# peeterjoot/physicsplay

modified: basicStatMechLecture18.tex

 @@ -5,7 +5,7 @@ \input{../blogpost.tex} \renewcommand{\basename}{basicStatMechLecture18} \renewcommand{\dirname}{notes/phy452/} -\newcommand{\keywords}{Statistical mechanics, PHY452H1S} +\newcommand{\keywords}{Statistical mechanics, PHY452H1S, Fermi gas, specific heat, density of states, graphene, relativisitic gas, chemical potential, energy, Fermi distribution, hole, electron} \input{../peeter_prologue_print2.tex} \beginArtNoToc @@ -113,7 +113,7 @@ \section{Disclaimer} }. \end{dmath} -Here we've extended the integration range without changing much. FIXME: justify for self. Taking derivatives with respect to temperature we have +Here we've extended the integration range to $-\infty$ since this doesn't change much. FIXME: justify this to myself? Taking derivatives with respect to temperature we have \begin{dmath}\label{eqn:basicStatMechLecture18:200} \frac{\delta e}{T} @@ -182,16 +182,47 @@ \section{Disclaimer} \lr{\frac{2m}{\hbar^2}} ^{3/2} \frac{\hbar }{\sqrt{2m}} \lr{6 \pi^2 \rho}^{1/3} -= \inv{4 \pi^2} += +\inv{4 \pi^2} \lr{\frac{2m}{\hbar^2}} \lr{6 \pi^2 \frac{N}{V}}^{1/3} \end{dmath} -FIXME: don't see how this leads to the board result? +Giving + +\begin{dmath}\label{eqn:basicStatMechLecture18:480} +\frac{C}{N} += +\frac{\pi^2}{3} +\frac{V}{N} +\inv{4 \pi^2} +\lr{\frac{2m}{\hbar^2}} +\lr{6 \pi^2 \frac{N}{V}} +^{1/3} +\kB (\kB T) += +\lr{\frac{m}{6 \hbar^2}} +\lr{\frac{V}{N}}^{2/3} +\lr{6 \pi^2} +^{1/3} +\kB (\kB T) += +\lr{\frac{ \pi^2 m}{3 \hbar^2}} +\lr{\frac{V}{\pi^2 N}}^{2/3} +\kB (\kB T) += +\lr{\frac{ \pi^2 m}{\hbar^2}} +\frac{\hbar^2}{2 m \epsilon_{\mathrm{F}}} +\kB (\kB T), +\end{dmath} + +or \begin{dmath}\label{eqn:basicStatMechLecture18:300} +\myBoxed{ \frac{C}{N} = \frac{\pi^2}{2} \kB \frac{ \kB T}{\epsilon_{\mathrm{F}}}. +} \end{dmath} This is illustrated in \cref{fig:lecture18:lecture18Fig4}.