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\title{Superconductivity - Assignment 5}
\author{
	Kees van Kempen (s4853628)\\
	\texttt{k.vankempen@student.science.ru.nl}
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\section{$T_c$ upper limit in BCS}
In BCS theory, the formation of Cooper pairs is mediated by phonons.
There is a phonon-electron interaction quantified by the dimensionless quantity 
\[
	\lambda := Vg(\epsilon_F)
\]
with $V$ Cooper's approximate potential and $g(\epsilon_F)$ the density of states near the Fermi surface for the electrons.
A thorough discussion can be found in Annett's book \cite[chapter 6]{annett} and in the slides of week 6 of this course.
The binding energy of the Cooper pairs (i.e. the energy gain of forming these pairs) is
\[
	-E = 2\hbar\omega_De^{-1/\lambda}.
\]
In the weak coupling limit of the BCS theory, the case we have considered so far, it is assumed that $\lambda << 1$.
It is when this assumption breaks down, BCS does not work and we find an upper limit to the critical temperature $T_c$.
We will look at a way to express the critical temperature in terms we can derive, and than look at the values that maximize this critical temperature whilst still following BCS theory.


\section{Energy gap $\Delta$ et al.}


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