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Crystal structures and Miller indices

Which value is closest to the interplanar spacing of the (111) planes of aluminum (FCC, lattice parameter 0.405 nm)?

  1. AAbout 0.135 nm
  2. BAbout 0.234 nm
  3. CAbout 0.701 nm
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Answer: B. About 0.234 nm

The interplanar spacing in a cubic crystal is d = a/√(h² + k² + l²), so for (111), d = 0.405/√3 ≈ 0.234 nm. About 0.135 nm is the mistake of taking a/3 without the square root, and about 0.701 nm is the mistake of multiplying a by √3, which gives the length of the body diagonal. The spacing gets smaller as the indices get larger, and in XRD, planes with smaller d diffract at higher angles. This formula applies only to cubic crystals; tetragonal and hexagonal crystals need formulas that treat the lattice parameters a and c separately.

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More Crystal structures and Miller indices questions

  1. Which statement about the ideal hexagonal close-packed (HCP) structure is correct?
  2. In a cubic crystal, which value is closest to the angle between the [100] and [111] directions?
  3. 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.
  4. In a face-centered cubic (FCC) metal, how many atoms touch a given atom at the nearest-neighbor distance (the coordination number)?
  5. What is the net number of atoms in a body-centered cubic (BCC) unit cell?