Bean's critical state model
Encyclopedia
Bean's critical state model, introduced by C. P. Bean in 1962, gives a macroscopic
explanation of the irreversible magnetization
behavior (hysteresis
) of hard Type-II superconductors.
in magnetization measurements. C. P. Bean postulated for the Shubnikov phase an extraordinary shielding process due to the microscopic structure of the materials. He assumed lossless transport with a critical current density Jc(B) (Jc(B→0) = const. and Jc(B→∞) = 0). An external magnetic field is shielded in the Meissner phase (H < Hc1) in the same way than 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 to Hc1.
Macroscopic
The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...
explanation of the irreversible magnetization
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...
behavior (hysteresis
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...
) of hard Type-II superconductors.
Assumptions
Hard superconductors often exhibit hysteresisHysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...
in magnetization measurements. C. P. Bean postulated for the Shubnikov phase an extraordinary shielding process due to the microscopic structure of the materials. He assumed lossless transport with a critical current density Jc(B) (Jc(B→0) = const. and Jc(B→∞) = 0). An external magnetic field is shielded in the Meissner phase (H < Hc1) in the same way than 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 to Hc1.
Explanation of the irreversible magnetization
To understand the origin of the irreversible magnetization, let us assume a hollow cylinder in an external magnetic field parallel to the cylinder axis. In the Meissner phase, a screening current flows within the London penetration depth. Exceeding Hc1, 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, a current with the density Jc flows. At low fields (H < H0), the vortices do not reach the inner surface of the hollow cylinder and the interior stays field-free. For H > 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 for H0 < H < H1 constant. For H > 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 remnant magnetization of a ferromagnet). With an opposed external field H0, the internal magnetic field finally reaches 0T (H0 equates the coercive field of a ferromagnet).Extension of Bean’s model
Bean assumed a constant critical current meaning that H << Hc2. Kim et al. extended the model assuming 1/J(H) proportional to H, 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.See also
- Ideally hard superconductorIdeally hard superconductorAn ideally hard superconductor is a type II superconductor material with an infinite pinning force . In the external magnetic field it behaves like an ideal diamagnet if the field is switched on when the material is in the superconducting state, so-called "zero field cooled" regime...
- Superconductor
- Type-II superconductorType-II superconductorA Type-II superconductor is a superconductor characterized by the formation of vortex lattices in magnetic field. It has a continuous second order phase transition from the superconducting to the normal state within an increasing magnetic field....
- HysteresisHysteresisHysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...