Model of Flux Trapping in Cooling Down Process

The flux trapping that occurs in the process of cooling down of the superconducting cavity is studied. The critical fields $B_{c2}$ and $B_{c1}$ depend on a position when a material temperature is not uniform. In a region with $T\simeq T_c$, $B_{c2}$ and $B_{c1}$ are strongly suppressed and can be smaller than the ambient magnetic field, $B_a$. A region with $B_{c2}\le B_a$ is normal conducting, that with $B_{c1}\le B_a < B_{c2}$ is in the vortex state, and that with $B_{c1}> B_a$ is in the Meissner state. As a material is cooled down, these three domains including the vortex state domain sweep and pass through the material. In this process, vortices contained in the vortex state domain are trapped by pinning centers distributing in the material. A number of trapped fluxes can be evaluated by using the analogy with the beam-target collision event, where beams and a target correspond to pinning centers and the vortex state domain, respectively. We find a number of trapped fluxes and thus the residual resistance are proportional to the ambient magnetic field and the inverse of the temperature gradient. The obtained formula for the residual resistance is consistent with experimental results. The present model focuses on what happens at the phase transition fronts during a cooling down, reveals why and how the residual resistance depends on the temperature gradient, and naturally explains how the fast cooling works.