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vatlychatran
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In the book Introduction to Superconductivity by M. Tinkham, there is an integral as shown below. Could you help me to derive the result? Thanks a lot!
"3.2 ORIGIN OF THE ATTRACTIVE INTERACTION
We now must examine the origin of the negative [tex]V_{kk'}[/tex] needed for superconductivity. If we take the bare Coulomb interaction [tex]V(\vec{r})= e^2/r[/tex] and carry out the computation of [tex]V(\vec{q})[/tex]
[tex]V(\vec{q}) = V(\vec{k}-\vec{k'}) = V_{\vec{k}\vec{k'}} = \Omega^{-1}\int V(\vec{r})e^{i\vec{q}.\vec{r}}d\vec{r}[/tex]
we find
[tex]V(\vec{q}) =\frac{4\pi e^2}{\Omega q^2}[/tex]"
"3.2 ORIGIN OF THE ATTRACTIVE INTERACTION
We now must examine the origin of the negative [tex]V_{kk'}[/tex] needed for superconductivity. If we take the bare Coulomb interaction [tex]V(\vec{r})= e^2/r[/tex] and carry out the computation of [tex]V(\vec{q})[/tex]
[tex]V(\vec{q}) = V(\vec{k}-\vec{k'}) = V_{\vec{k}\vec{k'}} = \Omega^{-1}\int V(\vec{r})e^{i\vec{q}.\vec{r}}d\vec{r}[/tex]
we find
[tex]V(\vec{q}) =\frac{4\pi e^2}{\Omega q^2}[/tex]"