9780030839931, Chapter 16, Problem 4P

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Chapter 16, Problem 4P Problem Consider the Boltzmann equation (16.13) for a metal in a static uniform electric field, with a collision term (16.18) appropriate to elastic impurity scattering. (a) Assuming a nonequilibrium distribution function of the form g(k) = f(k) + δg(k), (16.36) where is the equilibrium Fermi, function and is of order E, derive to linear order in E an integral equation obeyed by and snow that.conductivity can be written in the form = (16.37) where u(k) is a solution to the integral equation v(k) = dk' u(k')]. (16.38) (b) Let (k) and y(k) be any two functions of k. Define (16.40) so that (16.37) can be written compactly as = (16.40) Define (16.41) Show that {a, = and that Eq. (16.38) implies (16.42) so that the conductivity can also be written in the form = (16.43) (c) Prove for arbitrary a and Y that {α, (16.44) (Hint: Prove that + a + 0 for arbitrary and choose a that minimizes the left- hand side of this inequality.) (d) With the choice a = их, deduce that OXX satisfies the inequality (16.45) for arbitrary functions Y. (e) Suppose that W= W(1) + W(2) Let Y be where u is the solution to (16.38). Let σ(1) and σ(2) be the conductivities one would have if only W(1) or W(2) were present. Deduce from (16.45), as it applies to σ(1) and σ(2), that 1 (16.46) Step-by-step solution There is no solution to this problem yet. Get help from a Chegg subject expert. Ask an expert