Chemistry · Ch 11 — The Solid State

Efficiency of Packing in Body-Centred Cubic Structures

11.7.2

Efficiency of Packing in Body-Centred Cubic Structures

In a body-centred cubic (bcc) structure the central atom touches the two corner atoms lying along the body diagonal (Fig. 1.25). Let the edge length be $a$ .

First, from the face diagonal in triangle $EFD$ :

b^2 = a^2 + a^2 = 2a^2 \quad\Rightarrow\quad b = \sqrt{2}\,a

Then the body diagonal $c$ , from triangle $AFD$ :

c^2 = a^2 + b^2 = a^2 + 2a^2 = 3a^2 \quad\Rightarrow\quad c = \sqrt{3}\,a

Since three spheres touch along the body diagonal, its length equals $4r$ :

\sqrt{3}\,a = 4r \quad\Rightarrow\quad a = \frac{4r}{\sqrt{3}} \qquad\left(\text{equivalently } r = \frac{\sqrt{3}}{4}\,a\right)

A bcc cell contains 2 atoms, of total volume $2 \times \tfrac{4}{3}\pi r^3$ , and the cube volume is $a^3 = \left(\dfrac{4}{\sqrt{3}}r\right)^3$ . Therefore: …

Figure 1.25Body-centred cubic unit cell (spheres along the body diagonal are shown with solid boundaries).

What this figure shows. A bcc unit cell for the packing-efficiency derivation: three spheres touching along the body diagonal (corner-centre-corner) shown solid. Face diagonal b and body diagonal c are marked with triangles EFD and AFD used to derive c = sqrt(3) a and 4r = sqrt(3) a. …

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