# Metal Structures

Discussion Questions

• What metals have the closest packing structures?
• What metals have the body centered cubic structure?
• Does any metal adopt the simple cubic structure?
• What are the features of rare earth metals?

## Structure of Metals

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 model metal structures. On this page, we give a brief discussion on metal structures. Namely, metals have closest packing (fcc and hcp) structures, the body centered cubic (bcc), simple cubic (sc) and complicated closest packing structures. We will only scratch the surface on the subject of metal structures, but the basic concepts presented here let you understand why metals behave they way they do (properties).

### What metals have the closest packing structures?

In solids, two types of closest packing of spheres have been discussed. They are the face-centered closest packing (fcc), which is often called cubic closest packing (ccp), and hexagonal closest packing (hcp). Since the periodic table is a useful tool for representing information, the structure types of metals can be displayed using a periodic table.

Figure 1: In the above diagram, the ccp (fcc) structures are represented by a red circle, whereas the hcp structures are represented by a black hexagon.

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 noble metals ($$\ce{Cu}$$, $$\ce{Ag}$$, $$\ce{Au}$$, $$\ce{Ni}$$, $$\ce{Pd}$$, $$\ce{Pt}$$, $$\ce{Rh}$$, and $$\ce{Ir}$$) have fcc type structures. Among the group 2 elements, only $$\ce{Ca}$$ and $$\ce{Sr}$$ have the fcc structure, whereas $$\ce{Be}$$ and $$\ce{Mg}$$ have hcp structures. So do $$\ce{Zn}$$, $$\ce{Cd}$$, $$\ce{Sc}$$, $$\ce{Y}$$, $$\ce{Lu}$$, $$\ce{Ti}$$, $$\ce{Zr}$$, $$\ce{Hf}$$, $$\ce{Tc}$$, $$\ce{Re}$$, $$\ce{Ru}$$, $$\ce{Os}$$ 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 aspects, 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

Gold has a very high density, $$\rho$$, of 19.3 g/mL, and it has the fcc structure. If the gold atoms were spheres as modeled by the fcc structure, what is the atomic radius?

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

\begin{align} \rho &= \dfrac{4 \times 197.0}{6.023 \times 10^{23} \times \mathrm a^3}\\ &= \mathrm{19.3\: g / cm^3} \end{align}
\begin{align} \mathrm a &= 4.0774\times10^{-8}\: \mathrm{cm}\\ &= 2 \times 2^{1/2}\: \mathrm r \end{align}
$$\mathrm{r = 1.442 \times 10^{-8}\: cm}$$

Please work out these formulas and numbers yourself.

DISCUSSION
A handbook did give the radius of gold as 144 pm (1 pm = 1012).

Exercises 1

1. Copper has a specific gravity of 8.92; evaluate its atomic radius.
2. The atomic radius of silver $$\ce{Ag}$$ is listed as 145 pm. Evaluate its density.

### What metals have the body centered cubic structure?

Alkali metals ($$\ce{Li}$$, $$\ce{Na}$$, $$\ce{K}$$, $$\ce{Rb}$$, and $$\ce{Cs}$$) 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 than the fcc or hcp types as shown by Example 1.

Example 2

What is the fraction of the volume occupied by spheres for a bcc type structure?

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

\begin{align} body\: diagonal &= 4 r\\ &= 3^{1/2} a \end{align}

$$a = \dfrac{4 r}{3^{1/2}}$$

$$\mathrm{V_{sphere}} = \dfrac{4}{3}\pi r^3$$

$$\mathrm{V_{cell} = a^3} = \dfrac{64\times r^3}{3^{3/2}}$$

\begin{align} \textrm{Fraction of volume occupied by spheres} &= \mathrm{\dfrac{2\times V_{sphere}}{V_{cell}}}\\ &= \dfrac{3^{1/2}\pi}{8}\\ &= \mathrm{0.68\: or\: 68\%} \end{align}

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.

### Does any metal adopt the simple cubic structure?

Polonium ($$\ce{Po}$$) has been reported to have a simple cubic crystal structure. From a packing point of view, this type of arrangement is not stable, and this 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 encloses a whole atom. You can work out the fraction of space occupied by spheres in such an arrangement to be $$\pi$$/6 (0.52), which is much less than the bcc structure type.

### What are the features of rare earth metals?

If you go back to the periodic table of structure types, you will see that the rare earth elements have fcc (ccp), hcp, bcc, and another type marked by hc (4 H). In other words, these 14 elements exemplify many types of structures. We have covered the other types than the hc (4 H) type pretty well. Actually, the hc (4 H) type has a complicated packing sequence such as ABAC, ABCB, etc. That is why they are designated as (4 H). Actually, the structure of $$\ce{Sm}$$ (samarium) has a very complicated sequence of ACACBCBAB ACACBCBAB ....

## Questions

1. What is the volume of a sphere with a radius of 1.0 cm?

Hint: 4.19 cc

Skill -
Give the formula of volume for a sphere with radius r.

2. What is the volume of a cube with an edge length of 2.0 cm?

Hint: 8.0 cc

Discussion
The fraction of space occupied by spheres in a simple cubic arrangement is 4.19/8.0 = 0.52.

3. What is the number of nearest neighbours in a bcc structure?

Hint: 8

Skill -
Be prepared to answer simple "matter of fact" questions regarding structure types.

4. From the previous question, you know that there are 8 nearest neighbours in a bcc structure. The next nearest neighbor is at a distance a (unit cell edge length). How many neighbours have a distance a from an atom of the bcc type structure?

Hint: 6

Skill -
Be prepared to answer simple "matter of fact" questions regarding structure types.

5. Give three metals that have the bcc structure.

Hint: Any three of: $$\ce{Na}$$, $$\ce{K}$$, $$\ce{Fe}$$ (iron), $$\ce{Pb}$$ (lead) etc.

Discussion
Give some examples for the ccp (fcc) and hcp types.