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A novel algorithm, stretchedNMF, is introduced for non-negative matrix factorization (NMF), accounting for signal stretching along the independent variable's axis. It addresses signal variability caused by stretching, proving beneficial for analyzing data such as powder diffraction at varying temperatures. This approach provides a more meaningful decomposition, particularly when the component signals resemble those from chemical components in the sample. The stretchedNMF model introduces a stretching factor to accommodate signal expansion, solved using discretization and Block Coordinate Descent algorithms. Initial experimental results indicate that the stretchedNMF model outperforms conventional NMF for datasets exhibiting such expansion. An enhanced version, sparse-stretchedNMF, optimized for powder diffraction data from crystalline materials, leverages signal sparsity for accurate extraction, especially with small stretches. Experimental results showcase its effectiveness in analyzing diffraction data, including success in real-time chemical reaction experiments. [ABSTRACT FROM AUTHOR]

The synthesis of complex oxides at low temperatures brings forward aspects of chemistry not typically considered. This study focuses on perovskite LaMnO 3 , which is of interest for its correlated electronic behavior tied to the oxidation state and thus the spin configuration of manganese. Traditional equilibrium synthesis of these materials typically requires synthesis reaction temperatures in excess of 1000 °C, followed by subsequent annealing steps at lower temperatures and different p (O 2 ) conditions to manipulate the oxygen content postsynthesis (e.g., LaMnO 3+ x ). Double-ion exchange (metathesis) reactions have recently been shown to react at much lower temperatures (500-800 °C), highlighting a fundamental knowledge gap for how solids react at lower temperatures. Here, we revisit the metathesis reaction, LiMnO 2 + LaOX, where X is a halide or mixture of halides, using in situ synchrotron X-ray diffraction. These experiments reveal low reaction onset temperatures (ca. 450-480 °C). The lowest reaction temperatures are achieved by a mixture of lanthanum oxyhalide precursors: 2 LiMnO 2 + LaOCl + LaOBr. In all cases, the resulting products are the expected alkali halide salt and defective La 1-ϵ Mn 1-ϵ O 3 , where ϵ = x /(3 + x ). We observe a systematic variation in defect concentration, consistent with a rapid stoichiometric local equilibration of the precursors and the subsequent global thermodynamic equilibration with O 2 (g), as revealed by computational thermodynamics. Together, these results reveal how the inclusion of additional elements (e.g., Li and a halide) leads to the local equilibrium, particularly at low reaction temperatures for solid-state chemistry.