Metallic Bonding in the Solid State:

For metals the atoms have low electronegativities; therefore the electrons are delocalized over all the atoms.  We can think of the structure of a metal as an arrangement of positive atom cores in a sea of electrons. For a more detailed picture see "Conductivity of Solids".

 
To maximize the bonding in a metal it makes sense to pack as many atoms around each other as possible, maximize the number of nearest neighbors (called the "coordination number") and  minimize the volume.

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Close packed layers of atoms.

If we treat the atoms as spheres and consider all the atoms in the solid to be of equal size (as is the case for elemental metals), the most efficient form of packing is the close packed layer.  This is illustrated below where it is clear that close-packing of spheres is more efficient than, for example, square packing.

Below on the left is a square packed array compared to the more densely packed close packed array.
Within the square packed layer the coordination # of each atom is 4, in the close packed layer it is 6.

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To build our 3-dimensional metal structures we now need to stack the close packed layers on top of each other.  There are several ways of doing this.  The most efficient space saving way is to have the spheres in one layer fit into the "holes" of the layer below.

If we call the first layer "A", then the second layer ("B") is positioned as shown on the left of the diagram below.  The third layer can then be added in two ways.  In the first way the third layer fits into the holes of the B layer such that the atoms lie above those in layer A.  By repeating this arrangement one obtains ABABAB... stacking or hexagonal close packing.
 

 Hexagonal Close Packing. (HCP)

A more extended side view of the packing is shown below:

The {ABAB... } type stacking of close packed layers is called Hexagonal Close Packing (hcp) because the smallest lattice repeat is a Primitive Hexagonal unit cell.

In the  primitive hexagonal cell we have 1 atom at each of the corners of the cell (each is "worth" 1/8) and 1 atom within the cell giving us 2 atoms/unit cell.
The coordination number of the atoms in this structure is 12.  They have 6 nearest neighbors in the same close packed layer, 3 in the layer above and 3 in the layer below.
This is one of the most efficient methods of packing spheres (the other that is equally efficient is cubic close packing, see below).  In both cases the spheres fill 74% of the available space.

HCP is a very common type of structure for elemental metals.  Examples include Be, Mg, Ti, Zr, etc.
 
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While for the HCP structure the third close packed layer was positioned above the first , an alternate method of stacking is to place the third layer such that it lies in an unique position, in this way an "ABCABC..."  close packed layer sequence can be created, see below.  This method of stacking is call Cubic Close Packing (ccp)

How many atoms are there in the fcc unit cell ?
8 at the corners (8x1/8 = 1), 6 in the faces (6x1/2=3),  giving a total of 4 per unit cell.

Again there are many examples of ccp (fcc) (ABCABC) metal structures, e.g. Al, Ni, Cu, Ag, Pt.

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Density Calculations

In the fcc cell the atoms touch along the face diagonals, but not along the cell edge:

Length face diagonal = a(2)^1/2 = 4r
 
Use this information to calculate the density of an  fcc metal.

Example calculation.
Al has a ccp arrangement of atoms.  The radius of Al = 1.423Å ( = 143.2pm).  Calculate the lattice parameter of the unit cell and the density of solid Al (atomic weight = 26.98).

Solution:
Because Al is ccp we have an fcc unit cell.  Cell contents: 4 atoms/cell  [8 at corners (each 1/8), 6 in faces (each 1/2)]

Lattice parameter:  atoms in contact along face diagonal, therefore 4r_Al = a(2)^1/2

a = 4(1.432Å)/(2)^1/2 = 4.050Å.

Density (= r_Al) = Mass/Volume = Mass per unit cell/Volume per unit cell  g/cm³

Mass of unit cell = mass 4 Al atoms = (26.98)(g/mol)(1mol/6.022x10²³atoms)(4 atoms/unit cell) = 1.792 x 10^-22 g/unit cell

Volume unit cell = a³ = (4.05x10^-8cm)³ = 66.43x10^-24 cm³/unit cell

Therefore  r_Al = {1.792x10-22g/unit cell}/{66.43x10^-24 cm³/unit cell} = 2.698 g/cm³

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Other common types of metal structures
 

1. Body Centered Cubic (BCC)

Not close packed - atoms at corners and body center of cube.  # atoms/unit cell = 2.
Coordination number = 8 Æ less efficient packing (68%)
The atoms are only in contact along the body diagonal.
For a unit cell  edge length a, length body diagonal  = a(3)^1/2.  Therefore 4r = a(3)^1/2
 
Examples of BCC structures include one form of Fe, V, Cr, Mo, W.

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2. Simple Cubic

Again not close packed - primitive or simple cubic cell with atoms only at the corners.  # atoms/unit cell = 1.
Coordination number = 6 Æ least efficient method of packing (52%)
The atoms are in contact along the cell edge.  Therefore a = 2r.
A very rare packing arrangement for metals, one example is a form of Polonium (Po)

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Summary of Packing types for Metallic Structures.
 

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