40-EFT.WP.Materials.Superconductivity v1.0 | Chapter 4: Free Energy & Field Equations — GL–London–BCS Confluence

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Chapter 4: Free Energy & Field Equations — GL–London–BCS Confluence

I. Objectives & Applicability
This chapter states the minimal free-energy functional and field equations for unconventional superconductors within this volume’s context, incorporating T_fil and grad T_fil to harmonize GL–London–BCS usage. Relations are provided in experiment-ready form with explicit boundary conditions and scale hierarchy. All formulas/symbols/definitions are in English and must pass check_dim; cross-volume citations follow the “volume + version + anchor (P/S/M/I)” convention.

II. GL Free Energy & Stability (Minimal Usable Form)

• S40-1 (Free-energy functional)
  F = ∫_V [ a|psi|^2 + b|psi|^4 + κ_ij (D_i psi)^* (D_j psi) + (1/2μ0_ref)|B|^2       + λ_1 T_fil |psi|^2 + λ_2 (∂_i T_fil)(psi^* D_i psi + c.c.) ] dV,
  with D_i = (∂_i - i q A_i), B = curl A, i,j ∈ {x,y,z}; coefficients a,b,κ_ij,λ_1,λ_2 may vary slowly with T_temp, frequency band, and materials variables.
• S40-2 (Variations & GL field equations)
  Variation w.r.t. psi^* and A yields
  a psi + 2b|psi|^2 psi - D_i(κ_ij D_j psi) + λ_1 T_fil psi + λ_2 (∂_i T_fil) D_i psi = 0,
  curl B = μ0_ref J, with
  J_i = q κ_ij Im(psi^* D_j psi) + J_i^{(T)}, and J_i^{(T)} the tension-induced current contribution from the λ_2 term (to be calibrated by the Chapter 10 inference chain).
• S40-3 (Stability & signs)
  Require b > 0; κ_ij positive-definite; a(T_temp,T_fil) = a_0 (T_temp/T_c - 1) + a_T[T_fil]. A spontaneous order parameter occurs when a_eff = a + λ_1 T_fil < 0.
• S40-4 (Anisotropy & effective-mass tensor)
  Writing κ_ij = (ħ^2/2) (m^{-1})_ij defines an effective mass; the coherence-length tensor satisfies (in principal axes) ξ_i^2 ≍ κ_ii / |a_eff|.

III. London Approximation & Penetration Depth

• S40-5 (London limit)
  In weak fields and away from defects, set |psi| → |psi_0| (const.) and D_i psi ≈ - i q A_i psi, giving
  J_i ≈ - q^2 |psi_0|^2 κ_ij A_j + J_i^{(T)}.
• S40-6 (Anisotropic London equation)
  Combining with curl B = μ0_ref J yields
  curl ( Λ · curl B ) + B = curl ( Λ · μ0_ref J^{(T)} ),
  where Λ = (λ_L^2)_tensor and
  (λ_L^{-2})_{ij} = μ0_ref q^2 |psi_0|^2 κ_ij. Neglecting J^{(T)} recovers the standard anisotropic London equation.
• S40-7 (Critical fields & observables)
  In principal axes, approximate
  μ0_ref H_c1 ~ (Φ0_ref / 4π λ_{L,⊥}^2) ln(λ_{L,⊥}/ξ_⊥),
  μ0_ref H_c2(ê) ~ Φ0_ref / (2π ξ_⊥(ê) ξ_∥(ê)),
  with directional dependence set jointly by κ_ij and tension-induced anisotropy.

IV. BCS / Strong-Correlation Approximations & Coefficient Mapping

• S40-8 (Microscopic → GL mapping)
  Expand Delta(k) = ∑_ℓ Δ_ℓ φ_ℓ(k) over pairing basis {φ_ℓ}. In weak/intermediate coupling,
  a = N(0) (T_temp/T_c - 1) + a_T[T_fil], b ∝ N(0)⟨|φ_ℓ|^4⟩/β,
  κ_ij ∝ ⟨v_i v_j⟩_FS / α, while λ_1, λ_2 act as vertex-like corrections modulated by T_fil, grad T_fil. Material-specific integrals are consolidated in Chapter 11 dataset cards.
• S40-9 (Nodes & anisotropy renormalization)
  The tensor η_ij[T_fil, grad T_fil] renormalizes symmetric parts of κ_ij and can mix channels; when grad T_fil · e_c ≠ 0 with interface inversion breaking, odd/even-parity mixing enters effectively via the λ_2 coupling.

