IHJPAS. 36(1)2023 123 This work is licensed under a Creative Commons Attribution 4.0 International License Using the Size Strain Plot Method to Specity Lattice Parameters . Abstract X-ray diffractometers deliver the best quality diffraction data while being easy to use and adaptable to various applications. When X-ray photons strike electrons in materials, the incident photons scatter in a direction different from the incident beam; if the scattered beams do not change in wavelength, this is known as elastic scattering, which causes amplitude and intensity diffraction, leading to constructive interference. When the incident beam gives some of its energy to the electrons, the scattered beam's wavelength differs from the incident beam's wavelength, causing inelastic scattering, which leads to destructive interference and zero-intensity diffraction. In this study, The modified size-strain plot method was used to examine the pattern of x-ray diffraction lines (101),(121),(202),(042), and (242) for calcium titanate(CaTiO3) nanoparticles in this study. To calculate the new variables, the size strain plot method was created., X-ray line analysis and calculation of crystal size and lattice tension of calcium titanate oxide nanoparticles. It is used to calculate the crystal volume (44.7 nm) and to calculate the determination of network parameters such as the texture modulus (Tc), macro stress (MS), specific surface area (SSA), and dislocation density(Ξ·), respectively. Keywords:Calcium titanate,size strain plot,Texture Coefficient 1. Introduction The most important method for determining crystal structure is X-ray diffraction (XRD). The size of crystalline blocks and the degree of crystalline structural deformation in nanocrystals can be determined using this method. Methods for analyzing nanocrystalline materials using X-rays are currently being developed. [1-2]. X-ray diffraction is the most important method for determining crystal structure (XRD). This method may determine the size of crystalline blocks and the degree of crystalline structural deformation in nanocrystals. X-ray analysis methods for nanocrystalline Doi.org/10.30526/36.1.2891 Article history: Received 16 June 2022, Accepted 21 Augest 2022, Published in January 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Marwah Talib Jalil Department of Phesics,College of Education for Pure Sciences, Ibn Al - Haitham, University of Baghdad, Baghdad, Iraq. marwa.Taleb1204a@ihcoedu.uobaghdad.edu.iq Khalid Helal Harbbi Department of Phesics,College of Education for Pure Sciences, Ibn Al - Haitham, University of Baghdad, Baghdad, Iraq. Khalid.h.h@ihcoedu.uobaghdad.edu.iq https://creativecommons.org/licenses/by/4.0/ mailto:marwa.Taleb1204a@ihcoedu.uobaghdad.edu.iq mailto:Khalid.h.h@ihcoedu.uobaghdad.edu.iq IHJPAS. 36(1)2023 124 materials are currently being developed. The X-ray profile analysis is an average method widely used in determining crystalline size. In the Size-strain plot technique, which assumes that the Gaussian function governs the (strain profile), the significant angular vertices are less critical. [3]. Calcium titanate (CaTiO3) is a material of Perovskites and one of the most prevalent structural families. It is present in various compounds with diverse properties, uses, and significance. Paul Scherer developed Schererer's Formula in 1918, which computes the crystal size (D) of nanomaterials using the entire width of the X-ray diffraction pattern at half is the most significant value of the peaks [4]. The diffraction pattern's maxima are widened by a factor which is