The zeolitic compound was synthesized by mixing 1.64 g of sodium aluminate with 50 mL of an aqueous sodium hydroxide solution at a concentration of 0.103 g/mL. After 10 minutes of stirring, 11.1 mL of sodium silicate was added to the previous solution under vigorous magnetic stirring to form the gel. After 24 hours of aging, the formed aluminosilicate gel was transferred into an autoclave, which was then placed in an oven at 105˚C for 8 days. After crystallization, the obtained product was recovered, washed with distilled water until pH = 9, and dried at 95˚C for 24 hours.

The synthesized product was analyzed using several techniques. X-ray diffraction (XRD) measurements were carried out using an X’Pert3 Panalytical powder diffractometer employing Cu Kα radiation with wavelengths of (λKα1 = 1.54059 Å; λKα2 = 1.54441 Å). The data were collected under the following experimental conditions: 5 ≤ 2θ ≤ 50˚, Δ2θ = 0.02˚, 45 kV and 100 mA. Phase identification was performed using the QualX software with the COD database. Rietveld profile refinement was carried out using the FullProf program (version: March 2021). The profile function used was the Thompson-Cox-Hastings pseudo-Voigt convoluted with axial divergence asymmetry. The average crystallite size and microstrain were estimated using the Scherrer, Monshi-Scherrer, Williamson-Hall, Halder-Wagner, Size-Strain Plot, and modified Warren-Averbach methods. The instrumental broadening of the XRD peaks was determined using a crystalline silica standard, for which an instrumental FWHM, β_instrumental, of 0.071˚ was measured for the (1 1 1) reflection at 2θ = 28.4˚. The instrumental contribution was corrected using a Gaussian deconvolution procedure according to β correted 2 = β obs 2 − β instrumental 2 , where β_obs is the observed FWHM. The profiles of the individual XRD reflections were fitted in OriginPro 2024 software using a Gaussian function to determine the peak positions and FWHM values. Only well-resolved reflections without significant overlap with neighboring peak were considered for the microstructural analysis and subsequent fitting procedures. The morphology and local elemental chemical composition were investigated using a Regulus 8100 scanning electron microscope coupled with energy-dispersive spectroscopy. Fourier-transform infrared analysis was performed on a Thermo Scientific Nicolet iS50 FTIR spectrophotometer scanning the range from 500 to 4000 cm⁻¹. The elemental composition was determined using a Thermo Fisher Scientific X-ray fluorescence spectrometer.

2.3.1. Scherrer and Monshi-Scherrer Methods

The Scherrer method is the simplest and most rapid approach for estimation the average crystallite size, as it neglects the contribution of lattice microstrian to X-ray diffraction peak broadening. This size is calculated using Equation (1) [14][19][20].

The Monshi-Scherrer method is a modified form of the Scherrer equation. In this approach, Equation (1) is linearized to yield the following expression Equation (2) [20]:

With D: average crystallite size (nm); K: Scherrer constant (K = 0.9); λ: X-ray wavelenght (nm); β: full width at half maximum (FWHM) corrected for instrumental broadening, in radians; θ: Bragg’s angle in radians.

2.3.2. Williamson-Hall Methods

Williamson-Hall method considers X-ray diffraction peak broadening as a combinaison of crystallite size and lattice microstrain contributions. These microstrain are attributed to the presence of point defects, grains boundaries, and stacking faults [21][22]. This broadening is defined as follows:

Three models are derived from the Williamson-Hall method [20][22][23], namely:

a) Uniform Deformation Model

This model assumes that lattice microstrain is uniformly distributed in all crystallographic directions throughout the crystal, resulting in isotropic peak broadening. It is expressed by Equation (4) [13]:

where ε is the lattice microstrain.

b) Uniform Stress Deformation Model

Unlike UDM, USDM takes into account the anisotropy of Young’s modulus, thereby providing a more realistic description of the lattice deformation conditions. The stress associated with lattice microstrain is assumed to be uniform across all crystallographic directions, while explicitly incorporating the contribution of microstrain within the crystals. The following Equation (5) represents Hooke’s law, which describes a linear relationship between strain and stress:

where σ is the stress, ε is the anisotropic microstrain and Y_hkl is the modulus of elasticity or Young’s modulus. By substituting ε with its expression from relation 5, the Williamson-Hall model can be rewritten as Equation (6) [24]:

For cubic systems, Young’s modulus is given by Equation (7)

With:

where:

C₁₁, C₁₂, C₄₄ and S₁₁, S₁₂, S₄₄ are the stiffness constants and the elastic compliances of cubic analcime zeolites, respectively. The values of the elastic constants C_ij have been reported in references [25][26]; those used in this study are as follows: C₁₁ = 112.5 GPa, C₁₂ = 33.4 GPa, C₄₄ = 27.9 GPa [25].

c) Uniform Deformation Energy Density Model

Unlike the UDM and USDM approaches, which assume isotropy and a linear relationship between strain and stress, respectively, the UDEDM approach considers the anisotropic deformation of the crystalline lattice in all crystallographic directions. This deformation is caused by the strain energy density [14][22][27]. This model, therefore, relies on the anisotropy of Young’s modulus Y_hkl. Equation (8) gives Hooke’s law relating the anisotropic strain energy density (u) to the stress. After rearrangement, the Williamson-Hall model can be expressed as Equation (9)

2.3.3. Size-Strain Plot Method

The Size-Strain Plot method allows the estimation of crystallite size and lattice strain by assuming that the broadening of the diffraction profile results from a combination of Lorentzian and Gaussian functions Equation (10), associated with crystallite size and microstrain, respectively.

where β_L and β_G are the full widths at half maximum of the Lorentzian and Gaussian functions, respectively. The relationship used to estimate the crystallite size and lattice strain by the Size-Strain Plot method is expressed by Equation (11) [10][14][28].

