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Bean's critical state model, introduced by C. P. Bean in 1962, gives a
To understand the origin of the irreversible magnetization: assume a hollow cylinder in an external magnetic field parallel to the cylinder axis. In the Meissner phase, a screening current is within the London penetration depth. Exceeding ''Hc1'',
Bean's critical state model, introduced by C. P. Bean in 1962, gives a macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic.
Overview
When applied to physical phenomena a ...
explanation of the irreversible magnetization behavior ( hysteresis) of hard Type-II superconductors
In superconductivity, a type-II superconductor is a superconductor that exhibits an intermediate phase of mixed ordinary and superconducting properties at intermediate temperature and fields above the superconducting phases.
It also features the ...
.
Assumptions
Hard superconductors often exhibit hysteresis 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: assume a hollow cylinder in an external magnetic field parallel to the cylinder axis. In the Meissner phase, a screening current is within the London penetration depth. Exceeding ''Hc1'', vortices
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
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 density ''Jc''. 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 remanent magnetization of a ferromagnet
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
). With an opposed external field ''H0,'' the internal magnetic field finally reaches 0T (''H0'' equates the coercive field
Coercivity, also called the magnetic coercivity, coercive field or coercive force, is a measure of the ability of a ferromagnetic material to withstand an external magnetic field without becoming demagnetized. Coercivity is usually measured in ...
of a ferromagnet
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
).
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.References
See also
* * * * {{DEFAULTSORT:Bean's Critical State Model Superconductivity Magnetic hysteresis