(Solved): 3) Density of States. In three dimensions the density of states \\( g(E) d E \\) depends on energy. ...

3) Density of States. In three dimensions the density of states \\( g(E) d E \\) depends on energy. Derive an expression for the density of states in a \\( 2 \\mathrm{D} \\) system. 4) Velocities in the Free Electron Theory, (a) In free electron theory show that the speed \\( v_{\\mathrm{F}} \\) of an electron at the Fermi surface of a metal is \\( v_{F}=\\frac{\\hbar}{m}\\left(3 \\pi n^{2}\\right)^{\\frac{1}{3}} \\quad \\) where \\( n \\) is the electron density. (b) Show that the mean drift speed of an electron in an applied electric field \\( E \\) is \\[ v_{d}=|\\sigma E /(n e)| \\] and show that the electrical conductivity \\( \\sigma \\) is related to the mean free path of electrons by \\[ \\sigma=\\frac{n e^{2} \\lambda}{m v_{F}} \\] (c) Copper has valence one. The density of atoms in copper is \\( n=8.4510^{28} \\mathrm{~m}^{-3} \\). The conductivity of copeer is \\( \\sigma=5.910^{7} \\Omega^{-1} \\mathrm{~m}^{-1} \\) at \\( 300 \\mathrm{~K} \\). (i)Assuming that free electron theory applies to copper, calculate \\( v_{\\mathrm{d}} \\) and \\( v_{\\mathrm{F}} \\) for copper at \\( 300 \\mathrm{~K} \\) in an electric field of \\( 1 \\mathrm{~V} / \\mathrm{m} \\) and comment on their relative magnitudes. (ii) Estimate \\( \\lambda \\) for copper at \\( 300 \\mathrm{~K} \\) and compare to the mean spacing between atoms.