Supplementary Information: Williamson–Hall analysis (200–900 °C) and phase-selective fits for Mn_3O_4 ⇄ Mn_2O_3 transformationtled

This Supplementary Information provides Williamson–Hall (W–H) size–strain analysis for Mn–O nanoparticles annealed in air from 200 to 900 °C (4 h per temperature, independently annealed aliquots). Two analyses are included:Apparent W–H analysis for the full temperature series (200–900 °C) using multiple reflections from each diffractogram under the uniform deformation model (UDM), based on βcosθ versus 4sinθ. A non-negative least-squares linear fit was used to ensure physically meaningful positive intercept and slope. The intercept yields kλ/D_WH and the slope yields the apparent microstrain ε. At 500 °C (mixed phase), apparent parameters are composite metrics and are reported only for completeness.Phase-selective W–H analysis for the transformation window (500–800 °C). Peaks were assigned to Mn_3O_4 or Mn_2O_3 by nearest-neighbor matching to reference peak-position lists derived from near-single-phase patterns (Mn_3O_4: 200–400 °C; Mn_2O_3: 700–800 °C) using a ±0.35° tolerance in 2θ. Phase-selective W–H fits were computed when at least 4 peaks were assigned to a given phase.Contents:Table S1: Apparent W–H outputs for 200–900 °CFigures S1–S8: Apparent W–H plots for 200–900 °CTable S2: Phase-selective W–H outputs for 500–800 °CFigures S9–S13: Phase-selective W–H plots for 500–800 °CThis file supports the microstructural interpretation and methodological consistency of the XRD line-broadening analysis reported in the associated manuscript on Mn_3O_4 ⇄ Mn_2O_3 phase transformation under heat treatment.