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The application of oxygen-deficient perovskites (ODPs) has attracted interest as anode materials for lithium-ion batteries for their unique properties. One such material, CaSrFe0.75Co0.75Mn0.5O6-δ, has been studied extensively. The structure of CaSrFe0.75Co0.75Mn0.5O6-δ was investigated using various techniques, including Rietveld refinements with X-ray diffraction and neutron diffraction. Additionally, iodometric titration and X-ray photoelectron spectroscopy were employed to study the oxygen-deficiency amount and the transition metal’s oxidation states in the material. As an anode material, CaSrFe0.75Co0.75Mn0.5O6-δ exhibits promising performance. It delivers 393 mAhg−1 of discharge capacity at a current density of 25 mAg−1 after 100 cycles. Notably, this capacity surpasses both the theoretical graphite anode capacity (372 mAhg−1) and that of the calcium analog reported previously. Furthermore, the electrochemical performance of CaSrFe0.75Co0.75Mn0.5O6-δ remains highly reversible across various current densities ranging from 25 to 500 mAg−1. This suggests the material’s excellent stability and reversibility during charge–discharge cycles, showing its probable application as an anode for lithium-ion batteries. The mechanism of lithium intercalation and deintercalation within CaSrFe0.75Co0.75Mn0.5O6-δ has also been discussed. Understanding this mechanism is crucial for optimizing the battery’s performance and ensuring long-term stability. Overall, this study highlights the significant potential of oxygen-deficient perovskites, particularly CaSrFe0.75Co0.75Mn0.5O6-δ, for applications as an anode material for lithium-ion batteries, offering enhanced capacity and stability compared with traditional graphite-based anodes. 

This study investigates the thermoelectric properties of Ca2Fe2O5 over a temperature range of 7˚C to 50˚C. The experiment measured the voltage generated by temperature differences across two sides of the material, with a focus on the voltage response at temperatures both below and above room temperature. Results indicate that at lower temperatures (7˚C to 15˚C), the voltage generated by the temperature difference was higher, though not directly proportional to the magnitude of the temperature gradient. The highest voltage recorded for the smallest temperature difference in this range was 109 mV, observed between 14.6˚C and 17.6˚C (smallest temperature difference, 3˚C). Similarly, at temperatures above room temperature, the voltage generated was relatively lower, peaking at 125 mV between 9˚C and 44˚C (higher temperature difference). These results suggest complex behavior of Ca2Fe2O5’s thermoelectric response, with non-linear relationships between voltage and temperature differences at both low and high temperatures. 

The perovskite oxides CaMnO3-δ, Ca0.5Sr0.5MnO3-δ, and SrMnO3-δ were synthesized in air using a solid-state method, and their structural, electrical, and electrocatalytic properties were studied in relation to their oxygen evolution reaction (OER) performance. Iodometric titration showed δ values of 0.05, 0.05, and 0.0, respectively, indicating that Mn is predominantly in the 4+ oxidation state across all materials, consistent with prior reports. Detailed characterization was performed using X-ray diffraction (XRD), scanning electron microscopy (SEM), iodometric titration, and variable-temperature conductivity measurements. Four-point probe DC measurements revealed that CaMnO3-δ (δ = 0.05) has a semiconductive behavior over a temperature range from 25 °C to 300 °C, with its highest conductivity attributed to polaron activity. Cyclic voltammetry (CV) in 0.1 M KOH was employed to assess OER catalytic performance, which correlated with room-temperature conductivity. CaMnO3-δ exhibited superior catalytic activity, followed by Ca0.5Sr0.5MnO3-δ and SrMnO3-δ, demonstrating that increased conductivity enhances OER performance. The conductivity trend, CaMnO3-δ > Ca0.5Sr0.5MnO3-δ > SrMnO3-δ, aligns with OER activity, underscoring a direct link between electronic transport properties and catalytic efficiency within this series. 

This study presents a detailed investigation of the microstructure of the oxygen-deficient perovskite material Ca2FeGaO6-δ using Scanning Electron Microscopy (SEM). The material exhibits significant porosity and irregular grain morphology, with variations in grain size and growth. Unlike conventional perovskite structures, Ca2FeGaO6-δ shows non-uniform grain development, which can be attributed to the presence of oxygen vacancies (δ ). SEM analysis reveals that the irregularities in grain size and shape, coupled with the porous nature of the material, are likely to influence its functional properties. These findings provide valuable insights into the structural features of Ca2FeGaO6-δ , offering a foundation for understanding its potential applications in catalysis, sensors, and other technologies. The study highlights the critical role of microstructural characteristics in determining the material’s performance. 

This study presents a novel and facile technique for the rapid and sensitive detection of zinc (Zn) in foods and drinking water. The need for a reliable method to monitor Zn levels in consumables is crucial due to its significance in both nutritional assessment and environmental safety. The proposed technique integrates state-of-the-art sensing technology with an easy-to-implement approach, aiming to provide an efficient solution for Zn detection. The methodology involves the utilization of complexation of Zn2+ ion with resorcinol and use of UV-vis spectrophotometry, which demonstrates high sensitivity towards Zn2+ ions. It detected zinc up to 10-5M solution. 

The thermal conductivity of CaSrFe2O6-δ, an oxygen-deficient perovskite, is a critical parameter for understanding its thermal transport properties and potential applications in energy conversion and electronic devices. In this study, we present an investigation of the thermal conductivity of CaSrFe2O6-δ at room temperature for its thermal insulation property study. Experimental measurement was conducted using a state-of-the-art thermal characterization technique, Thermtest thermal conductivity meter. The thermal conductivity of CaSrFe2O6-δ was found to be 0.574W/m/K, exhibiting a notable thermal insulation property. 

Materials with low thermal conductivity have been used in thermoelectrics, electronics, solar devices, and land base and aerospace gas turbines to prevent heat dissipation and provide safety and longevity of equipment. We report Ca2Fe2O6-δ, and Sr2Fe2O6-δ for their low thermal conductivities. These compounds are vacancy-ordered oxygen-deficient perovskites but with different vacancy arrangements. Ca2Fe2O6-δ has a brownmillerite type structure while Sr2Fe2O6-δ has a different structure. 

The crystal structure of CaSrFe0.75Co0.75Mn0.5O6−δ is investigated through neutron diffraction techniques in this study. The material is synthesized using a solid-state synthesis method at a temperature of 1200˚C. Neutron diffraction data is subjected to Rietveld refinement, and a comparative analysis with X-ray diffraction (XRD) data is performed to unravel the structural details of the material. The findings reveal that the synthesized material exhibits a cubic crystal structure with a Pm-3m phase. The neutron diffraction results offer valuable insights into the arrangement of atoms within the lattice, contributing to a comprehensive understanding of the material’s structural properties. This research enhances our knowledge of CaSrFe0.75Co0.75Mn0.5O6−δ, with potential implications for its applications in various technological and scientific domains. 

Making exact approximations to solve equations distinguishes applied mathematicians from pure mathematicians, physicists, and engineers. Perturbation problems, both regular and singular, are pervasive in diverse fields of applied mathematics and engineering. This research paper provides a comprehensive overview of algebraic methods for solving perturbation problems, featuring a comparative analysis of their strengths and limitations. Serving as a valuable resource for researchers and practitioners, it offers insights and guidance for tackling perturbation problems in various disciplines, facilitating the advancement of applied mathematics and engineering.