1. Sommerfeld Expansion in 2D a. Show that every term in the Sommerfeld expansion for density n vanishes except for the T =0 term. Also show that this procedure (erroneously) predicts the exact relationship μ =E F . b. By performing the exact integration analytically, derive the following result: μ +k B T*ln(1+exp(-μ /k B T))=E F c. Using a suitable expansion of the natural log, show that the correction from μ =E F is negligible. d. The conclusion in (a) included all orders of the expansion – so why is there a discrepancy with the exact solution? 2. Free electron gas with an anisotropic dispersion a. Suppose one has a free electron gas in 3D where the dispersion relation is given by
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Solid State Physics,
Fundamental physics concepts, Sommerfeld, free electron gas, Sommerfeld expansion, exact relationship µ=EF