Atomic radius of a face-centred cubic metal

Worked example

Find the metallic radius of Cu from its density, 8.954 g cm⁻³, given that it is face-centred cubic (Z = 4) with M = 63.55 g mol⁻¹.

1. 1. Density from a cubic unit cell: solve for Cell edge a

   $a = \sqrt[3]{\dfrac{ZM}{\rho N_\mathrm{A}}}$

   = 361.2 pm

2. 2. Multiply by a coefficient: 0.3535533905932738 × Cell edge a → r = a√2 / 4 (atoms touch along a face diagonal)

   = 127.7 pm

Answer: r(Cu) = 128 pm

Method

Rearrange ρ = ZM/(N_A a³) for a. In fcc, atoms touch along the face diagonal, which is four radii long: 4r = a√2, so r = a√2/4.

Common mistakes

• That uses the body diagonal (body-centred cubic). In face-centred cubic the atoms touch along a face diagonal: 4r = a√2.
• Atoms only touch along the edge in simple cubic packing. In fcc they touch along the face diagonal.

Formulas used

• Density from a cubic unit cell

  $\rho = \dfrac{ZM}{N_\mathrm{A}a^3}$

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