Packing efficiency of bcc, and why CsCl is 8:8 while NaCl is 6:6

Packing efficiency of bcc

Atoms per unit cell: $8 \times \tfrac{1}{8}$ (corners) $+ 1$ (body centre) $= 2$.

Atoms touch along the body diagonal: $\sqrt{3}\,a = 4r \Rightarrow a = \dfrac{4r}{\sqrt{3}}$.

$$\text{Packing efficiency} = \frac{2 \times \frac{4}{3}\pi r^3}{a^3} = \frac{\frac{8}{3}\pi r^3}{\left(\frac{4r}{\sqrt3}\right)^3} = \frac{\sqrt{3}\,\pi}{8} = 0.680 = \mathbf{68\%}$$

(32% is empty space; compare 74% for ccp/hcp and 52.4% for simple cubic.)

Radius ratio rule

A cation of radius $r^+$ keeps contact with $N$ anions of radius $r^-$ only if $r^+/r^-$ exceeds a limiting value: 0.414 to 0.732 for octahedral (6) and above 0.732 for cubic (8) coordination.

Salt	r+ (pm)	r− (pm)	r+/r−	Predicted	Found
CsCl	Cs+ 174	Cl− 181	0.96	8 (cubic)	8 : 8
NaCl	Na+ 102	Cl− 181	0.56	6 (octahedral)	6 : 6

The large Cs⁺ ion can touch eight Cl⁻ ions at the corners of a cube, gaining more Madelung energy. The small Na⁺ would "rattle" in a cubic hole, so it takes the octahedral holes of a ccp array of Cl⁻.

Is CsCl bcc?

No. The body-centre ion (Cs⁺) differs from the corner ions (Cl⁻), so the two sites are not equivalent. CsCl is a simple cubic lattice with a two-ion basis (Cl at 0,0,0 and Cs at ½,½,½): two interpenetrating simple cubic lattices.