A Model for Magnetic Instabilities in Hard Superconductors: The Adiabatic Critical State

Abstract

The isothermal critical‐state model of hard superconductors is extended to include the effects of heating when the applied field is changed suddenly and magnetic flux enters adiabatically into the bulk. We consider the following specific situation. A semi‐infinite slab of superconductor is cooled in a magnetic field lying in its surface plane. Next, the external field is raised isothermally by an amount H_s. This excess field decreases linearly to a depth δ= 10H_s/4πJ_c from the surface. Finally, the field is raised by an infinitesimal amount ΔH in a time short compared to the thermal diffusion time and long compared to the electromagnetic diffusion time. Each element of volume exposed to the changing field receives a thermal impulse proportional to the local‐flux‐change times J_c. This thermal impulse, in turn, lowers the critical current and allows more flux to penetrate. We find that if H_s exceeds some critical value H_fj, then the isothermal critical state is not the only allowed state of the superconductor. This instability field is given in terms of the critical current density J_c, derivative of the critical current density with temperature, ∂J_c/∂T, and the volume specific heat C by the formula H_fj= [−π³CJ_c/(∂J_c/∂T)]^1/2. The application of the incremental field ΔH can initiate an avalanching process, or a flux jump, that terminates in an adiabatic critical state. Immediately following the flux jump the internal field, the induced supercurrent, and the temperature rise at each position are associated in a self‐consistent way with the avalanche of flux that has entered the superconductor. In this framework a flux jump is viewed as a switching from the isothermal critical state to an adiabatic critical state. The magnitude of the jump is related to J_s and is calculated.