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Crystal structures and Miller indices

If the lattice parameter of α-iron (BCC) is 0.2866 nm, which value is closest to the atomic radius in the hard-sphere model?

  1. AAbout 0.124 nm
  2. BAbout 0.101 nm
  3. CAbout 0.143 nm
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Answer: A. About 0.124 nm

In BCC, atoms touch along the body diagonal, so √3a = 4r gives r = √3a/4 = 1.732 × 0.2866/4 ≈ 0.124 nm. About 0.101 nm comes from applying the FCC relation r = √2a/4, and about 0.143 nm from applying the simple cubic relation r = a/2. FCC γ-iron has a larger lattice parameter of about 0.36 nm, but its atomic radius differs by only a few percent; most of the difference in lattice parameter comes from the difference in atomic arrangement (structure).

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