40-EFT.WP.Materials.Superconductivity v1.0 | Chapter 6: Critical Windows — T_c, H_c1, H_c2, xi, lambda_L, kappa

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Chapter 6: Critical Windows — T_c, H_c1, H_c2, xi, lambda_L, kappa

I. Definitions & Measures (Windowed Formulation)

1. S60-1 (Critical window W_c)
   Define windows over the control space χ = { T_temp, H, ε, p, x, d, … } where observable proxies of the order parameter (e.g., ρ_s, λ_L^{-2}, ξ^{-2}) cross thresholds and can be robustly extracted. Typical 2D slices:
   • Temperature window: W_c^{(T)} = { T_temp : |a_eff(T_temp)| ≤ a_th }, with a_eff = a + λ_1 T_fil.
   • Field window: W_c^{(H)} = { H : H_c1(ê) < H < H_c2(ê) }, where ê encodes field–crystal orientation.
2. S60-2 (Coherence window W_coh)
   For a frequency/geometry pair, define the measurable propagation/phase-coherence domain W_coh(ω, geom) = { χ : L_coh(ω, χ) ≥ L_min(geom) }. In microwave/THz measurements this is constrained via the two T_arr conventions and n_eff(ω, χ) (implemented in Chapter 7’s data contract).
3. S60-3 (Primary quantities & tensorization)
   • xi_i: coherence-length principal components, xi_i^2 ≍ κ_ii / |a_eff| (Chapter 4).
   • lambda_{L,i}: penetration-depth principal components, (λ_L^{-2})_{ij} = μ0_ref q^2 |psi_0|^2 κ_ij.
   • kappa = λ_L/ξ; in anisotropic media use kappa_i = λ_{L,i} / xi_i.
   • H_c1(ê), H_c2(ê): direction-dependent lower/upper critical fields defining the sides of W_c^{(H)}.

II. Theoretical Relations & Scaling (GL near T_c + Tension Corrections)

• S60-4 (Near-T_c scaling)
  In the GL regime t = 1 − T_temp/T_c ≪ 1: |psi|^2 ∝ t, λ_{L,i}(T) ∝ t^{-1/2}, xi_i(T) ∝ t^{-1/2}, and kappa_i(T) ≍ const.
• S60-5 (Critical-field relations in principal axes)
  μ0_ref H_c1(ê_i) ≈ (Φ0_ref / 4π λ_{L,⊥}^2) ln(λ_{L,⊥} / ξ_⊥);
  μ0_ref H_c2(ê) ≈ Φ0_ref / (2π ξ_⊥(ê) ξ_∥(ê)).
  Directional dependence is set jointly by κ_ij and tension-induced anisotropy from T_fil, grad T_fil.
• S60-6 (Angular H_c2)
  For axial symmetry: H_c2(θ) = H_c2(0) / √(cos^2 θ + γ^{-2} sin^2 θ), with γ = ξ_∥ / ξ_⊥.
• S60-7 (Fluctuations & Ginzburg number)
  The fluctuation-dominated width grows with Gi ∝ [ k_B T_c / ( μ0_ref H_c^2(0) ξ_⊥^3(0) ) ]^2. Extraction windows W_c^{(T)} should avoid strong-fluctuation zones, or reweight by coherence (see M6-1).

