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According to Sommerfeld the derivative of the density of states $g'(\varepsilon)$ apears in several thermodynamic quantities. Will this also be the case if one use the correct dispersion relation of crystal? If yes don't we encounter the divergence in these quantities when we pass a van Hove singularity (where $g'(\varepsilon)\rightarrow \infty $)?

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For the logarithmic and weaker singularities that van Hove singularities give rise to in two and three dimensional crystals, $g'(\epsilon)$ will be discontinuous at the critical frequency of any van Hove singularity, so yes, you are quite right. Which quantities did you have in mind? –  Chay Paterson Jun 27 '13 at 18:14
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