Statistical Mechanics · Problem 8.4

Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions — R. K. Pathria

Physics — Statistical Mechanics · fermi-gas-compressibility

Problem

(a) Show that the isothermal compressibility κT\kappa_T and the adiabatic compressibility κS\kappa_S of an ideal Fermi gas are given by κT=1nkTf1/2(z)f3/2(z),κS=35nkTf3/2(z)f5/2(z),\kappa_T = \frac{1}{nkT}\frac{f_{1/2}(z)}{f_{3/2}(z)}, \quad \kappa_S = \frac{3}{5nkT}\frac{f_{3/2}(z)}{f_{5/2}(z)}, where n(=N/V)n(= N/V) is the particle density in the gas. Check that at low temperatures κT≃32nεF[1−π212(kTεF)2],κS≃32nεF[1−5π212(kTεF)2]\kappa_T \simeq \frac{3}{2n\varepsilon_F}\left[1 - \frac{\pi^2}{12}\left(\frac{kT}{\varepsilon_F}\right)^2\right], \quad \kappa_S \simeq \frac{3}{2n\varepsilon_F}\left[1 - \frac{5\pi^2}{12}\left(\frac{kT}{\varepsilon_F}\right)^2\right] (b) Making use of the thermodynamic relation CP−CV=T(∂P∂T)V(∂V∂T)P=TVκT(∂P∂T)V2C_P - C_V = T\left(\frac{\partial P}{\partial T}\right)_V \left(\frac{\partial V}{\partial T}\right)_P = TV\kappa_T\left(\frac{\partial P}{\partial T}\right)_V^2 show that CP−CVCV=49CVNkf1/2(z)f3/2(z)≃π23(kTεF)2(kT≪εF).\frac{C_P - C_V}{C_V} = \frac{4}{9}\frac{C_V}{Nk}\frac{f_{1/2}(z)}{f_{3/2}(z)} \simeq \frac{\pi^2}{3}\left(\frac{kT}{\varepsilon_F}\right)^2 \quad (kT \ll \varepsilon_F). (c) Finally, making use of the thermodynamic relation γ=κT/κS\gamma = \kappa_T/\kappa_S, verify the results of Problem 8.3.

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