Solid state Physics (1st Edition) Chapter 2, Problem 2P ☐ (2 Bookmarks) Show all steps: ON Problem Thermodynamics of the Free and Independent Electron Gas (a) Deduce from the thermodynamic identities Cv = (24) = (³³). T ат (2.96) From Eqs. (2.56) and (2.57), and from the third law of thermodynamics (s→ 0 as T→ 0) that the entropy density, s = S/V is given by : kn dk [C- 1)ur)—) +J/\t」「 Where f (E(K)) is the Fermi function (Eq. (2.56)). (2.97) (b) Since the pressure P satisfies Eq. (B.5) in Appendix B, P=-(u - Tsun), deduce from (2.97) that dk 17 √ In (1 + exp[- P knT (h²k²/2m) KBT - μ (2.98) Show that (2.98) implies that P is a homogeneous function of u and T of degree 5/2; that is, P(2µ, 2T) = 25/² P(µ, T) (2.99) Ρίμ, for any constant A. (c) Deduce from the thermodynamic relations in Appendix B that OP OP = S. (2.100) = n₁ δμ/τ от (d) By differentiating (2.99) with respect to A show that the ground-state relation (2.34) holds at any temperature, in the form P = u(2.101) (e) Show that when KBT << &F, the ratio of the constant-pressure to constant-volume specific heals satisfies - 1 = 3 π² (KBT 2 &F + 0 (KBT EF (f) Show, by retaining further terms in the Sommerfeld expansions of u and n, that correct to order 73 the electronic heat capacity is given by C₁ == -KB² Tg(&F) 3 90 g(EF) g" (EF) (2.102) [1()] - 21 g(εF)