Mathematical analysis of a flux-jump model in superconductivity

Abstract

Type II superconductors can trap a transient magnetic field and become "cryomagnets" that are very useful for applications. During this process, flux jumps i.e. sudden jumps of the total magnetization occur and hinder the properties of these magnets. To understand the electrodynamics of these systems and in particular flux jumps, we analyzed mathematically a model based on Maxwell's equations and temperature in a 1D configuration. When a magnetic pulse is applied to a superconductor, three effects occur, from fastest to slowest: Joule heating, magnetic relaxation and temperature diffusion. Adimensionalising the problem, we obtain a nonlinear diffusion for the magnetic field coupled to a forced diffusion equation for the temperature with only two parameters. Two regimes occur, depending on temperature: for medium temperature the heat capacity of a sample can be assumed constant while for low temperature it depends on temperature causing a nonlinear temperature evolution. Flux jumps can be explained using the fixed points of the equations. We found that they occur for pulses of duration close to the magnetic relaxation time and mostly at low temperature because of the nonlinear dependance. Flux trapping is maximal for medium amplitude long duration pulses and low to medium temperatures, so these conditions are optimal to produce better cryomagnets.