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Bean's critical state model, introduced by C. P. Bean[1][2] in 1962, gives amacroscopic explanation of the irreversiblemagnetization behavior (hysteresis) of hardType-II superconductors.
Hard superconductors often exhibithysteresis in magnetization measurements. C. P. Bean postulated for theShubnikov phase an extraordinary shielding process due to the microscopic structure of the materials. He assumed lossless transport with a critical current densityJc(B)(Jc(B→0) = const. andJc(B→∞) = 0). An external magnetic field is shielded in the Meissner phase (H < Hc1) in the same way as in a soft superconductor. In the Shubnikov phase(Hc1 < H < Hc2), the critical current flows below the surface within a depth necessary to reduce the field in the inside of the superconductor toHc1.
To understand the origin of the irreversible magnetization: assume a hollow cylinder in an external magnetic field parallel to the cylinder axis.[3] In the Meissner phase, a screening current is within the London penetration depth. ExceedingHc1,vortices start to penetrate into the superconductor. These vortices are pinned on the surface (Bean–Livingston barrier). In the area below the surface, which is penetrated by the vortices, is a current with the densityJc. At low fields(H < H0), the vortices do not reach the inner surface of the hollow cylinder and the interior stays field-free. ForH > H0, the vortices penetrate the whole cylinder and a magnetic field appears in the interior, which then increases with increasing external field. Let us now consider what happens, if the external field is then decreased: Due to induction, an opposed critical current is generated at the outer surface of the cylinder keeping inside the magnetic field forH0 < H < H1 constant. ForH > H1, the opposed critical current penetrates the whole cylinder and the inner magnetic field starts to decrease with decreasing external field. When the external field vanishes, a remnant internal magnetic field occurs (comparable to the remanent magnetization of aferromagnet). With an opposed external fieldH0, the internal magnetic field finally reaches 0T (H0 equates thecoercive field of aferromagnet).
Bean assumed a constant critical current meaning thatH << Hc2. Kimet al.[4] extended the model assuming1/J(H) proportional toH, yielding excellent agreement of theory and measurements on Nb3Sn tubes. Different geometries have to be considered as the irreversible magnetization depends on the sample geometry.[5]