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modified: basicStatMechLecture18.tex

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commit 2dc61fbbaaa140b25f279865127a6ddab12b416c 1 parent 82b3116
@peeterjoot authored
Showing with 35 additions and 4 deletions.
  1. +35 −4 notes/blogit/basicStatMechLecture18.tex
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39 notes/blogit/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}.
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