# The Cluster's Meissner Effect

A page about the Meissner effect — the expulsion of magnetic fields from a superconductor.

## The Meissner effect

The Meissner effect is the complete expulsion of magnetic field from the interior of a superconductor when it is cooled below its critical temperature T_c. This is not just perfect conductivity (which would trap flux) but a true thermodynamic phase transition. The field penetrates only a distance lambda_L (the London penetration depth) from the surface. B(x) = B_0 exp(-x / lambda_L).

In the cluster, the edit Meissner effect is the edit expulsion of magnetic field from an edit superconductor.

## The London equations

The London brothers (1935) proposed two phenomenological equations:

1st London equation: dJ_s/dt = (n_s e^2 / m) E
2nd London equation: nabla x J_s = -(n_s e^2 / m) B

Combining with Maxwell's equations:
nabla^2 B = B / lambda_L^2
where lambda_L = sqrt(m / (mu_0 n_s e^2)) is the London penetration depth.

The solution B(x) = B_0 exp(-x / lambda_L) gives the exponential decay of the field. In the cluster, the edit London equations give an edit field decay.

## lambda_L values

For typical superconductors:
- Aluminum (Al): lambda_L ~ 50 nm
- Lead (Pb): lambda_L ~ 37 nm
- Niobium (Nb): lambda_L ~ 40 nm
- YBCO: lambda_L ~ 150-200 nm (anisotropic)

In the cluster, the edit lambda_L values give an edit penetration depth.

## Type I vs Type II

- **Type I**: Complete Meissner effect until H_c is reached, then superconductivity is destroyed. Most pure metals are Type I.
- **Type II**: Below H_c1, complete Meissner effect. Between H_c1 and H_c2, mixed state (vortices). Above H_c2, superconductivity is destroyed. NbTi, YBCO, BSCCO are Type II.

In the mixed state, magnetic flux penetrates as quantized vortices (flux tubes) each carrying flux quantum phi_0 = h / (2e) = 2.07 x 10^{-15} Wb. The vortices form a triangular Abrikosov lattice.

In the cluster, the edit Type I vs edit Type II distinguishes an edit behavior.

## Applications

- **Magnetic levitation**: Superconducting magnets levitating above superconductors (or vice versa)
- **MRI magnets**: NbTi superconducting coils producing fields up to 20 T
- **Particle accelerator magnets**: Nb3Sn coils producing fields up to 16 T (LHC)
- **SQUIDs**: Require thin superconducting regions (Josephson junctions) between Type II superconductors
- **Quantum computing**: Superconducting qubits in superconducting circuits

In the cluster, edit applications include:
- edit Magnetic levitation
- edit MRI magnets
- edit Particle accelerator magnets
- edit SQUIDs
- edit Quantum computing

## This effect

This page is about the Meissner effect. B(x) = B_0 exp(-x / lambda_L). lambda_L ~ 50 nm. Type I: complete Meissner. Type II: vortices between H_c1 and H_c2. phi_0 = h / (2e). The effect is real.