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Consider an ideal gas of N fermions in equilibrium with a reservoir of energy at temper- ature T. The average number of fermions in a state with energy e is given by 1 n(e) : E-μL exp kBT +1 = A€ ¹/2, where kg is the Boltzmann constant. The density of states is given by D(e) where A is a positive constant. a) Explain the physical meaning of μ. Write down an equation that determines its value. [3] b) What is the ground state of this system for T=0? Determine the Fermi energy EF in terms of A and N. Under certain conditions, the average number of fermions with energy e can be approxi- mated by a piece-wise function 1 €<μ-kBT, n'(e) = μ+kBT-€ 2kBT µ = kBT < € < µ+kBT₂ € > μ+ KBT. c) Sketch this function. Discuss when n'(e) is a good approximation to the actual n(e). [3] d) Using n' (e), derive an algebraic equation that determines µ. [5] e) Assume that µ = ep (1 +A(T)²) and perform a series expansion of the result from part d) to first non-trivial order. [4] f) Use the result of part e) to show that A = -1/12. [2] g) Use the form of n'(e) to estimate the heat capacity of the gas as T →0. [4] =