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Just to make it clear, this is related to solid state physics.

QUESTION

How is the effective mass of electrons in uniform magnetic field related to the specific heat of an electron gas?

DETAILS

This is a homework question. I just don't know where to start. We have the energy of electrons nearby the minimum of the band \begin{equation} E(\pmb{k})=E(\pmb{k}_0)+\frac{\hbar ^2}{2}(\pmb{k}-\pmb{k}_0)^T\hat{m}^{-1}(\pmb{k}-\pmb{k}_0), \end{equation}

where

\begin{equation} \hat{m}^{-1}= \begin{pmatrix} m_T^{-1}& 0& 0\\ 0& m_T^{-1}& 0\\ 0& 0& m_L^{-1} \end{pmatrix} , \end{equation} where $m_T$ and $m_L$ are transversal and longitudal effective mass. The magnetic field is $$\pmb{B}=(B_x,B_y,0).$$ If it helps (although I don't see whether it can), the first task was to find the cyclotron frequency, which I managed \begin{equation} \omega_c=\frac{eB}{\sqrt{m_T m_L}}. \end{equation}

I think that somewhere in the process equation \begin{equation} c_V=\frac{1}{3}(\pi k_B)^2g(E_F)T \end{equation} will be used, but I don't see how.

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