# 5 problems on free electron gas

**Tutor description**

1. For a linear diatomic lattice dispersion curve the maximum frequency of the acoustic phonon branch is 1012 Hz. The forbidden gap is 4 meV wide. Calculate the lowest allowed frequency of the optical branch. 1. Using the Drude model of an electron gas, calculate the relaxation time for copper at room temperature given that its free electron density is approximately 8.5 x 10 28 m-3 and that its resistivity is 1.5 x 10-8 m. 2.2. Calculate the Fermi energy of copper (in eV) using the value for the free electron density given above. The free electron density of caesium is approximately 9 x 1027 m-3. Evaluate the Fermi energy of this metal. 3.3. For a three dimensional free electron gas (as covered in the lecture notes) derive the expression for the Fermi energy, EF, by explicit integration of the density of states. 4. Show that at T = 0 the internal energy, U, of a three dimensional electron gas of density n is 3nEF/5 5.The density of states of a two-dimensional free electron

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