Gold crystallises as a face-centred cubic lattice with a one-atom basis, (Chapter 1)

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction · Lattice dynamics and thermal properties of solids

Condensed Matter Physics, Ch Einstein, Debye and the heat capacity of solids

Problem

Gold crystallises as a face-centred cubic lattice with a one-atom basis, a=4.078 A˚a = 4.078\ \mathrm{\mathring{A}} (Chapter 1). From this cell, Chapter 2 gives the distances from the centre of gold's Brillouin zone to its two nearest kinds of face as 1.334 A˚−11.334\ \mathrm{\mathring{A}^{-1}} (a {111}\{111\} face) and 1.541 A˚−11.541\ \mathrm{\mathring{A}^{-1}} (a {200}\{200\} face). ((a)) Using n=4/a3n = 4/a^3, find gold's Debye wavevector qDq_D, and check that it lies between the two zone-face distances above. ((b)) A classmate argues: "Since qDq_D is smaller than the {200}\{200\} zone-face distance, every lattice wave with wavevector between qDq_D and 1.541 A˚−11.541\ \mathrm{\mathring{A}^{-1}} must be forbidden in gold, so ωD\omega_D really is the fastest vibration the crystal can carry, at least along that direction." Explain what is wrong with this claim.

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