Kim model for magnetization of type-II superconductors

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Abstract
We have calculated the initial magnetization curves and complete hysteresis loops for hard type-II superconductors. The critical-current density Jc is assumed to be a function of the internal magnetic field Hi according to Kim’s model, Jc(Hi)=k/(H0+‖Hi‖), where k and H0 are constants. As is the case for other critical-state models, additional assumptions are that bulk supercurrent densities are equal to Jc, and that the lower critical field is zero. Our analytic solution is for an infinite orthorhombic specimen with finite rectangular cross section, 2a×2b (a≤b), in which a uniform field H is applied parallel to the infinite axis. Assuming equal flux penetration from the sides, we reduced the two-dimensional problem to a one-dimensional calculation. The calculated curves are functions of b/a, a dimensionless parameter p=(2ka)1/2/H0, and the maximum applied field Hm. The field for full penetration is Hp=H0[(1+p2)1/2−1]. A related parameter is H*m=H0[(1+2p2)1/2−1]. Hysteresis loops were calculated for the different ranges of Hm : Hm0 when p≪1, the width of the hysteresis loop ΔM may be used to deduce Jc as a function of H: Jc(H)=ΔM(H)/[a(1−a/3b)].