X-ray Diffraction Crystal Space-Group Classification (PWDR) L1-519

Materials ScienceCrystal symmetry classification from XRD lattice + structure-factor recoveryδ=5 · advancedL_DAG = 6.8📋 Stub — not mineable

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Unclaimed Principle — open for contribution

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Forward model E

S t a g e 1 (L 1 - 518) : 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, F_{h} k l) . S t a g e 2 : cr y s t a l_{s} y s t e m = l a tt i c e_{p} a r am e t e r_{d} ec i s i o n (a, b, c, an g l es); B r a v ai s = ce n t er in g_{d} ec i s i o n (sy s t e ma t i c_{a} b se n ces); p o in t_{g} r o u p = L a u e_{c} l a s s_{d} ec i s i o n (F_{h} k l sy mm e t r i es) + W i l so n_{p} l o t (ce n t r osy mm e t r y); s p a c e_{g} r o u p = sy s t e ma t i c_{a} b se n c e_{d} ec i s i o n (s p ec i f i c a x es) p er I n t er na t i o na l T ab l es V o l A .

X-ray Diffraction Space-Group Classification (PWDR): wraps L1-518 XRD analytical core with the deterministic crystallographic-symmetry decision tree from International Tables for Crystallography Vol A. Stage 1 (analytical, from L1-518): recover lattice parameters (a, b, c, alpha, beta, gamma), atomic positions, and systematic-absence pattern. Stage 2 (deterministic threshold): apply hierarchical decision tree (crystal system -> Bravais lattice -> point group -> space group) to assign one of 230 space groups per Hermann-Mauguin notation. Difficulty tier delta = 5. Mismatch parameters: lattice_parameter_uncertainty, systematic_absence_threshold_uncertainty, twinning_artifact, modulation_or_superlattice, pseudosymmetry_ambiguity.

L-DAG

L.xray_source -> L.bragg_diffraction -> L.structure_factor -> L.lattice_parameter_recovery -> L.systematic_absence_test -> L.space_group_decision_tree -> int.angular

L.xray_sourceL.bragg_diffractionL.structure_factorL.lattice_parameter_recoveryL.systematic_absence_testL.space_group_decision_treeint.angular

Well-posedness W

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

Existence and uniqueness inherited from L1-518 with additional constraint: pseudosymmetry can cause space-group ambiguity (e.g., P2_1/n vs Pn — distinguishing requires additional statistics or chemistry constraint). Stability inherits L1-518's high condition number plus additive systematic_absence_threshold_uncertainty. Joint Hadamard well-posedness for the space-group classification established by Hahn 2005 (International Tables Vol A), Cromer-Liberman 1970 (anomalous scattering for centrosymmetry tests), Le Bail 2005 (powder indexing), Kabsch 2010 (XDS for space-group determination), Spek 2003 (PLATON ADDSYM).

Solvability C

Solver class:: linear-operator + decision-tree [International Tables Vol A space-group decision tree] | direct methods + symmetry-search [Le Bail indexing + space-group enumeration] | linear-operator + deep neural prior [SpaceGroupNet]
Convergence rate q:: 1
Complexity:: O(N_reflections * 230) for full space-group enumeration; faster via hierarchical decision tree O(N_reflections * log(230))

Specs (0)

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