Magnetization of type-II superconductors in the modified Bean model

Abstract

Within the construct of the modified Bean model which takes into consideration the surface barrier Δ H and a nonzero value of the lower critical field H c1, we have calculated the initial magnetization curves and full hysteresis loops of type-II superconductors immersed in an external field H = H dc + H ac cos( ωt), where H dc (≥ 0) is a dc bias field and H ac (> 0) is an ac field amplitude. We denote the maximum and minimum values of H by H A(= H dc + H ac) and H B(= H dc − H ac). We consider an infinitely long cylinder with radius a, and the applied field along the cylinder axis. Magnetization equations M( H) for full hysteresis loops are derived for four different ranges of H A: 0 < H A ≤ H c1 + Δ H, H c1 + Δ H ≤ H A ≤ H c1 + Δ H + H p, H c1 + Δ H + H p ≤ H A ≤ H c1 + Δ H + 2 H p, H c1 + Δ H + 2 H p ≤ H A. Here H p is the field for full penetration. Each of these four cases is further classified for several ranges of H B. To describe completely the descending and ascending branches of full hysteresis loops for all cases, 83 stages of H are considered. To verify the present derivations, all the equations were confirmed to be continuous at their end points. Some typical hysteresis loops computed using the appropriate magnetization equations are demonstrated. From the results, we recognize the role of Δ H and H c1 on the hysteresis loops. Δ H does not cause any essential deformation of the M( H) curves, but merely expands them up and down with the increse of Δ H. On the contrary, H c1 introduces a step-like feature into the hysteresis loops, resulting in drastic change in their shape. Present derivations would be a useful tool for analyses of the magnetization curves of type-II superconductors.