What Is the Bean Model?

The Bean model, also known as the Bean critical-state model, is a theoretical framework describing the distribution of magnetic flux and current density inside type-II superconductors subjected to external magnetic fields. Proposed by C.P. Bean in 1962 and 1964, it posits that the shielding current density inside a superconductor is either at its critical value J_c or zero, with no intermediate states. The model captures the irreversible, hysteretic magnetization behavior that arises because flux lines (Abrikosov vortices) are pinned at material defects and inhomogeneities rather than flowing freely.

Type-II superconductors admit magnetic flux above a lower critical field H_c1 in the form of quantized flux tubes surrounded by circulating supercurrents. These vortices are driven inward by gradients in the applied field but are resisted by pinning forces at grain boundaries, precipitates, and other microstructural features. The Bean model treats this competition as producing a critical current density that the material can sustain everywhere it carries current, with the resulting flux gradient determining how far the magnetic field penetrates the sample.

Critical State and Flux Penetration

In Bean's formulation, when an external field is applied to a superconducting slab, flux enters from the surface and penetrates inward only as far as the critical current can sustain. The field profile inside the sample is piecewise linear, with slope equal to μ₀J_c in regions where current flows and zero slope in the flux-free interior. As the applied field increases, penetration deepens; when the field exceeds a characteristic full-penetration value, flux and current reach the center of the sample. This geometric picture, described rigorously in analysis of the Bean model available on arXiv, enables quantitative predictions of magnetization as a function of applied field for simple geometries including slabs, cylinders, and discs.

Magnetization and Critical Current Measurement

One of the most practical consequences of the Bean model is a direct relationship between the width of the magnetization hysteresis loop and the critical current density J_c. By measuring the difference in magnetization between increasing and decreasing field sweeps using a magnetometer, researchers can infer J_c without requiring electrical contacts on the sample. This technique is standard practice in the characterization of high-temperature superconductors such as YBCO (yttrium barium copper oxide) and BSCCO (bismuth strontium calcium copper oxide). The extended Bean model, treating ellipsoidal geometries and demagnetization effects, extends this methodology to single crystals, as described in extended Bean critical-state analysis on ScienceDirect. Variational and numerical formulations of the model address geometries where analytical solutions are not available, as shown in ResearchGate's treatment of the Bean model's variational formulation.

Limitations and Extensions

The Bean model assumes J_c is independent of the local magnetic field magnitude, a reasonable approximation for low-temperature superconductors where strong pinning holds J_c nearly constant up to moderate fields. For high-temperature superconductors and at elevated temperatures, J_c decreases with field, and modified models such as the Kim model introduce a field-dependent critical current. Beyond static fields, the Bean model underpins calculations of AC loss, the energy dissipated per cycle in superconducting components carrying alternating currents or experiencing time-varying applied fields.

Applications

The Bean model has applications in a range of superconducting technology areas, including:

• Characterization of critical current density in bulk and thin-film superconductors
• Design and loss estimation for superconducting power cables and transformers
• Magnetic levitation and flux-pinning-based bearing analysis
• Fault current limiters for power grid protection
• Magnetic shielding design for sensitive instrumentation