V. Linearized GL & Anisotropy of H_c2

• S40-10 (Near-transition linearization)
  For small |psi|, drop quartic terms:
  - D_i(κ_ij D_j psi) + a_eff psi = 0.
  In uniform field, a Landau-level–like analysis gives
  H_c2(θ,φ) = Φ0_ref / [ 2π √(det(Ξ_⊥(θ,φ))) ],
  where Ξ encodes anisotropy from κ_ij and tension; in axial symmetry this reduces to
  H_c2(θ) = H_c2(0)/√(cos^2 θ + γ^{-2} sin^2 θ), γ = ξ_∥/ξ_⊥.
• M4-1 (Fitting workflow for H_c2)
  With directional scans {H_c2(θ,φ,T_temp)}, parameterize Ξ and jointly invert κ_ij(T_temp) and admissible ranges of λ_1, λ_2 using the measurement matrix together with λ_L, ξ data.

VI. Boundary Conditions & Multiscale Matching

• S40-11 (Interface BC — de Gennes extension)
  Along normal n̂:
  n̂_i κ_ij D_j psi + ζ_1 (n̂ · grad T_fil) psi = 0,
  and n̂ × (B_2 - B_1) = μ0_ref K_s. If κ_ij or T_fil jump across an interface, continuity and flux conservation constrain A, psi.
• S40-12 (Thin-film/layered multiscale)
  When thickness d enters kernel scales (Chapter 3 K_T, K_G), use distinct κ_ij(d) and λ_L(d) for in-plane vs. out-of-plane; nonlocal terms appear explicitly via convolution kernels over d.
• M4-2 (Boundary-parameter calibration)
  Fit ζ_1 and interface-equivalent parameters via joint Josephson/tunneling spectra and magnetization loops; feed results to Chapter 12 implementation bindings.

VII. Normalization & Dimensionless Form (for Simulation & Inference)

• S40-13 (Non-dimensionalization)
  Choose scales ξ_0^2 = κ_0/|a_eff|, psi_0^2 = |a_eff|/2b, Ã = (q ξ_0/ħ) A, r̃ = r/ξ_0, obtaining
  F/F_0 = ∫ [ -|ψ̃|^2 + (1/2)|ψ̃|^4 + K̃_ij (D̃_i ψ̃)^* (D̃_j ψ̃) + β̃_1 T̃_fil |ψ̃|^2 + β̃_2 (∂̃_i T̃_fil)(ψ̃^* D̃_i ψ̃ + c.c.) + (1/2)|B̃|^2 ] d^3 r̃,
  with corresponding dimensionless GL equations ready for Chapter 10 SimStack.
• M4-3 (Simulation–experiment alignment)
  Use {K̃_ij, β̃_1, β̃_2} as primary parameters; optimize experiment design (band/geometry/noise budget) via the sensitivity spectrum of the measurement matrix y = M(θ).

VIII. Minimal Reconciliation with Observables

• S40-14 (Observable–parameter map)
  λ_{L,i}^{-2} ∝ μ0_ref q^2 |psi_0|^2 κ_ii, ξ_i^2 ∝ κ_ii/|a_eff|; relations for H_c1, H_c2 vs. {λ_L, ξ} follow S40-7. The presence of J^{(T)} yields extra phase/amplitude coupling in microwave/THz transmission (Chapter 7 via T_arr data contracts).
• S40-15 (Parameter correlation & identifiability)
  Where κ_ij and λ_2 are jointly weakly identifiable, prioritize combined vortex-lattice orientation/locking experiments with angle-resolved H_c2(θ) to break degeneracies; identifiability metrics and condition numbers are given in Chapter 8.

IX. Cross-Volume References & Chapter Anchors

• Cross-volume (fixed style):
  EFT.WP.Core.Equations v1.1 Ch.2 S20-* (path/arrival-time conventions);
  EFT.WP.Core.Metrology v1.0 Ch.1–3,5 (dimensions/units/uncertainty & constants);
  EFT.WP.Core.Tension v1.0 S72-, EFT.WP.Core.Density v1.0 S92- (energy consistency checks).
• Anchors (this chapter):
  S40-1–S40-15 (as above); M4-1 (angle-resolved H_c2 fitting), M4-2 (boundary-parameter calibration), M4-3 (dimensionless simulation–experiment alignment).

X. Summary
With a tension-coupled minimal GL functional at its core, this chapter establishes consistent links among London limits, linearized GL, and microscopic BCS mappings, and provides experiment-ready reconciliations with λ_L, ξ, and H_c1/H_c2. Together with Chapter 3’s “tension-landscape ↔ pairing” rules and the T_arr/measurement-matrix conventions in Chapters 7–8, it closes the path to Chapter 10’s inversion and model comparison via standard equations and parameterization gateways.