inversely proportional to crystallite size, and the extra broadening is measured. The formula can be used to estimate the size of a powder specimen if the crystallites are small enough [5]. Calcium Titanate is a colorless, diamagnetic solid, often coloured owing to impurities. It is a chemical compound of titanium, oxygen, and calciumPerovskite is a mineral with the formula CaTiO3. Calcium titanate (CaTiO3) is semi-conductive, photorefractive, and ferroelectric. It is reduced to give ferrotitanium alloys or titanium metal [6]. Multiple methods can prepare calcium titanate. CaTiO3 has been synthesized at high temperatures using mixtures of calcium carbonate (CaCO3), titanium dioxide (TiO2), and calcium oxide (CaO). It also uses forgiving chemistry methods such as sol-gel or solvothermal methods, hydrothermal or organic-inorganic solution, or coprecipitation [7]. 2. size strain plot Data from high-angle reflections are given less weight in this strategy. Because XRD data is of lower quality and peaks overlap at greater angles with higher diffracting, the isotropic broadening is improved. The Gaussian function depicts the strain profile, while the Lorentzian function depicts crystallite size [8]., according to this assumption. The equation was also used to express the real broadening of this method. π›½β„Žπ‘˜π‘™ = 𝛽𝐿 + 𝛽𝐺 (1) L and G Peak broadening is represented by Lorentz and Gaussian functions, respectively. size-strain plot method can be determined by the following equation: (π‘‘β„Žπ‘˜π‘™ π›½β„Žπ‘˜π‘™ π‘π‘œπ‘ πœƒ) 2 = π‘˜ 𝐷 (𝑑2β„Žπ‘˜π‘™ π›½β„Žπ‘˜π‘™ π‘π‘œπ‘ πœƒ) + ( πœ€ 2 )2 (2) For spherical particles, the form is expressed as 3/4, where K is a constant In comparison to other methods, the Size-Strain plot method has a significant benefit, because it gives the great angle peaks are given less weight, [9]. As a result, despite the fact that the Gaussian function is believed to control (the strain profile), the great angle peaks are given less weight. [3]. If (πœ€ = 0) D = k(𝑑2β„Žπ‘˜π‘™ π›½β„Žπ‘˜π‘™ π‘π‘œπ‘ πœƒ) (π‘‘β„Žπ‘˜π‘™ π›½β„Žπ‘˜π‘™ π‘π‘œπ‘ πœƒ) 2 (3) If(D=∞) ( πœ€ 2 )2 = (π‘‘β„Žπ‘˜π‘™ π›½β„Žπ‘˜π‘™ π‘π‘œπ‘ πœƒ) 2 = y_intercept (4) Herein case, Equation (2) was used to calculate the crystallite size and lattice strain. As indicated in the table, (dhkl 2Ξ²hkl cos ΞΈ) on X-axis and (dhklΞ²hklcosΞΈ) 2 on the y- axis is calculated for each line shape as shown in Table (1). Using the marks of plotting the relationship, the crystallite size and lattice strain were calculated. between (dhkl 2Ξ²hkl cos ΞΈ) and (dhklΞ²hklcosΞΈ) 2. IHJPAS. 36(1)2023 125 Table 1. The results the Size Strain Plot method of (dhkl 2Ξ²hkl cos ΞΈ) and (dhkl Ξ²hkl cos ΞΈ) 2 for all the profile lines Peaks 𝟐𝜽(degree) ΞΈ(degree) π’„π’π’”πœ½ 𝜷(deg) 𝜷 (𝒓𝒂𝒅) d (nm) (dΞ²cosΞΈ) 2 X10 -6 (101) 23.26247 11.63123 0.979465 0.219002 0.00382 0.38207 2.04398 (121) 33.25642 16.62821 0.958181 0.392082 0.00683 0.26918 3.11219 (202) 47.48688 23.74344 0.915357 0.292729 0.00510 0.19131 0.79966 (042) 59.30423 29.65211 0.869045 0.731695 0.01276 0.15570 2.98290 (242) 69.53593 34.76796 0.821468 0.310271 0.00541 0.13508 0.36071 The slope and intercept of the plot were calculated in Figure (1), using equation (2) to calculate the framework strain from the y-intercept and the crystallite size of the slope (2). Figure 1. Plot of Size-Strain plot method Table 2. Using Size –Strain plot to calculate crystallite size and lattice strain Table (3) Calculation of the lattice strain, stress, the crystallite size, and the energy after modifying ΖΈ=2( intercept)1/2 U (KJ/m3) Οƒ (G Pa) D(nm) 1.2108X10-3 0.1158X10-3 0.18430463 44.71 3. Determination of the lattice parameters: K Ξ» (nm) Y( intercept) X1 X2 0.89 0.15406 0.366544014X10-6 3.0936432X10 -4 5.95625427X10-4 slope D( nm)=K/slope ΖΈ=2(intercept)1/2 Y1 Y2 0.306616686X10-2 44.71 1.21085X10-3 1.31262383X10-6 2.19034815X10-6 IHJPAS. 