Statistical Mechanics · Problem 8.9

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

Physics — Statistical Mechanics · fermi-gas-sommerfeld-expansion-higher-order

Problem

Making use of another term of the Sommerfeld lemma (E.15), show that in the second approximation the chemical potential of a Fermi gas at low temperatures is given by

\mu \simeq \varepsilon_F\left[1 - \frac{\pi^2}{12}\left(\frac{kT}{\varepsilon_F}\right)^2 - \frac{\pi^4}{80}\left(\frac{kT}{\varepsilon_F}\right)^4\right] \tag{8.1.35a}

and the mean energy per particle by

\frac{U}{N} \simeq \frac{3}{5}\varepsilon_F\left[1 + \frac{5\pi^2}{12}\left(\frac{kT}{\varepsilon_F}\right)^2 - \frac{\pi^4}{16}\left(\frac{kT}{\varepsilon_F}\right)^4\right]. \tag{8.1.37a}

Hence determine the

T^3

-correction to the customary

T^1

-result for the specific heat of an electron gas. Compare the magnitude of the

T^3

-term, in a typical metal such as copper, with the low-temperature specific heat arising from the Debye modes of the lattice. For further terms of these expansions, see Kiess (1987).

Related theory

Heat Work In Isobaric Gas Expansion

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