# Superconducting gap, temperature

Tinkham (page 63) states that the temperature dependence of the gap energy of a superconductor $\Delta(T)$ can be calculated using the following integral: http://i45.tinypic.com/w1s13t.png

How can this actually be carried out? I am not sure how to approach this problem or re-arrange the equation for finding $\Delta(T)$ numerically.

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## 1 Answer

I have not tried it on this specific equation, but in principle you can solve problems like this by a combination of numerical integration and a root finding algorithm. For a given variable $x$ you wish to determine for a fixed value $v$ of the integral, the root finding algorithm will find a solution to the equation

$v-integral(x)=0.$

The root finding algorithm will try to match the variable in such a way that the equation is satisfied, while the integral is evaluated numerically. In your example, $\Delta$ corresponds to the variable $x$ while $1/N(0)V$ takes on the role of v.

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How can that be done in a programming language though? – ElizabethPor Feb 27 at 0:20
I would start with math software like mathematica, since it contains useful algorithms for numerical integration and root-finding. – Frederic Brünner Feb 27 at 1:19
This has been done in Mathematica. I'll link you a screen shot: i.stack.imgur.com/0Km6p.png. Problem is, when I try to do this in another language (i.e. Python: pastie.org/6347014), I cannot ask Python to declare and find $\Delta$ without getting a `x is not declared error`. – ElizabethPor Feb 27 at 11:31
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