Critical State of Superconducting Solenoids

Abstract

The critical state is a term introduced by Bean [¹] to describe the magnetic properties of a bulk type II superconductor. In this state every region of the superconducting material carries the maximum induced critical current density. Kim, Hempstead, and Strnad [²] extended this idea through experiments on the resistive states of cylindrical tubes of Nb₃Sn and Nb-25% Zr to show that a Lorentz force type equation relates the field and the current density as determined by the local value of field in the critical state. Their empirical relation is

$$\alpha = J_c (H + B_o )$$

(1)

where J _c is the current density, H is magnetic field, and α and B ₀ are temperature-sensitive constants depending on the microstructure of the material. α essentially is the high-field (H≫B ₀) Lorentz force and is a measure of the maximum current-carrying capacity. The physical origin of B ₀ is not understood. Kim et al. [²] measured the temperature dependence of α and also found that the persistent currents decayed as the logarithm of time. The theoretical verification of these observations was given by Anderson [^3,4] in Ms flux creep theory. The core of this theory is thermally activated motion of flux bundles aided by the Lorentz force to overcome the pinning effects of physical defects. The importance of the Lorentz force on the flux lines already had been pointed out by Gorter [⁵]. In addition to the work cited, others have shown that (1) is satisfied for single crystal whiskers and strips of Nb₃Sn [^6–8].