We report on a comparison of methods based on XRD patterns for calculating crystal size. In this case, XRD peaks were extracted from hydroxyapatite obtained from cow, pig, and chicken bones. Hydroxyapatite was synthesized through the thermal treatment of natural bones at 950 °C. XRD patterns were selected by adjustment of X-Pert software for each method and for calculating the size of the crystals. Methods consisted of Scherrer (three models), Monshi–Scherrer, three models of Williamson–Hall (namely the Uniform Deformation Model (UDM), the Uniform Stress Deformation Model (USDM), and the Uniform Deformation Energy Density Model (UDEDM)), Halder–Wanger (H-W), and the Size Strain Plot Method (SSP). These methods have been used and compared together. The sizes of crystallites obtained by the XRD patterns in each method for hydroxyapatite from cow, pig, and chicken were 1371, 457, and 196 nm in the Scherrer method when considering all of the available peaks together (straight line model). A new model (straight line passing the origin) gave 60, 60, and 53 nm, which shows much improvement. The average model gave 56, 58, and 52 nm, for each of the three approaches, respectively, for cow, pig, and chicken. The Monshi–Scherrer method gave 60, 60, and 57 nm. Values of 56, 62, and 65 nm were given by the UDM method. The values calculated by the USDM method were 60, 62, and 62 nm. The values of 62, 62, and 65 nm were given by the UDEDM method for cow, pig, and chicken, respectively. Furthermore, the crystal size value was 4 nm for all samples in the H-W method. Values were also calculated as 43, 62, and 57 nm in the SSP method for cow, pig, and chicken tandemly. According to the comparison of values in each method, the Scherrer method (straight line model) for considering all peaks led to unreasonable values. Nevertheless, other values were in the acceptable range, similar to the reported values in the literature. Experimental analyses, such as specific surface area by gas adsorption (Brunauer–Emmett–Teller (BET)) and Transmission Electron Microscopy (TEM), were utilized. In the final comparison, parameters of accuracy, ease of calculations, having a check point for the researcher, and difference between the obtained values and experimental analysis by BET and TEM were considered. The Monshi–Scherrer method provided ease of calculation and a decrease in errors by applying least squares to the linear plot. There is a check point for this line that the slope must not be far from one. Then, the intercept gives the most accurate crystal size. In this study, the setup of values for BET (56, 52, and 49 nm) was also similar to the Monshi–Scherrer method and the use of it in research studies of nanotechnology is advised.

Young’s modulus (E) is one of the most important parameters in the mechanical properties of solid materials. Young’s modulus is proportional to the stress and strain values. There are several experimental and theoretical methods for gaining Young’s modulus values, such as stress–strain curves in compression and tensile tests, electromagnetic-acoustic resonance, ultrasonic pulse echo and density functional theory (DFT) in different basis sets. Apparently, preparing specimens for measuring Young’s modulus through the experimental methods is not convenient and it is time-consuming. In addition, for calculating Young’s modulus values by software, presumptions of data and structures are needed. Therefore, this new method for gaining the Young’s modulus values of crystalline materials is presented. Herein, the new method for calculating Young’s modulus of crystalline materials is extracted by X-ray diffraction. In this study, Young’s modulus values were gained through the arbitrary planes such as random (hkl) in the research. In this study, calculation of Young’s modulus through the relationship between elastic compliances, geometry of the crystal lattice and the planar density of each plane is obtained by X-ray diffraction. Sodium chloride (NaCl) with crystal lattices of FCC was selected as the example. The X-ray diffraction, elastic stiffness constant and elastic compliances values have been chosen by the X’Pert software, literature and experimental measurements, respectively. The elastic stiffness constant and Young’s modulus of NaCl were measured by the ultrasonic technique and, finally, the results were in good agreement with the new method of this study. The aim of the modified Williamson–Hall (W–H) method in the uniform stress deformation model (USDM) utilized in this paper is to provide a new approach of using the W–H equation, so that a least squares technique can be applied to minimize the sources of errors.

