2. Noninteracting fermions at very low temperature: Sommerfeld approximation Consider a system of non-interacting Fermi particles, with an (intensive) density of states D(e) which is a smooth function of e such that D(e) > 0 for e≈EF. Sommerfeld showed that at low temperature, the integral of the product of the Fermi function F(e) = [e³(−) + 1]¯¹ with a smooth function (€) of the energy can be expanded as follows: 7π4 360 ** deF(e)ø(c) = ["_deo(e) + —ª² (kBT)²¢' (µ) + (kBT)" (µ) + O(kËT/µ)б. (3) (a) (b) (c) (d) Apply the Sommerfeld expansion to obtain expressions for the energy per unit volume u = (E)/V and for the particle density n = N/V, containing terms up to quadratic order in the temperature. Assuming that the number of particles and the temperature are fixed, compute the chem- ical potential and the energy per unit volume u = (E)/V in terms of EF, kBT, and D(€), up to quadratic order in the temperature. Make sure that your results do not depend on μ. Find the contribution to the heat capacity per unit volume cy due to these fermions, to leading order in the temperature T. Show that cy = T for a certain power k, and determine the "Sommerfeld Parameter" . Consider now the special case of non-relativistic free fermions of spin 1/2 in dimension 3. Compute F in terms of n, h, and m. Simplify your expressions for u(T), μ(T) and y, and write them in terms of EF, kB, n, and T (but without any explicit dependence on ħ or m).

2. Noninteracting fermions at very low temperature: Sommerfeld approximation Consider a system of non-interacting Fermi particles, with an (intensive) density of states D(e) which is a smooth function of e such that D(e) > 0 for e≈EF. Sommerfeld showed that at low temperature, the integral of the product of the Fermi function F(e) = [e³(−) + 1]¯¹ with a smooth function (€) of the energy can be expanded as follows: 7π4 360 ** deF(e)ø(c) = ["_deo(e) + —ª² (kBT)²¢' (µ) + (kBT)" (µ) + O(kËT/µ)б. (3) (a) (b) (c) (d) Apply the Sommerfeld expansion to obtain expressions for the energy per unit volume u = (E)/V and for the particle density n = N/V, containing terms up to quadratic order in the temperature. Assuming that the number of particles and the temperature are fixed, compute the chem- ical potential and the energy per unit volume u = (E)/V in terms of EF, kBT, and D(€), up to quadratic order in the temperature. Make sure that your results do not depend on μ. Find the contribution to the heat capacity per unit volume cy due to these fermions, to leading order in the temperature T. Show that cy = T for a certain power k, and determine the "Sommerfeld Parameter" . Consider now the special case of non-relativistic free fermions of spin 1/2 in dimension 3. Compute F in terms of n, h, and m. Simplify your expressions for u(T), μ(T) and y, and write them in terms of EF, kB, n, and T (but without any explicit dependence on ħ or m).

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