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ass3: Do question 8 according to slide 16

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Kees van Kempen
2022-03-15 18:44:29 +01:00
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superconductivity_assignment3_kvkempen.tex
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@ -57,18 +57,19 @@ $\kappa = \frac{\lambda}{\xi} = 34 > \frac{1}{\sqrt{2}}$,
which means we are indeed dealing with a type-II superconductor. which means we are indeed dealing with a type-II superconductor.
As $B_{c1} < B_E < B_{c2}$, the cylinder is in the vortex state. As $B_{c1} < B_E < B_{c2}$, the cylinder is in the vortex state.
From the previous set of assignments, we know what the currents in the cylinder look like. From the previous set of assignments, we know what the currents in the cylinder look like.
From free energy considerations, we have found in lecture 4 that for type-II superconductors, it is favorable to allow flux quanta inside the superconductor in this vortex state.
In this derivation, the contribution of one flux quantum is considered, but the consideration holds for many vortices, until they start to interact and repel eachother.
At that point, the vortex-vortex interaction orders the vortices in a lattice.
When the vortex cores start to overlap, there are no superconducting regions left, thus the material enters the normal conducting state.\footnote{I wanted to paint a complete picture althought it is not needed to answer the question.}
Minimizing the free energy over the flux shows the energy is lowered for determined thresholds $B_{c1} < B_E < B_{c2}$.
The average field inside the cylinder is gives as Let's start with the result from said free energy considerations.
The average field inside the cylinder is given by the following self-consisting equation as
\[ \[
\langle \vec{B} \rangle = \frac{1}{V_{\text{cylinder}}} \int_{\text{cylinder}} \vec{B}(\vec{r}) d\vec{r} . B = B_E - \frac{\phi_0}{8\pi\lambda^2}\ln{\frac{\phi_0}{4\exp{(1)}\xi^2B}}.
\] \]
Plugging in the values for \ce{Nb3Sn}, $B_E = \SI{1}{\tesla}$, and $\phi_0 = \SI{2.0678}{\weber}$, $B$ is found as $B = \SI{.986}{\tesla} \approx B_E$ by intersection.
To determine this $\vec{B}$ inside the material, we first need to know how many vortices there are. %https://www.wolframalpha.com/input?i=B+%3D+1+-+%282.0678*10%5E-15%29%2F%288*pi*%28124*10%5E-9%29%5E2%29+*+ln%282.0678*10%5E-15%2F%284*exp%281%29*%283.65*10%5E-9%29%5E2+*+B%29%29
We assume that every vortex lets through only one flux quantum $\Phi_0$,
and that the vortices will arange themselves as far as possible from eachother.
If their distance then is large enough to assume there is no overlap between regions of finite $\vec{B}$ around them,
we can calculate the average field by just summing over the quanta and lastly over the field that penetrates the material in the outside of the cylinder.
For this latter calculation, we can use the field for a type-I superconductor.
\section{Superconducting wire} \section{Superconducting wire}