36(1)2023 126 3.1 Texture Coefficient (Tc): The texture coefficient Tc was used to quantify the XRD results (hkl). The following equation can be used to compute this factor for each direction. 𝑇𝑐(β„Žπ‘˜π‘™) = 𝐼(β„Žπ‘˜π‘™)/𝐼0(β„Žπ‘˜π‘™) (1/𝑁)[βˆ‘ 𝐼(β„Žπ‘˜π‘™)/𝐼0(β„Žπ‘˜π‘™)]𝑁 (5) The obtained and standard intensities of the (hkl) plane are represented by I(hkl) and I0(hkl), respectively, and N denotes the number of diffraction peaks [10]. To determine whether one orientation stands out from the others, the TC formulation compares the power ratios of surfaced coatings to non-textured models. The link is familiarized by comparing each ratio's average value of all replication intensity ratios. The total number of reflections should be equal to the sum of the TC values, so the TC per reflection cannot exceed n. A TC value is greater than the one indicating a texture [11]. 3.2 Micro strains (MS): During thin film growth, microstrains form as a result of pressure or stretching in the lattice, causing it to deviate from the lattice constant. As a result, strain stretching is caused by changing the displacement of the atoms in relation to their reference lattice posit ion. The strain was calculated using the equation: < 𝑀𝑆 >= 𝛽 π‘π‘œπ‘ πœƒ 4 (6) Where: Ξ²=FWHM of the intensity of the peak in radian;ΞΈ= Bragg angle [11]. 3.3 Specific Surface Area (SSA) Because of the large surface-to-volume ratio with decreasing particle size, Surface states will be crucial in nanoparticles [12]. A crucial characteristic is specific surface area. It is a quantitative value derived from information that can be used to identify the kind and characteristics of a material. It is especially important in adsorption, heterogeneous catalysis, and surface reactions. SSA denotes the Surface Area (SA) per mass according to [13] (Zhang et al., 2016). The specific surface area and the surface-to-volume ratio of materials increase dramatically as their size decreases [14]. 𝑆𝑆𝐴 = π‘†π΄π‘π‘Žπ‘Ÿπ‘‘ π‘‰π‘π‘Žπ‘Ÿπ‘‘ βˆ—π‘‘π‘’π‘›π‘ π‘‘π‘¦ (7) 𝑆 = 6 βˆ— 103 𝐷𝑝 (8) Where (S) is the specific outward area, (D) is the sizespherical shaped; and (Ξ‘) is the density of CaTiO3 3.4 Dislocation density () The Dislocation density () can be calculated from equation [15].  = 1 𝐷2 (𝑙𝑖𝑛𝑒𝑠/π‘›π‘š2) (9) 4. Result and Discussion IHJPAS. 36(1)2023 127 Using equation (5), we calculated the outline lines in the CaTiO3 powder x-ray diffraction design (5). When TC >1, it is confirmed that the selected levels' crystal development will proceed in this direction. The improvement of the material's crystal formation is correlated with the value of this factor when TC < 1 is polycrystalline but in a non-uniform direction. The best situation for surface expansion is if TC=1. Table (4) shows the Texture Coefficient results for all outline lines. Table 4. The results of (𝑇𝑐) for all outline lines. Peaks I I (h kl ) I (h kl)/I0(h kl) Tc (1 01) 10 400.6596 40.06596 1.341837 (1 2 1) 100 1802.3441 18.0234 0.6036 (2 0 2) 40 916.6432 22.9168 0.7675 (0 42) 15 608.5128 40.5675 1.3586 (2 42) 20 554.4279 27.7213 0.9284 Sum 149.2951376 AV: 1 Micro strains were premeditated using equation (6) for the outline lines in the CaTiO3 powder x- ray diffraction design. Table (5) shows the Micro strains results for the outline lines . Table 5. The results