, whereas the hcp structures are represented by black hexagon. Some examples of these structures are also given on page 77 in Inorganic Chemistry by T.W. Swaddle, Academic Press.
Copper is the most common mentioned metal that has the fcc structure. This element is one of the noble metals. It has been widely used for door knobs and other tools, and has been widely recognized. Yet, most nobel metals, Cu, Ag, Au, Ni, Pd, Pt, Rh, and Ir have fcc type structures. Among the group 2 elements, only Ca and Sr have the fcc structure, whereas Be and Mg have hcp structures. So do Zn, Cd, Sc, Y, Lu, Ti, Zr, Hf, Tc, Re, Ru, Os and most rare earth elements. These are mentioned to bring your attention to these two common types of structures, and you are encouraged to at least be able to give a few examples for each type.
Amazingly, the hcp and fcc structure are very similar in many aspect, but nature knows best. The structures adopted by various metals occur by their design. When crystallization takes place, the atoms arrange themselves according to their structure types.
Example 1
Solution
Assume the radius of gold spheres to be r, and the edge
of the face centered unit cell to be a. There are four gold
(atomic mass 197.0, Avogadro's number = 6.023x1023) atoms
per unit cell, thus
= 19.3 g / cm3
a = 4.0774*10-8 cm
= 2 * 21/2 r
r = 1.442 *10-8 cm
Discussion
A handbook did give the radius of gold as 144 pm (1 pm = 1012).
Copper has a sepcific gravity of 8.92, evaluate its atomic radius.
The atomic radius of silver Ag is listed as 145 pm. Evaluate its density.
What metals have the body centered cubic structure?
Alkali metals, Li, Na, K, Rb, and Cs all have the body centered cubic (bcc) structure. In addition, the vanadium and chromium groups also have the bcc structure. Furthermore, at room temperature, iron has a bcc structure.
This type of structure has two atoms per unit cell, and it is slightly less densely packed as the fcc or hcp types as shown by Example 1.
Example 2
Solution
Assume the radius of spheres be r, and the edge
of the body centered unit cell to be a. There are two
spheres per unit cell, thus
= 31/2 a
a = 4 r / 31/2
Vsphere = 4/3pr3
Vcell = a3 = 64*r3 / 33/2
Fraction of volume occupied by spheres = 2*Vsphere/Vcell
= 31/2p/8
= 0.68 or 68%.
Discussion
The fraction of 0.68 is slightly less than those (0.74) of closest packed
structures. Thus, the bcc structures are less densely packed according to the
hardsphere model.
Is there any metal adopting the simple cubic structure?
Yes, polonium (Po) has been reported to have a simple cubic crystal structure.
From packing point of view, this type of arrangement is not stable, and
this structure type is not common.
The cubic unit cell contains only one sphere, and the edge length is exactly equal to the diameter of the sphere. In the diagram, we choose the origin to be the center of an atom, but if you choose the origin to be the center among 8 spheres, your cube enclose a whole atom.
You can work out the fraction of space occupied by spheres in such an arrangement to be p/6 (0.52), much less than the bcc structure type.