In 1966 B. D. Josephson derived the macroscopic equilibrium field equations for metals from thermodynamics rather than from Maxwell's equations, showing that in the mixed state of a type-II superconductor the macroscopic current density satisfies the force-balance equation (1/c)(j×B)+P=0(1/c)(\mathbf{j}\times\mathbf{B})+\mathbf{P}=0, where P\mathbf{P} is the pinning-force density, and rederived it from a stress tensor. The paper itself acknowledges that this macroscopic theory rests on unproved microscopic ground: the elastic (shear) stresses of the flux-line lattice are omitted from the force balance; P\mathbf{P} is defined only operationally by δF1=−∫P⋅ξ dV\delta F_1=-\int\mathbf{P}\cdot\boldsymbol{\xi}\,dV with no derivation from material microstructure; surface critical currents are asserted to be fixed by microscopic considerations that are not carried out; the locality approximation for the free energy is not corrected in rapidly varying regions; and the mechanism limiting the critical current in longitudinal-field (force-free) geometry, where the Lorentz force vanishes, is explicitly stated to be unelucidated. Later literature closed the lattice-elasticity gap (elastic vortex-matter theory is now standard) and built statistical, collective, and strong-pinning frameworks plus flux-cutting and helical-instability mechanisms, but a general first-principles theory of P\mathbf{P} from specified microstructure and a quantitatively settled force-free critical-current mechanism remain open. The problem is to complete these microscopic and elastic foundations of the 1966 macroscopic theory.

Background

A type-II superconductor — one whose Ginzburg–Landau parameter κ=λ/ξ\kappa=\lambda/\xi exceeds 1/21/\sqrt{2}, where ξ\xi is the coherence length over which the superconducting order parameter varies and λ\lambda is the magnetic-field penetration depth — does not expel magnetic fields completely once the field exceeds a lower critical value. Instead it enters the mixed state: magnetic flux penetrates as Abrikosov vortices (also called flux lines), thin filaments of suppressed superconductivity of radius roughly ξ\xi, each carrying one quantum of magnetic flux Φ0=hc/2e\Phi_0=hc/2e and circled by persistent supercurrents, which usually arrange themselves in a triangular flux-line lattice. Ginzburg–Landau (GL) theory is the standard phenomenological theory of superconductivity in terms of a complex order parameter, from which ξ\xi and λ\lambda derive. Material inhomogeneities (defects, precipitates, dislocations — pinning centers) can bind flux lines; this pinning lets the material carry a dissipationless transport current up to a critical current density jcj_c. The macroscopic current density j\mathbf{j} exerts on the vortex system the Lorentz force density (j×B)/c(\mathbf{j}\times\mathbf{B})/c (Gaussian units; B\mathbf{B} is the magnetic induction, i.e. the spatially averaged microscopic field), balanced by the pinning force, giving the force-balance equation (1/c)(j×B)+P=0(1/c)(\mathbf{j}\times\mathbf{B})+\mathbf{P}=0 with P\mathbf{P} the pinning-force density. Together with Maxwell's equations this underlies the critical-state models used to predict magnetization, AC losses, and current capacity of superconducting magnets.

Josephson's 1966 paper put this framework on a thermodynamic footing: it defined the macroscopic fields B\mathbf{B}, H\mathbf{H}, M\mathbf{M} and current density j\mathbf{j} for metals in equilibrium by variational principles (not assuming Maxwell's equations), derived the force-balance equation for the mixed state without invoking forces on flux lines, rederived it from a magnetic stress tensor, and extended the treatment to thin films and the superconducting surface sheath (a surface layer that remains superconducting above the bulk upper critical field, where a two-dimensional mixed state of flux spots occurs).