III. Extraction Workflows (Mx-6-*, linked to the Measurement Matrix)

• M6-1 (Unified T_c extraction)
  Input: ρ(T), C_p(T), λ_L^{-2}(T), ρ_s(T), I–V(α) (for 2D).
  Steps: (1) derive per-proxy T_c^{(k)} with uncertainty u_k; (2) build weighted composite T_c = Σ w_k T_c^{(k)}, w_k ∝ u_k^{-2} with a consistency test; (3) for 2D films, also report T_BKT and T_c − T_BKT.
  Output: T_c, the set {T_c^{(k)}}, and consistency stats (recorded in data cards).
• M6-2 (Extracting λ_L)
  Input: cavity/resonator f, Q, phase, TDO, μSR, microwave/THz transmission T_arr.
  Steps: enforce W_coh(ω); resonator & μSR yield bulk-mean λ_L(T); transmission uses the T_arr conventions with n_eff(ω) to decouple geometry.
  Output: lambda_{L,i}(T) with confidence intervals.
• M6-3 (Extracting H_c2&xi)
  Input: magnetoresistance R(H,T,θ,φ), critical-current decay, calorimetric/magnetization anomalies.
  Steps: near T_c, use slope −dH_c2/dT|_{T_c} to estimate ξ(0); fit H_c2(θ,φ) to invert ξ_⊥, ξ_∥.
  Output: H_c2(ê,T), xi_i(T), γ(T).
• M6-4 (Extracting H_c1)
  Input: low-field M(H) loops, first-vortex entry imaging.
  Steps: remove demagnetization/geometry factors; convert first-entry field H_p to H_c1.
  Output: H_c1(ê,T) with a full uncertainty budget.
• M6-5 (Composing kappa)
  Input: λ_L, ξ, H_c1, H_c2.
  Steps: solve consistently using redundant relations (kappa = λ_L/ξ, approximate H_c1,H_c2 formulas).
  Output: kappa_i(T) plus a consistency index.
• M6-6 (Anisotropy inversion)
  Input: angle-resolved H_c2(θ,φ), λ_L(θ).
  Output: κ_ij(T) and principal axes; write the Jacobian J = ∂y/∂θ into Chapter 8’s measurement matrix y = M(θ).
• M6-7 (Thin-film/2D special case)
  Input: thickness scans { λ_L(d), H_c2(d) }, I–V scaling.
  Output: T_BKT, Gi_{2D}, and first estimates of nonlocal-kernel scales {K_T, K_G} (fed back to Chapters 3–4).

IV. Uncertainty Synthesis & Power Analysis (Metrology)

• S60-8 (Error propagation)
  For any derived f(x_1,…,x_n), the combined standard uncertainty
  u_c^2(f) = Σ (∂f/∂x_i)^2 u^2(x_i) + 2 Σ_{i<j} (∂f/∂x_i)(∂f/∂x_j) cov(x_i,x_j), with expanded uncertainty U = k · u_c.
• S60-9 (Identifiability & condition number)
  With J the Jacobian of y = M(θ), build the Fisher matrix F = J^T Σ^{-1} J; cond(F) together with priors governs parameter identifiability.
• M6-8 (Power analysis)
  Given a target relative-uncertainty threshold τ and resource bounds (field/temperature time), optimize the scan grid over χ and frequency allocations to maximize tr(F) or minimize U.

V. Band & Geometry Effects (Links to Chapters 7/8)

• S60-10 (Coherence-weighted fusion)
  Within W_c ∩ W_coh, weight multi-band observations y(ω) by w(ω) ∝ L_coh(ω) or Fisher-information density; outside W_coh, down-weight to weak constraints or treat as systematic terms.
• M6-9 (Data contract & fields)
  All derived quantities must record unit, u(x)/U, and references/see (explicit volume + version + anchor) in the dataset card, and declare which T_arr convention is used (delta_form).

VI. Testable Predictions & Design Guidance (Coupled to Tension Landscapes)

• Slope co-variation: At fixed {x, δ}, uniform strain/pressure makes the near-T_c slope of H_c2(T) co-vary with the slope of λ_L^{-2}(T); the effect scales with ∂T_fil/∂ε.
• Angular drift: Rotation of grad T_fil rotates the principal axes of γ(T); the H_c2(θ) extrema co-drift with the vortex-lattice orientation φ_0 from Chapter 5.
• Thickness scaling: When d ~ ξ, log–log slopes of { λ_L(d), H_c2(d) } deviate jointly from local models; the deviation is monotone in the nonlocal-kernel radius.
• BKT gap: In 2D samples, T_c − T_BKT widens with |grad T_fil|; tension-landscape engineering gives a controllable handle on T_BKT.

VII. Cross-Volume References & Anchors (This Chapter)

• Cross-volume (fixed style):
  EFT.WP.Core.Metrology v1.0 Ch.1–3,5 (dimensions/units/uncertainties & constants);
  EFT.WP.Core.Equations v1.1 Ch.2 S20-* (two T_arr conventions with explicit path/measure declarations);
  This volume Chapter 4 (κ_ij, ξ, λ_L, and H_c1/H_c2 relations) and Chapter 7 (arrival-time implementation in low-T EM measurements).
• Anchors (S/M):
  S60-1—S60-10; M6-1—M6-9.

VIII. Summary
By unifying critical and coherence windows, this chapter standardizes the definitions, extraction, and uncertainty synthesis of T_c, H_c1, H_c2, xi, lambda_L, and kappa, and provides an executable interface to the measurement matrix y = M(θ). It also delivers tension-landscape–linked predictions. Results feed directly into Chapter 7 (propagation & arrival time), Chapter 8 (identifiability & experiment design), and Chapter 10 (inversion & model comparison).