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There are two methods to estimate the particle size from X-ray diffraction data: the Debye equation and the Scherrer formula. The main goal of this study is to describe the methodology of particle size estimation on the base of two these methods and to apply it to TiO2 powder to determine the diameters and the mass content of anatase and brookite components. The studied nano-dispersed TiO2 powder was synthesized by the sol-gel method. The proposed method of particle size estimation consists of several steps: 1. Approximation of diffraction peaks by Gaussians and calculation of initial values of particle size with the use of the Scherrer formula; 2. Iterations with the use of the Debye equation to obtain more accurate particle size values; 3. Calculation of the mass content of different components corresponding to the minimum R-factor.

Paul Scherrer and Peter Debye developed powder X-ray diffraction together, but it was Scherrer who figured out how to determine the size of crystallites from the broadening of diffraction peaks.

In the present experiment, we presented the nanocrystalline pure and silver (0.2, 0.4, 0.6, and 0.8 M%)-doped hexagonal molybdenum trioxide rods synthesized by using facile and cost-effective hydrothermal method and analyzed the effects of Ag contents of different microstructural and optical properties. X-ray diffraction (XRD) patterns confirmed the hexagonal crystal structure of the nanorods, supported by FTIR spectra. The Debye–Scherrer formula, Williamson–Hall (W–H), Halder–Wagner (H–W), and size–strain plot (SSP) techniques were applied to investigate different crystallographic characteristics such as crystallite size and lattice strain of h-MoO 3 nanorods by evaluating the broadening of XRD peaks. The different relevant structural parameters of the resultant h-MoO 3 nanorods connected to XRD analysis such as dislocation density, lattice parameters, unit cell volume, and stacking fault have also been evaluated. The formation of nanorod shape and the presence of Ag contents were confirmed by field emission scanning electron microscope and energy-dispersive spectroscopy, respectively. The optical bandgap was estimated via both Kubelka–Munk (K–M) and Tauc’s rules. The estimated values of the bandgap were found to be in the range of 2.83–3.04 eV. The optical bandgap increased with the increased of Ag contents up to 0.4 M% and afterword decreased up to 0.8 M%. The similar variation trend of optical bandgap was observed for both methods.

Based on the desire to improve material properties, the effects of temperature have begun to be investigated. It was found that for nano-sized powder materials, such as ferrites, the structural properties like crystal structure and grain size, as well as many magnetic and electrical properties depending on them, change with the calcination temperature. Considering these changes, the effect of calcination temperature on the structural, magnetic, and electrical properties of MZA ferrites (Mg 0.75 Zn 0.25 Al 0.2 Fe 1.8 O 4 ) prepared by co-precipitation was investigated in this study. The produced MZA ferrites were calcined at three different temperatures (600, 700, and 800 °C). The X-ray diffraction results showed that the samples exhibited a cubic spinel structure. It was found that the crystal sizes (D_sch) calculated using the Debye-Scherrer equation increased with increasing calcination temperature (22.47, 33.53, and 42.53 nm). From the Williamson–Hall (W–H) plots, crystal sizes were calculated almost same as Debye–Scherrer crystal sizes. The nano-sized particles were examined by scanning electron microscope (SEM). Elemental analysis was performed using EDX. ν 1 and ν 2 absorption bands and O–H and C–H vibrations were detected in the FTIR spectra. Magnetic measurements were carried out at room temperature and in the range of ± 60 kOe under the applied field. Magnetic results are explained by superparamagnetism. Dielectric measurements were performed at room temperature and a frequency range of 20 Hz to 10 MHz. The dielectric properties can be explained by Maxwell–Wagner theory. Impedance spectroscopy study revealed that the relaxation mechanism is consistent with the Cole–Cole model. In AC conductivity studies at room temperature, it was found that the sample calcined at 600 °C would be suitable for energy storage devices.

Ni 0.7− x Zn 0.3 Mg x Fe 2 O 4 spinel ferrite nanoparticles were synthesized via self-combustion. Structural, morphological, elastic, optical, and photoluminescence properties were examined using X-ray diffraction (XRD), Fourier transform infrared (FTIR), Field emission scanning electron microscope (FESEM), Transmission electron microscope (TEM), selected area electron diffraction (SAED) pattern, Ultraviolet–Visible (UV–Vis) and Photoluminescence (PL) spectroscopy. XRD confirmed a cubic lattice with a preferred (311) orientation. Increasing Mg 2+ ion concentration enhanced lattice parameters due to its larger ionic radius compared to Ni 2+ . Crystallite sizes were analyzed using Debye-Scherrer, Williamson-Hall, Halder-Wagner, and Size-Strain methods. XRD data revealed an unusual cation distribution over tetrahedral and octahedral sites. FTIR spectra confirmed the spinel ferrite structure with prominent peaks for these sites. TEM images showed spherical nanoparticles, and SAED pattern confirmed the cubic structure. Band gaps from Tauc’s plots ranged between 2.52 and 2.78 eV. PL spectra showed three sub-bands: near band edge emission, violet band, and blue band.

