Metallic Crystals - The Structures of Pure Metals

Calculating Packing Densities

To calculate the particle packing density the spheres in the unit cell are counted up. The body-centered cubic structure contains (1 + 8·⅛ = 2) formula units per cell; the face-centered cubic unit cell contains (6·½ + 8·⅛ = 4) formula units, giving it the higher packing density. To determine the packing efficiency, we procede as follows (all spheres are identical in size): If the edge length of the cubic cell is a, then the volume of the cell is: $V = a 3$.

In the case of the body-centered structure, the radius of the spheres is exactly ¼ of the body diagonal of the cube; for the face-centered cubic structure it is exactly ¼ of the diagonal of the face. This results in the following packing efficiencies:

Fig.1
Body-centered cubic
Fig.2
Face-centered cubic

Calculation

Tab.1
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 ≈ 0.68017 ≈ 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 ≈ 0.74048 ≈ 74.0 %$
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