Please solve below questions as much as you can.

Thank you for your help, and I'll make sure to rate.

7. (Optional) Density of states in a superconductor Consider electrons in a real metal whose Fermi surface is spherical with Fermi wavevector 16F and Fermi energy 61?. We are interested only in states near the Fermi-surface, where we can approximate their energies as 66 = 6 61:\" = hvp(k lap). 66 is the energy measured relative to the Fermi energy (it can be positive or negative), and Up is a parameter known as the Fermi velocity. (a) Find the Fermi-velocity for a free electron gas in d = 3. (b) For real systems, the energy dispersion 6(k) need not be quadratic in the momentum. Express the Fermi velocity dened above in terms of the derivative(s) %|E=kp of 60:). Sometimes the proportionality between 66 and (k kp) is expressed, not in terms of Up but in terms of an \"elfective mass\" m* for the excitations. Using the result you just obtained as a guide, write the equation for 66 in terms of a suitable dened m". (c) When the interaction which gives rise to superconductivity is turned on, the new energies 66' are related to the old ones, 66, by the equation 66' = sign(6e) \\/ 662 + A2. where A is a constant. Find the new density of states D(JE') and plot it together with the density of states D056) of a metal