In the drude (classical) model of an electron gas, the equation of motion for electrons subject to an electric field e is dv e dt t m is the drift velocity and r is the scattering time where define conductivity o in terms of the current density j and the electric field, write down expression for j in terms of electron number density n, and hence derive an expression for steady-state conductivity in the drude model in terms of t and n an 5 marks (b) describe the assumptions of the sommerfeld model, state the boundary condition which the electron wavefunction must sat isfy at the edge of the crystal, and give a wavefunction which satisfies these constraints in 3d 5 marks (c) in the debye model, the lattice specific heat can be written 3 cep/t cv (t)9nkb da (e -1) өр where 0p is the debye temperature you many find it useful to treat the integral by either taking lowest-order expansion in x, using e1 + x + o(x2), or using the result 4t (e 1) 15 0 (i) find an expression which describes the low-temperature behaviour t < 0p.[2 marks] 11 expression which describes the high-temperature behaviour t op.[4 marks] (ii) find an (iii sketch the cv(t), showing the low, intermediate, and high temperature behaviours.