Polycrystalline diffraction is a robust methodology employed to assess elastic strain within crystalline components. The Extended Caking (exCaking) method represents a progression of this methodology beyond the conventional azimuthal segmentation (Caking) method for the quantification of elastic strains using Debye–Scherrer 2D X-ray diffraction rings. The proposed method is based on the premise that each complete diffraction ring contains comprehensive information about the complete elastic strain variation in the plane normal to the incident beam, which allows for the introduction of a novel algorithm that analyses Debye–Scherrer rings with complete angular variation using ellipse geometry, ensuring accuracy even for small eccentricity values and offering greater accuracy overall. The console application of the exCaking method allows for the accurate analysis of polycrystalline X-ray diffraction data according to the up-to-date rules presented in the project repository. This study presents both numerical and empirical examinations and error analysis to substantiate the method’s reliability and accuracy. A specific validation case study is also presented to analyze the distribution of residual elastic strains in terms of force balance in a Ti-6Al-4V titanium alloy bar plastically deformed by four-point bending.

In the present study, as-synthesized pure ammonium manganese phosphate hydrate (AMP) is infused with rGOx variate (Xmg = 25, 50, 75, 100) and four different hybrid composites (AMPG1, AMPG2, AMPG3 and AMPG4) have been synthesized by facile microwave route. The XRD results show two prominent peaks, one at 2θ = 10.04° (010) and other at 2θ = 31.3° (200) in all AMPGs. The Debye–Scherrer’s calculations show minimum crystallite size only for AMPG2 (80.9 nm). The Raman study confirms the rGO presence in AMPG2. The XPS confirms the existence of Mn as Mn2+ in AMPG2. The SEM/HR-TEM shows a cluster of uniform rectangular flake slabs only for AMPG2. The CV reveals that pure AMP and AMPGs exhibit pseudocapacitance. The GCD shows higher specific capacitance of 705 F g−1 at a current density of 1 A g−1 for AMPG2. The AMPG2//rGO hybrid device at 3 M aqueous H2SO4 shows higher specific capacitance of 336 F g−1 at 1 A g−1 in the potential window 0–1.8 volts, and even after 5000 cycles, the device retained 80% of its specific capacitance. The reason may be due to mapping of optimal concentration of rGO (50 mg) with PO43- and NH4+ of AMP by forming strong coordination for better activation sites for ion mobility. The energy and power densities of AMPG2//rGO device are 151 Wh kg−1 and 448 W kg−1 at 1 A g−1, which are reported for the first time for high-energy supercapacitor applications.Graphic abstract

Herein, pure ZnS and Sn-doped (1–5 wt%) ZnS nanoparticles (NPs) were prepared by simple chemical co-precipitation method. The average crystallite sizes of synthesized NPs determined using Debye–Scherrer formula were found to be between 3 and 5 nm. Morphological features of ZnS NPs were determined by field emission scanning electron microscopy (FE-SEM). The microscopy result showed that the samples had microspherical structure and were assembled by tiny particles. UV–Vis studies were performed for the determination of bandgap of the samples. The compositional information of photocatalysts was determined by energy-dispersive X-ray spectroscopy. Photocatalytic activity of NPs was evaluated by taking methylene blue (MB) and methyl orange dyes as model pollutants and their comparative degradation behavior has been discussed. The Sn-doped ZnS NPs displayed excellent photodegradation efficiency of 93% for MB dye compared to 49% for ZnS.

Germanium sulfide (GeS) nanoparticles (NPs) were synthesized using hydrothermal method. The chemical composition was determined through energy-dispersive X-ray analysis, while scanning electron microscopy and transmission electron microscopy were employed to study the morphology. X-ray diffraction analysis confirmed the cubic structure of the GeS NPs. The crystallite size and lattice strain of cubic GeS NPs were found to be 29.65 nm and 1.24 × 10 - 3 respectively using Debye-Scherrer method. The energy levels of Ge and S were studied using X-ray photoelectron spectroscopy to determine the binding states and chemical composition. Thermo-gravimetric analysis was performed to calculate the kinetic parameters for thermal stability. Zeta potential of GeS NPs was - 22.4 mV indicating the prevention from aggregation. The optical band gap of the nanoparticles was measured to be 1.66 eV. Their photo-catalytic activity was evaluated at different powder concentrations, and the relevant parameters were estimated using the Langmuir-Hinshelwood kinetic model. The rate constant ‘kf’ was obtained to be 0.0018 min - 1 , 0.0020 min - 1 , 0.0035 min - 1 for powder concentration of 1 g/L, 2 g/L and 3 g/L, respectively. Additionally, the potential applications of GeS NPs as pressure sensors and infrared sensors were comprehensively investigated.

The sol–gel technique was used in the chemical synthesis and characterizations based on structural, morphological, optical and electrical studies of pure and Ag-doped zinc oxide (ZnO) nanoparticles. X-ray diffraction, scanning electron microscopy, energy Dispersive X-ray spectrometry, transmission electron microscope, ultraviolet spectroscopy, photoluminescence and FT-IR analysis were used to perform the characterization of the morphological analysis, optical studies, phase purity and crystalline size. The Powder X-ray diffraction results proved polycrystalline nature of ZnO with a hexagonal wurtzite structure. Debye-Scherrer’s formula was used to evaluate the average crystallite size of pure and Ag-doped ZnO. Their values have been determined to be 14 and 18 nm respectively. To examine the various functional groups FTIR was utilized. The unique aggregation of the particles was stated by the scanning electron microscopy investigation and transmission electron microscope analysis was used to substantiate the nanosphere formation. Here, the estimated optical band gap value for pure and Ag-doped ZnO nanoparticles was 3.22 and 3.17 eV, respectively. UV–visible spectroscopy was used to perform this process. Photoluminescence studies have proved the Ag-doped ZnO sample of the blue shift emission bands. At different frequencies and temperatures, under specific conditions, the dielectric properties like dielectric constant, dielectric loss and AC conductivity of Ag-doped ZnO nanoparticles were analyzed. Graphical Abstract