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Chemistry · Ch 1 — Solid State

Packing efficiency of metal crystal in simple cubic lattice

1.7.1

Packing efficiency of metal crystal in simple cubic lattice

The packing efficiency of a metal crystal in the simple cubic lattice is obtained by the following steps.

Step 1 : Radius of sphere : In a simple cubic unit cell the corner spheres touch one another along the edge of the cube (Fig. 1.9), so the edge length is two radii:

a=2rorr=a2...(1.6)a = 2r \quad\text{or}\quad r = \frac{a}{2} \qquad \text{...(1.6)}

Figure 1.9Face of a simple cubic unit cell: four touching spheres at the corners of a square with the square outline joining their centres, the edge marked a by a dimension arrow and the two radii along the top edge each marked r, illustrating a = 2r.
Fig. 1.9 — Face of a simple cubic unit cell: four touching spheres at the corners of a square with the square outline joining their centres, the edge marked a by a dimension arrow and the two radii along the top edge each marked r, illustrating a = 2r.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. A 2-D view of one face of a simple cubic unit cell: four identical spheres in a 2-by-2 arrangement touching along the edges, with a square outline joining their centres. A double-headed arrow across the top spans one edge and is labelled aa; the two radii running from the top two sphere centres to their touching point are each labelled rr -- showing that the edge e …

Step 2 : Volume of sphere : The volume of a sphere is 43πr3\tfrac{4}{3}\pi r^3. Substituting r=a/2r = a/2,

volume of one particle=43π(a2)3=πa36...(1.7)\text{volume of one particle} = \frac{4}{3}\pi\left(\frac{a}{2}\right)^{3} = \frac{\pi a^3}{6} \qquad \text{...(1.7)}

Step 3 : Total volume of particles : A simple cubic unit cell contains only 1 particle, so the total volume occupied by the particles in one unit cell is πa36\dfrac{\pi a^3}{6}.

Step 4 : Packing efficiency

Packing efficiency=πa3/6a3×100=100 π6=100×3.1426=52.36 %\text{Packing efficiency} = \frac{\pi a^3/6}{a^3}\times 100 = \frac{100\,\pi}{6} = \frac{100\times 3.142}{6} = 52.36\,\% …

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