BCC chromium: packing density and coordination number

Lattice and basis

• Lattice: an infinite, regular, periodic arrangement of points in space, each point having identical surroundings. It is a purely geometrical framework.
• Basis: the atom or group of atoms placed at each lattice point.

$$\text{Crystal} = \text{Lattice} + \text{Basis}$$

For chromium, the lattice is BCC and the basis is a single Cr atom.

BCC structure of chromium

One Cr atom at each of the 8 corners and one at the body centre. Atoms touch along the body diagonal.

        o-----------o
       /|          /|
      / |         / |      o = Cr atom at corner
     o-----------o  |      @ = Cr atom at centre
     |  |   @    |  |
     |  o--------|--o      atoms touch along
     | /         | /       body diagonal
     |/          |/
     o-----------o

Packing density

Atoms per cell: $N = 8\times\frac{1}{8} + 1 = 2$.

Along the body diagonal: $\sqrt{3}\,a = 4R \Rightarrow a = \dfrac{4R}{\sqrt{3}}$.

$$\begin{aligned} \text{APF} &= \frac{N\cdot\frac{4}{3}\pi R^3}{a^3} = \frac{2\times\frac{4}{3}\pi R^3}{\left(\frac{4R}{\sqrt{3}}\right)^3} \\ &= \frac{\frac{8}{3}\pi R^3\times3\sqrt{3}}{64R^3} = \frac{\sqrt{3}\,\pi}{8} = 0.68 \end{aligned}$$

So 68 % of the volume of the cell is filled by atoms. (For Cr, $R \approx 0.125$ nm gives $a = 4R/\sqrt{3} \approx 0.289$ nm.)

Coordination number

The body-centre atom touches all 8 corner atoms, and each corner atom touches the centre atoms of the 8 cells sharing that corner. So the coordination number is 8.

Answer: packing density $= 0.68$, coordination number $= 8$.