(Solved): (i) The conduction electron contribution to the heat capacity of a metal is o ...

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(i) The conduction electron contribution to the heat capacity of a metal is only about \( 1 \% \) of the classical prediction. Explain why this is the case, a sketch may aid your answer. (ii) At \( \mathrm{T}=0 \mathrm{~K} \) the Fermi energy \( \left(E_{f}\right) \) corresponds to a maximum momentum \( \left(k_{f}\right) \), the Fermi surface. Derive an expression for the Fermi energy that depends only on the electron concentration (iii) The density of states for a 3D metal is given as \[ D(E)=\frac{V}{2 \pi^{2}}\left(\frac{2 m}{\hbar^{2}}\right)^{\frac{3}{2}} E^{\frac{1}{2}} . \] show that the simplified density of states at the Fermi energy \( \left(D\left(E_{f}\right)\right) \) is \[ D\left(E_{f}\right)=\frac{3}{2} \frac{N}{E_{f}} . \] (iv) In the Drude model when electrons flow in a conductor they experience a force of the following form \[ F=m\left(\frac{d v}{d t}+\frac{v}{\tau}\right)=-e E . \] Clearly explain all of the terms in the equation and show that the mean drift velocity stabilises as \[ v=-\mu E \] where \( \mu \) is the electron mobility. (v) In a Hall sensor what does the difference in the direction of the Hall field indicate about the charge carriers?