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I have been doing some readings on superconductivity and have come across the London penetration depth. I somewhat understand how it comes into play with the London equation and Ampère's law.

Right now, I am having a hard time finding out how one would, if one is given a material that is said to be a superconductor and knowing only the critical temperature, be able to determine the $\lambda_L$ of this material.

I know that it is possible to use μSR, but how exactly would one go about doing that? Also, is that the only experimental method if one does not know anything else about the material?

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There is a good introductory explanation to μSR and using it to measure the penetration depth here.

Spin-polarized muons in condensed matter physics

It basically works because the magnetic field distribution in a type-II superconductor depends on the penetration depth and the μSR technique can be used to measure the field distribution in a material.

Another method is to use a tunnel diode oscillator.

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