# Metallic Crystals - The Structures of Pure Metals

## Exercises

Exercise : Packing densities

The figures depict the unit cell of a body-centered cubic and face-centered cubic structure, respectively. Which of these two structures has the higher packing density?

correct

wrong

partly correct

 Fig.1Body-centered cubic The body-centered cubic structure contains ( ) formula units per unit cell. The face-centered cubic structure contains ( ) formula units, giving it the higher packing density. Fig.2Face-centered cubic The face-centered cubic structure contains ( ) formula units, giving it the higher packing density.
Instruction : Packing densities

What portion of the entire space is occupied by the volume of the spheres in the above unit cells? (orange and blue spheres are identical in size)

The edge length of the cubic cell is a, the volume of the cells is $V=a 3$ .

In the body-centered structure, the radius of the spheres is equal to exactly 1/4 of the body diagonal; in the face-centered structure it is 1/4 of the face diagonal. This results in the following packing efficiencies:

Tab.1
Calculation
Body-centered cubicFace-centered cubic
$2 · V s p h e r e = 2 · 4 3 π · ( r s p h e r e ) 3 r s p h e r e = 3 ⋅ a 4 2 · V s p h e r e V c e l l = 2 ⋅ 4 3 π ⋅ ( 3 ⋅ a 4 ) 3 ⋅ 1 a 3 = 3 ⋅ π 8 = 0.6798 ≈ 68.0 %$$4 ⋅ Vsphere = 4 ⋅ 4 3 π ⋅ ( rsphere ) 3 rsphere = 2 ⋅ a 4 4 ⋅ Vsphere Vcell = 4 ⋅ 4 3 π ⋅ ( 2 ⋅ a 4 ) 3 ⋅ 1 a 3 = 2 ⋅ π 6 = 0.7401 ≈ 74.0 %$
Instruction : Crystal systems

Iridium crystallizes in a face-centered cubic lattice.Sketch the unit cell.

Fig.3
Face-centered cubic
Exercise : Crystal systems

Iridium crystallizes in a face-centered cubic lattice.

The number of iridium atoms in each unit cell is .

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