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De Haas-van Alphen Effect of Free Electron Gas Revisited

The free electron gas in a uniform magnetic field at low temperature is restudied. The grand partition function previously obtained by Landau's quantitative calculation contains three parts, which are all approximate. An improved calculation is presented, in which two of the three parts are obtained in exact forms. A simple remedy for Landau and Lifshitz's qualitative calculation in the textbook is also given, which turns the qualitative result into the same one as obtained by the improved quantitative calculation. The chemical potential is solved approximately and the thermodynamic quantities are calculated explicitly in both a weak field and a strong field. The thermodynamic quantities in a strong field obtained here contain both non-oscillating and oscillating corrections to the corresponding results derived from Landau's grand partition function. In particular, Landau's grand partition function is not sufficiently accurate to yield our nonzero results for the specific heat and the entropy. An error in the Laplace-transform method for the problem is corrected. The results previously obtained by this method are also improved.