Taking into account X-ray diffraction, one of the well-known methods for calculating the stress-strain of crystals is Williamson-Hall (W–H). The W-H method has three models, namely (1) Uniform deformation model (UDM); (2) Uniform stress deformation model (USDM); and (3) Uniform deformation energy density model (UDEDM). The USDM and UDEDM models are directly related to the modulus of elasticity (E). Young’s modulus is a key parameter in engineering design and materials development. Young’s modulus is considered in USDM and UDEDM models, but in all previous studies, researchers used the average values of Young’s modulus or they calculated Young’s modulus only for a sharp peak of an XRD pattern or they extracted Young’s modulus from the literature. Therefore, these values are not representative of all peaks derived from X-ray diffraction; as a result, these values are not estimated with high accuracy. Nevertheless, in the current study, the W-H method is used considering the all diffracted planes of the unit cell and super cells (2 × 2 × 2) of Hydroxyapatite (HA), and a new method with the high accuracy of the W-H method in the USDM model is presented to calculate stress (σ) and strain (ε). The accounting for the planar density of atoms is the novelty of this work. Furthermore, the ultrasonic pulse-echo test is performed for the validation of the novelty assumptions.

Calcium titanate-CaTiO₃ (perovskite) has been used in various industrial applications due to its dopant/doping mechanisms. Manipulation of defective grain boundaries in the structure of perovskite is essential to maximize mechanical properties and stability; therefore, the structure of perovskite has attracted attention, because without fully understanding the perovskite structure and diffracted planes, dopant/doping mechanisms cannot be understood. In this study, the areas and locations of atoms and diffracted planes were designed and investigated. In this research, the relationship between Young’s modulus and planar density of unit cell, super cells (2 × 2 × 2) and symmetry cells of nano CaTiO₃ is investigated. Elastic constant, elastic compliance and Young’s modulus value were recorded with the ultrasonic pulse-echo technique. The results were C₁₁ = 330.89 GPa, C₁₂ = 93.03 GPa, C₄₄ = 94.91 GPa and E = 153.87 GPa respectively. Young’s modulus values of CaTiO₃ extracted by planar density were calculated 162.62 GPa, 151.71 GPa and 152.21 GPa for unit cell, super cells (2 × 2 × 2) and symmetry cells, respectively. Young’s modulus value extracted by planar density of symmetry cells was in good agreement with Young’s modulus value measured via ultrasonic pulse-echo.

Recently, researchers have focused on the biocompatibility and mechanical properties of highly porous structures of biomaterials products. Porous composites are a new category of bioengineering that possess excellent functional and structural properties. In this study, the physical and mechanical properties of prepared doped silver (Ag)-hydroxyapatite (HA) by the mechanochemical and spark plasma sintering (SPS) methods were investigated. The influence of dopant on phase formation, structural properties, mechanical properties and morphological characteristics was investigated. Furthermore, in this case, as a new approach to produce a porous scaffold with an average size of >100 µm, the hair band was used as a mold. According to the Monshi–Scherrer method, the crystal size of scaffold was calculated 38 ± 2 nm and this value was in the good agreement with average value from transmission electron microscopy (TEM) analysis. In addition, the stress–strain compression test of scaffold was considered, and the maximum value of compressive strength was recorded ~15.71 MPa. Taking into account the XRD, TEM, Fourier-transform infrared (FTIR), scanning electron microscope (SEM) and energy dispersive X-Ray analysis (EDAX) analysis, the prepared scaffold was bioactive and the effects of doped Ag-HA and the use of polyvinyltrimethoxysilane (PVTMS) as an additive were desirable. The results showed that the effect of thermal treatment on composed of Ag and HA were impressive while no change in transformation was observed at 850 °C. In addition, PVTMS plays an important role as an additive for preventing the decomposition and creating open-microporous in the scaffold that these porosities can be helpful for increasing bioactivity.