Crystal structures and Miller indices
Check the coordination numbers and packing factors of SC, BCC, FCC and HCP, densities calculated from the unit cell, and Miller indices and interplanar spacings with multiple-choice questions. A numerical way to firm up the basics of crystallography.
Crystal structures and Miller indices: key points
Most metals take one of three crystal structures: body-centered cubic (BCC), face-centered cubic (FCC) or hexagonal close-packed (HCP). The coordination number is 8 for BCC and 12 for FCC and HCP, and the atomic packing factor is about 0.68 for BCC and about 0.74 for FCC and HCP. Knowing the number of atoms per unit cell (2 for BCC, 4 for FCC) and the lattice parameter, you can calculate the density.
Crystal planes and directions are described by Miller indices. The indices of a plane are the reciprocals of its intercepts on the axes, reduced to the smallest integer ratio. In cubic crystals the interplanar spacing is d = a/√(h² + k² + l²), and the [hkl] direction is perpendicular to the (hkl) plane. These are the foundation for understanding slip systems and XRD peaks.
All Crystal structures and Miller indices questions (12)
- In a face-centered cubic (FCC) metal, how many atoms touch a given atom at the nearest-neighbor distance (the coordination number)?
- What is the net number of atoms in a body-centered cubic (BCC) unit cell?
- 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)?
- 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.
- 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?
- 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?
- 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?
- Which correctly pairs the close-packed planes and close-packed directions of an FCC metal?
- Which value is closest to the interplanar spacing of the (111) planes of aluminum (FCC, lattice parameter 0.405 nm)?
- Which statement about the ideal hexagonal close-packed (HCP) structure is correct?
- In a cubic crystal, which value is closest to the angle between the [100] and [111] directions?
- 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.