X-ray Diffraction Crystal Structure Recovery (analytical core) L1-518

Materials SciencePowder / single-crystal X-ray diffraction lattice and structure-factor recoveryδ=5 · advancedL_DAG = 6.8📋 Stub — not mineable

📋

Unclaimed Principle — open for contribution

This Principle is declared in the catalog but has no reference solver, no pinned dataset, and is not registered on-chain. There is no reward pool. Submitting a cert against this Principle today will record the cert for reproducibility but pay zero PWM.

To claim it as a Bounty #7 contribution: open a PR adding (1) a reference solver, (2) ≥1 dataset pinned to IPFS, (3) updates to the L3 manifest with dataset CIDs. After verifier-agent triple-review, the founders' 3-of-5 multisig signs PWMRegistry.register() and the Principle becomes mineable.

Read the contribution guide →Open a claim issue →

Forward model E

B r a g g^{'} s l a w : n \cdot λ = 2 \cdot d \cdot s in (t h e t a); s t r u c t u r e f a c t or : F_{h} k l = j \sum f_{j} \cdot e x p (2 \cdot p i \cdot i \cdot (h x_{j} + k y_{j} + l z_{j})) \cdot e x p (- B_{j} \cdot (s in (t h e t a) / λ)^{2}); p o w d er in t e n s i t y : I (2 \cdot t h e t a) = K \cdot h \sum k l ∣ F_{h} k l ∣^{2} \cdot m_{h} k l \cdot L p (t h e t a) \cdot p r o f i l e_{p} se u d o_{v} o i g t (2 \cdot t h e t a - 2 \cdot t h e t a_{h} k l) + ba c k g r o u n d; j o in tl y in v er t I (2 \cdot t h e t a) \to (a, b, c, a l p ha, b e t a, g amma, a t o mi c_{p} os i t i o n s, occ u p an c i es, B_{f} a c t or s)

X-ray Diffraction Crystal Structure Recovery: analytical forward model for powder or single-crystal XRD. Bragg's law n*lambda = 2*d*sin(theta) governs diffraction angles; structure factor F_hkl = sum_j f_j * exp(2*pi*i*(h*x_j + k*y_j + l*z_j)) governs intensities. For powder XRD: I(2*theta) = K * sum_hkl |F_hkl|^2 * m_hkl * Lp(theta) * P_hkl(eta) * profile(2*theta - 2*theta_hkl); for single-crystal: I_hkl = K * |F_hkl|^2 * Lp(theta) * absorption_correction. Recovery is posed as the joint inverse problem that recovers (a, b, c, alpha, beta, gamma, atomic_positions, occupancies, thermal_factors) from measured I(2*theta) or {I_hkl}. The forward DAG has 7 primitives with one coupling constraint (n_c = 1): the structure-factor-to-intensity multiplicative coupling (via |F|^2). Difficulty tier delta = 5 (advanced) given the high condition number of structure-factor inversion (phase problem unresolved without anomalous scattering or direct methods). Mismatch parameters: instrumental_broadening, preferred_orientation_powder, sample_displacement, beam_divergence, monochromator_misalignment, background_subtraction_error.

L-DAG

L.xray_source -> L.bragg_diffraction -> L.structure_factor -> L.intensity_calculation -> L.profile_function -> L.detector_response -> int.angular

L.xray_sourceL.bragg_diffractionL.structure_factorL.intensity_calculationL.profile_functionL.detector_responseint.angular

Well-posedness W

Existence:: true
Uniqueness:: conditional
Stability:: conditional
κ:: 600

Existence guaranteed within declared Omega bounds. Uniqueness conditional: for known atomic composition + Bravais lattice + sufficient resolution, structure refinement is unique up to centrosymmetric equivalence; phase problem requires Patterson methods (heavy-atom), direct methods (Karle-Hauptman maximum-entropy), or anomalous-scattering for ab-initio. Stability conditional with kappa raw ~ 600 (limited by phase problem and reflection-overlap in powder XRD); kappa_eff ~ 80 with Rietveld refinement under structural constraints. Joint Hadamard well-posedness for the XRD forward established by Rietveld 1969 (foundational), Hill-Howard 1986 (Rietveld profile refinement), Karle-Hauptman 1956 (direct methods, Nobel 1985), McCusker 1999 (Rietveld guidelines), Young 1993 (Rietveld method textbook), Coelho 2018 (TOPAS), Toby-Von Dreele 2013 (GSAS-II).

Solvability C

Solver class:: linear-operator + nonlinear refinement [Rietveld profile refinement; least-squares structure refinement] | direct methods [Karle-Hauptman maximum-entropy phase recovery] | linear-operator + deep neural prior [DeepPattern, CrystalNet]
Convergence rate q:: 1
Complexity:: O(N_reflections * N_parameters^2) per Rietveld iteration; direct methods O(N_reflections * log(N_reflections)) per phase-cycle iteration

Specs (0)

No L2 specs registered yet for this principle.