of Micro strains all peaks of CaTiO3 nanoparticles (h k l) 2ΞΈ(degree) ΞΈ(degree) Cos ΞΈ Ξ² ( rad) X10-3 (1 01) 23.26247 11.631235 0.979465485 0.003820379439 0.93548245 (1 2 1) 33.25642 16.62821 0.958181797 0.006839663761 1.638410328 (2 02) 47.48688 23.74344 0.915357585 0.005106502732 1.168569002 (0 42) 59.3042323 29.65211615 0.869045281 0.012764026 2.77312914 (2 42) 69.5359388 34.7679694 0.821468132 0.005412508205 1.111550751 Sum 7.626141671 AV. 1.52 For all of the outline lines in the x-ray deflection pattern of CaTiO3 powder, the Specific Surface Area was calculated using equation (7). Table 1 shows the results of the Specific Surface Area for all of the outline lines. Table 6. The results of specific surface Area (SSA) D (nm) V part= 4/3D SSA(m 2 /g) 32.29477611 43.05 35.36 SA Intensity 6X10 3 3.94 It can be calculated Specific Surface Area from equation (9) for all the outline lines in the x-ray diffraction pattern of CaTiO3 powder. The results of Dislocation density () for all the outline lines are registered in Table (7). Table 7. The results of dislocation density () ( h kl ) D ( nm) =1/D 2 ( lines/nm 2 ) IHJPAS. 36(1)2023 128 (1 01) 36.6424298 0.744785942 (1 21) 20.92171259 0.002284575613 (2 02) 29.33361225 0.001162167982 (04 2) 12.36089207 0.006544860152 (2 42) 30.83831302 0.001051523997 5. Conclusions In this study, all the profile lines in the CaTiO3 powder's x-ray diffraction pattern had their micro strains calculated. A specific surface area was calculated for all the profile lines in the CaTiO3 powder x-ray diffraction pattern. For all profile lines in the x-ray diffraction, the dislocation density (h) can be calculated in the x-ray diffraction pattern of CaTiO3 powder. It can be used to determine the kind and characteristics of a material. All the profile lines in the CaTiO3 powder's x-ray diffraction pattern had their texture coefficient calculated. When TC>1, for(101),(042), it is confirmed that the preferred levels' crystal growth direction is in that direction. The improvement of the material's crystal growth is correlated with the value of TC, but TC <1 for (121), (202), and (242) is polycrystalline, albeit in a non-uniform direction. References 1.Bykkam, S.; Ahmadipour, M.; Narisngam, S.; Kalagadda, V. R.; Chidurala, S. C. Extensive studies on X-ray diffraction of green synthesized silver nanoparticles. Adv. Nanopart, 2015, 4, 1,1- 10. 2. Holder, C. F.; Schaak, R. E.; Tutorial on powder X-ray diffraction for characterizing nanoscale materials. Acs Nano, 2019,13, 7, 7359-7365. 3.Shukla, P.; Subhasisa N.; Guanjun W.; Xiaojun S.; and Jonathan L.; Surface property modifications of silicon carbide ceramic following laser shock peening. Journal of the European Ceramic Society, 2017,37,9, 3027-3038,. 4.Gralik, G.; Raupp-Pereira, F.; Hotza, D.; Labrincha, J. 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Result and Discussion Using equation (5), we calculated the outline lines in the CaTiO3 powder x-ray diffraction design (5). When TC >1, it is confirmed that the selected levels' crystal development will proceed in this direction. The improvement of the material's cry... Table (4) shows the Texture Coefficient results for all outline lines. Table 4. The results of (𝑇𝑐) for all outline lines. For all of the outline lines in the x-ray deflection pattern of CaTiO3 powder, the Specific Surface Area was calculated using equation (7). Table 1 shows the results of the Specific Surface Area for all of the outline lines. Table 6. The results of specific surface Area (SSA) Table 7. The results of dislocation density (()