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Crystal structures and Miller indices

Which statement about the ideal hexagonal close-packed (HCP) structure is correct?

  1. Ac/a ≈ 1.633, and its packing factor is about 0.74, the same as FCC
  2. Bc/a ≈ 1.633, and its packing factor is about 0.68, the same as BCC
  3. Cc/a ≈ 1.414, and its packing factor is about 0.74, the same as FCC
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Answer: A. c/a ≈ 1.633, and its packing factor is about 0.74, the same as FCC

HCP stacks close-packed planes in an ABAB… sequence, and when hard spheres touch ideally, c/a = √(8/3) ≈ 1.633. The coordination number is 12 and the packing factor is about 0.74, the same close packing as FCC, which stacks ABCABC…. About 0.68 is the packing factor of BCC, and 1.414 (√2) is the ratio of the face diagonal to the edge in FCC, not the c/a of HCP. Real HCP metals deviate from the ideal c/a: about 1.62 for Mg, about 1.59 for Ti and about 1.86 for Zn. Because they have few slip systems, HCP metals tend to be less ductile at room temperature than FCC metals.

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More Crystal structures and Miller indices questions

  1. In a cubic crystal, which value is closest to the angle between the [100] and [111] directions?
  2. At 912 °C, pure iron transforms from BCC α-iron (lattice parameter 0.2904 nm) to FCC γ-iron (lattice parameter 0.3647 nm). Which correctly describes the change in volume per atom? Both lattice parameters are values at 912 °C.
  3. In a face-centered cubic (FCC) metal, how many atoms touch a given atom at the nearest-neighbor distance (the coordination number)?
  4. What is the net number of atoms in a body-centered cubic (BCC) unit cell?
  5. 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)?