PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction · Free electrons, energy bands and electronic structure
Condensed Matter Physics, Ch Electrons in a periodic potential: Bloch's theorem and the opening of a gap
Problem
A simple metal has free-electron density n=2.65×1028m−3. One family of its lattice planes has spacing a=3.0A˚, and the periodic potential across those planes has amplitude ∣UG∣=0.40eV.
((a)) Using Bragg's law (Chapter 3) at normal incidence to these planes, find the wavevector of the first zone boundary for an electron travelling perpendicular to them.
((b)) Using Chapter 8's free-electron formula, kF=(3π2n)1/3, find the metal's Fermi wavevector, and say how close its electrons moving perpendicular to the planes come to being Bragg-reflected.
((c)) State which of the two zone-boundary standing waves has the lower energy, and why.
((d)) Find the size of the gap this potential opens at the boundary, and name the one respect in which this gap differs from the frequency gap the diatomic chain of Chapter 5 opened between its acoustic and optical branches.