The paper then states what the macroscopic theory cannot yet do. (1) When the shearing stresses of the flux-line lattice matter, the force-balance equation certainly needs modification — not derived there. (2) The pinning-force density P\mathbf{P} is defined only as a macroscopic average through δF1=−∫P⋅ξ dV\delta F_1=-\int\mathbf{P}\cdot\boldsymbol{\xi}\,dV for slow vortex displacements ξ\boldsymbol{\xi}; no microscopic theory computes it from material microstructure. (3) The critical value of the surface current is said to be determined by microscopic considerations — surface-barrier theory (the Bean–Livingston barrier, an energetic barrier preventing vortex entry at a surface) predates the paper but the critical values are not derived. (4) The free energy is treated as a local volume integral; nonlocality is argued only to smear boundary behavior over microscopic distances, and precise corrections in rapidly varying regions are not worked out. (5) In the longitudinal-field or force-free geometry, where the transport current flows parallel to B\mathbf{B} so that j×B=0\mathbf{j}\times\mathbf{B}=0 and the static force balance would allow an unlimited current, the paper states (footnote 9) that the exact mechanism determining the critical current density has not yet been elucidated and conjectures entangled flux lines; experimentally this geometry shows an enhanced critical current, helically wound vortices, and unusual electric-field behavior — the long-standing longitudinal magnetic field effect.

Subsequent decades produced the elastic theory of the vortex lattice (compression modulus c11c_{11}, dispersive tilt modulus c44(k)c_{44}(k), shear modulus c66c_{66}), statistical and collective pinning theory (weak random pinning acting on an elastically deforming lattice), strong-pinning theory (individual defects that deform the lattice strongly), flux-line cutting (the crossing and reconnection of vortices, allowing flux transport in the force-free geometry), vortex helical-instability scenarios, and large-scale time-dependent GL (TDGL) simulations — the components against which the status of each 1966 gap must be judged.

Problem Statement

Complete the microscopic and elastic foundations of Josephson's 1966 macroscopic equilibrium field/force-balance theory of the type-II superconducting mixed state, by supplying, within one coherent framework, the five linked ingredients the paper itself flags as missing. (1) Incorporate the flux-line lattice's elastic stresses (compression, tilt, and shear) into the macroscopic force-balance equation and stress tensor, reducing to the 1966 local equation (1/c)(j×B)+P=0(1/c)(\mathbf{j}\times\mathbf{B})+\mathbf{P}=0 when the lattice structure is negligible. (2) Derive the macroscopic pinning-force density P\mathbf{P} — defined in 1966 only operationally by δF1=−∫P⋅ξ dV\delta F_1=-\int\mathbf{P}\cdot\boldsymbol{\xi}\,dV for slowly varying vortex displacements ξ\boldsymbol{\xi} — from specified material microstructure (defect type, size, density, and vortex configuration), including its statistical averaging over disorder realizations and its domain of validity, in a way that predicts measurable critical currents rather than fitting them. (3) Compute, from microscopic (Ginzburg–Landau-level or deeper) considerations, the critical surface currents and surface-barrier values that the 1966 paper only attributes to microscopic physics. (4) Work out the nonlocal corrections to the field equations and boundary conditions — or give a controlled demonstration of their negligibility — in the rapidly varying (boundary and short-wavelength) regions where treating the free energy as a local volume integral fails. (5) Identify and quantify the mechanism that limits the transport current density in the longitudinal-field (force-free) geometry, where j×B=0\mathbf{j}\times\mathbf{B}=0 makes the static Lorentz-force balance vacuous — distinguishing, for instance, flux-line cutting from a vortex helical-lattice instability — and predict the observed critical currents and their angular, field, and size dependence. The five ingredients jointly support one objective — the microscopic basis that the 1966 macroscopic theory assumes — and are interrelated: all five (1)–(5) are posed jointly, and components already settled in the standard literature (for example the elastic moduli of the vortex lattice) may be incorporated from established results. The synthesis makes their relation to the remaining gaps explicit and yields quantitative predictions testable against measured critical currents and magnetization.

References

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