# Magnetic Suspectibility, Magnetic Field Expuslion, and Superconductors

I am reading a kit which tries to measure changes in magnetic susceptibility as a function of temperature, in the hope to capture the transition around the critical temperature. However, I am a little confused by this statement:

"This experiment is based on the expulsion of a magnetic field by the superconducting sample (the Meissner Effect). A current introduced into the coil will generate a magnetic field. When cooled below the Critical Temperature, the sample expels the induced field which can be seen by a distinct change in the inductance of the coil."

A coil surrounds a superconducting material applying $$H$$. The superconductor should create an $$M$$ that is equal to and opposite to the applied field $$H$$. From what I understand, lowering the temperature changes the nature of the material such that the magnetic susceptibility reaches $$-1$$. Why this happens is something I do not understand. If someone could give a brief explanation, I would appreciate it.

Second, how exactly is magnetic "expulsion" related to the inductance of the coil?

$$L = \frac{N^2 \mu_0 \mu_r A}{l} = \frac{N^2 \mu_0 (\chi + 1) A}{l}$$

Where $$\chi$$ is the magnetic susceptibility. Nothing in this equation is directly related to magnetic field expulsion [however that is quantified]. Could someone please explain what exactly is going on?

• The counter mechanism in Meissner effect is attributed to movements of electrons within the material, which will arrange themselves exactly to cancel out magnetic field inside that material. Experimentally, it is simple enough to demonstrate that superconducting materials will levitate or lock itself above or below a magnet, thus some physical mechanism must be happening to cause this. Commented Nov 18, 2022 at 5:29
• Does this change/moment in electrons ultimately change $\chi$ or does it have some other effect on inductance somehow? Commented Nov 18, 2022 at 5:40
• $LI = \Phi$, where $\Phi$ is the flux through the current loop (you have a coil but the general idea is still there, mutatis mutandae). If the flux is altered by the application of a counter-field from the superconducting sample, and the current is unaltered, the inductance changes.
– Tony
Commented Nov 18, 2022 at 5:54

One can answer this question with many layers of complexity.

Simple explanation

When the superconductor is cooled down below a critical temperature $$T_c$$, suddenly a dissipationless current flows around the edge of the sample and cancels out the magnetic field inside the sample. The magnetic susceptibility of the sample is $$\chi \approx 0$$ when $$T>T_c$$, and $$\chi = -1$$ when $$T, so the inductance changes abruptly lowering the temperature.

Phenomenological description

This effect has been phenomenologically described by London by the introduction of this equation: $$\nabla\times \vec{j}_s = - \frac{n e^2}{m} \vec{B},$$ where $$\vec{j}_s$$ is the "supercurrent" density, $$\vec{B}$$ the magnetic field, $$n$$ is the density of carriers, $$e$$ is their electric charge and $$m$$ is their mass. Taking the curl of Maxwell's equation $$\nabla\times\vec{B}=\mu_0\vec{j}_s$$ and considering the other Maxwell equation $$\nabla\cdot\vec{B}=0$$ and the London equation above, one gets the effective equation for the magnetic field inside the superconductor: $$\nabla^2\vec{B} = \frac{1}{\lambda^2}\vec{B}$$ where $$\lambda^2 = m/ne^2\mu_0$$. Solving this equation in simple situations one finds that the field decays inside the superconductor as $$\approx e^{-z/\lambda}$$, where $$z$$ is the distance from the surface.

Physical explanation

The London equation comes out of the blue, so we need to find the physical mechanism behind it. This is tricky and involves quantum mechanics, so I will just discuss it briefly. The idea comes from Bardeen, Cooper and Schrieffer and it's the following. Electrons (close to the Fermi level) in a crystal lattice feel an effective attractive interaction due to phonons, i.e. lattice vibrations. It is possible to prove that below the critical temperature, no matter how small the attraction is, some of the electrons form bound pairs (Cooper pairs) that are in a coherent quantum state (electrons now have a collective behavior, and they no longer behave as independent particles). Mathematically this means that instead of a bunch of particles, we are now describing a unique "blob" of particles by means of a unique complex function of the space coordinates: $$\Delta(\vec{r}) = |\Delta(\vec{r})|e^{i\theta(\vec{r})}$$. We can use this field to compute physical observables, such as the current density (or the energy spectral gap...); however it turns out that the phase $$\theta(\vec{r})$$ has no physical meaning, thus it is a gauge degree of freedom, just like the gauge degree of freedom of the electromagnetic vector potential $$\vec{A}(\vec{r})$$. We can use this simple observation to write down a Lagrangian (a function from which we can find the relevant equations) in a very natural way: in such a way that the two gauge degrees of freedom "cancel out". The equations following from this Lagrangian are indeed the London equation and the Maxwell equations that we have used above to explain the Meissner effect.

• Nice answer. Since Cooper pairs mechanism (or BCS theory in general) is applicable only up to an upper limit of < 50K, would this mean that most high temperature superconductors are likely using a different mechanism? Taking away phonon/regional caused pairing, there seem not many candidate mechanisms left other than Coulomb repulsion between the electrons, could this repulsion conceivably have caused spin additions among groups of electrons, causing them to behave as bosons and superconduct? Commented Nov 22, 2022 at 3:16
• You are right, phonon coupling is probably not the mechanism of pairing in high temperature superconductors, and as you said the only candidate is Coulomb repulsion. As far as I know there is still not a consensus on the mechanism of pairing in this case, even though there are many ideas in the literature. Commented Nov 22, 2022 at 9:50
• I wonder how much of a handicap it is not knowing the exact mechanism. Does BCS theory help in discovering/predicting new low temperature superconductors? Commented Nov 22, 2022 at 13:47