# 5 problems on free electron gas

Jun 2nd, 2016
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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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1. Using the Drude model of an electron gas, calculate the relaxation time forcopper at room temperature given that its free electron density isapproximately 8.5 x 10 28 m-3 and that its resistivity is 1.5 x 10-8 m.According to Drude model, the conductivity of the metal isne 2mm 2ne9.11 10 311.5 10 8 * 8.5 10 28 * (1.6 10 19 ) 2 2.8 10 14 s2. Calculate the Fermi energy of copper (in eV) using the value for the freeelectron density given above. The free electron density of caesium isapproximately 9 x 1027 m-3. Evaluate the Fermi energy of this metal.The relation between number density of electron gas and its Fermi wavenumber is3kn F231k F (3 2 n) 32EF 2kF2m22(3 2 n) 32m2( c ) 2(3 2 n) 322mc(0.197 10 6 ) 2 eV 2(3 2 * 8.5 10 28 ) 32 * 511000eV 3.79 10 20 * 1.85 10 202 7.02eVFor caesium,22(3 2 n) 32m2(c) 2(3 2 n) 322mc(0.197 10 6 ) 2 eV 2(3 2 * 9 10 27 ) 32 * 511000eV 3.79 10 20 * 4.14 10192 1.57eV3. For a three dimension

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