Bean's critical state model

C. P. Bean postulated for the Shubnikov phase an extraordinary shielding process due to the microscopic structure of the materials.

An external magnetic field is shielded in the Meissner phase (H < Hc1) in the same way as in a soft superconductor.

In the Shubnikov phase (Hc1 < H < Hc2), the critical current flows below the surface within a depth necessary to reduce the field in the inside of the superconductor to Hc1.

At low fields (H < H0), the vortices do not reach the inner surface of the hollow cylinder and the interior stays field-free.

Kim et al.[4] extended the model assuming 1/J(H) proportional to H, yielding excellent agreement of theory and measurements on Nb3Sn tubes.

Calculated magnetization curve for a superconducting slab, based on Bean's model. The superconducting slab is initially at H = 0. Increasing H to critical field H* causes the blue curve; dropping H back to 0 and reversing direction to increase it to -H* causes the green curve; dropping H back to 0 again and increase H to H* causes the orange curve.
A schematic of the magnetic field distribution in a superconducting cylinder during the change of external magnetic field H, based on Bean's model.