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Crystal structures and Miller indices

What is the net number of atoms in a body-centered cubic (BCC) unit cell?

  1. A2
  2. B4
  3. C9
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Answer: A. 2

Each corner atom is shared by 8 unit cells, so it contributes 1/8, while the body-center atom belongs entirely to that cell. The net number of atoms is therefore 8 × 1/8 + 1 = 2. Nine comes from counting each corner atom as a whole atom without considering sharing, and 4 is the value for FCC (corners 8 × 1/8 + face centers 6 × 1/2 = 4). The number of atoms per unit cell n goes straight into the density calculation ρ = nM/(N_A a³), so you need to know it reliably for each structure. SC has 1, and the hexagonal-prism cell of HCP has 6.

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More Crystal structures and Miller indices questions

  1. Treating atoms as hard spheres of equal size, what is the atomic packing factor of the FCC structure (the fraction of the unit-cell volume occupied by atoms)?
  2. Which correctly relates the lattice parameter a and the atomic radius r in the BCC structure? Assume hard-sphere atoms with nearest neighbors in contact.
  3. Copper has the FCC structure with a lattice parameter of 0.3615 nm and a molar mass of 63.55 g/mol. Taking Avogadro's number as 6.022×10²³ mol⁻¹, which value is closest to the calculated density of copper?
  4. 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?
  5. In a cubic crystal, a plane intercepts the x axis at 1a and the y axis at 2a, and is parallel to the z axis. What are its Miller indices?