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Question (a) In the free electron case the density of levels

Question

(a) In the free electron case the density of levels at the Fermi energy can be written in the form $g(epsilon )=mk_F/hbarpi^2$ . Show that the general form $small g_n(epsilon )=int_{S_n(epsilon)} frac{ds}{4pi^3} frac{1}{left | { triangledown epsilon_n(k) } right |}$ reduces to this when $small epsilon_n(k)= hbar^2k^2/ 2m$ and the (spherical) Fermi surface lies entirely within a primitive cell.

(b) Consider a band in which, for sufficiently small k, $small epsilon_n(k)= epsilon_0 + (hbar^2/ 2)(frac{k_x^2}{m_x} +frac{k_y^2}{m_y}+frac{k_z^2}{m_z})$ (as might be the case in a crystal of orthorhombic symmetry) where $small m_x, m_y,m_z$ are positive constants . Show that if $small epsilon_0$ that this form is valid, then $small g_n(epsilon)$ is proportional to $small (epsilon- epsilon_0)^frac{1}{2}$ , so its derivative becomes infinite (Van Hove Singularity) as $small epsilon$ approaches the band minimum. (Hint: Use the form $small g_n(epsilon)=int frac{dk}{4pi^3}delta (epsilon- epsilon_n(k))$ for the density of levels) Deduce from this that if the quadratic form for $small epsilon_n(k)$ remains valid up to $small epsilon_F$ , then $small g_n(epsilon_F)$ can be written in the obvious generalization of the free electron form: $small g_n(epsilon_F)=frac{3}{2}frac{n}{epsilon_F-epsilon_0}$ where n is the contribution of the electrons in the band to the total electronic density.

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