X-ray Diffraction Analysis and Williamson-Hall Method in USDM Model for Estimating More Accurate Values of Stress-Strain of Unit Cell and Super Cells (2 × 2 × 2) of Hydroxyapatite, Confirmed by Ultrasonic Pulse-Echo Test

Abstract

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.

Influence of samarium on the structural, magnetic, and gas sensing performance of cadmium zinc ferrites

“…The reduction of the unit cell (i.e. decrease in the lattice parameter) is due to the compression of the spinel lattice [47][48][49] and this is supported by the obtained negative strain of sample (S4) in table 3 since positive values correspond to tensile stress and strain while negative values correspond to compressive stress and strain [50]. The Sm 3+ cations occupy the grain boundaries beyond its solubility limit (x = 0.04) since rare-earth cations have finite solubility when incorporated in spinel ferrites due to its large ionic radius; they usually occupy octahedral sites substituting the smaller ions Fe 3+ [48,51,52].…”

Phase Transformations and Subsurface Changes in Three Dental Zirconia Grades after Sandblasting with Various Al2O3 Particle Sizes

“…The surface strain/stress induced by sandblasting was evaluated with the Williamson-Hall (W-H) method in the uniform deformation model (UDM) [ 21 ]. The physical broadening of XRD peak as a function of microstrain was considered according to Equation (1) [ 21 , 22 ]: where L is the nanocrystal size; K is the shape factor, usually taken as 0.89 for ceramic materials; λ is the wavelength of radiation in nanometer; ß is the full width at half maximum of the peak in radians; θ is the diffracted angle of the peak; ε is the strain induced by crystal deformation.…”

“…The surface strain/stress induced by sandblasting was evaluated with the Williamson-Hall (W-H) method in the uniform deformation model (UDM) [21]. The physical broadening of XRD peak as a function of microstrain was considered according to Equation (1) [21,22]:…”

Different solvents and organic modifiers for the control of crystallographic parameters in nano-crystallite hydroxyapatite for amplification of photocatalytic activity

“…Hexagonal HAp comprises P6 3 /m space groups by a sixfold symmetric axis arranged with a threedimensional helix as well as a reecting plane, whereas the usual lattice parameters are: (1) a = b = 9.42 Å and c = 6.88 Å; (2) cell volume = 530.301 (Å); 3 and (3) crystal density = 3.140 g cm −3 . 15,16 Nanocrystals show intrinsic strain arising from size internment that may be modied by adjusting synthesis factors such as pH and concentration, which impact optical as well as other features. 17 The photocatalytic properties of HAp have been under scrutiny due to its use with metals for dye degradation.…”