2.3.4. Halder-Wagner Method

In contrast to the size-strain plot method, which assumes that broadening due to crystallite size follows a Lorentzian function and that due to strain is described by a Gaussian function, the Halder-Wagner method assumes that XRD peak broadening is a symmetric Voigt function [29], i.e., a convolution of a Lorentzian function and a Gaussian function [30]. In this case, Equation (12) gives the full width at half maximum:

The relation used to determine the crystallite size and lattice strain, according to the Halder-Wagner method, is given by Equation (13) [22][31]:

2.3.5. Modified Warren-Averbach Method

The Warren-Averbach method, based on the Fourier analysis of XRD peak profiles, enables the separation and quantification of crystallite size and lattice microstrain [20][32]-[34]. A modified equation derived from this approach Equation (14) [20][35][36] was proposed without employing Fourier series analysis.

The XRD pattern of the synthesized product (Figure 1) reveals the presence of a pure phase consisting solely of analcime (COD card 00-900-8207), with space group I a -3 d and lattice parameter a = 13.74079 Å, characterized by peaks at 2θ (˚) = 15.76, 18.22, 24.20, 25.90, 30.48, 31.86, 33.20, 35.74, 36.96, 40.42, 41.52, 42.60, 44.68, 47.68, and 48.64. All peaks were indexed and correspond respectively to the reticular planes (2 1 1), (2 2 0), (3 2 1), (4 0 0), (3 3 2), (4 2 2), (4 3 1), (5 2 1), (4 4 0), (5 3 2), (6 2 0), (5 4 1), (6 3 1), (6 4 0), and (6 3 3). The most intense peak corresponds to the (4 0 0) plane; this feature has also been reported by [37][38].

Figure 1. XRD pattern of synthetized zeolite.

Figure 2 shows the observed and calculated profiles along with their difference profile. The Rietveld refinement yielded agreement factors of Rp = 12.1% and Rwp = 12.6%, indicating a satisfactory overall fit between the experimental and calculated profiles [39][40]. The low value of R_Bragg = 3.1% reflects the good quality of the structural model. However, the relatively high GoF value of 10.9 suggests a discrepancy between the experimental data and the refined model; this deviation may indicate the presence of contributions not accounted for in the refinement, such as preferred orientation effects, microstrain, or the presence of minor secondary phases [41][42]. The low value of Rexp = 3.8 indicates the high quality of the experimental data.

Figure 2. Experimental, calculated XRD pattern and their difference for Rietveld refinement of synthetized zeolite sample.

Although 15 reflections were indexed in the XRD pattern, only 12 sufficiently intense reflections were selected for the microstructural analysis. The (620), (541), and (631) reflections were excluded because of their very low intensities, which prevented reliable determination of their FWHM values.

3.3.1. Scherrer and Monschi-Scherrer Methods

The average crystallite size is obtained from the slope of the plot of lnβ as a function of ln(1/cosθ). The Scherrer and Monshi-Scherrer methods (Figure 3 and Figure 4) yield crystallite sizes of 56.46-98.84 nm (with an average of 75.47 nm) and 98.71 nm, respectively (Figure 3 and Figure 4).

Figure 3. Scherrer method.

Figure 4. Monshi-Scherrer method.

3.3.2. Williamson-Hall Method

a) Uniform deformation model (UDM)

The linear fit (Figure 5) of β h k l ⋅ cos θ versus 4 sin θ enabled the determination of the average crystallite size from the intercept (Kλ/D = 8.74588 × 10⁻⁴), while the latiice microstrain was directly obtained from the slope (ε = 9.31 × 10⁻⁴). The resulting average crystallite size was found to be 158.54 nm. The positive value of the microstrain indicates that the strain is of a tensile nature [11][22].

b) Uniform stress deformation model (USDM)

The Young’s moduli corresponding to the Miller indices of cubic analcime were calculated using Equation (7) and are reported in Table 1. The linear fit of the plot of β h k l ⋅ cos θ versus 4 sin θ / Y h k l , shown in Figure 6, enabled the average crystallite size to be estimated from the intercept (Kλ/D = 9.38071 × 10⁻⁴). The resulting average crystallite size is approximately 149.46 nm. The slope, corresponding to σ = 0.0698 GPa, was used to determine the lattice microstrain (ε = 9.024 × 10⁻⁴)

Table 1. Young’s moduli corresponding to the Miller indices of cubic analcime.

c) Uniform deformation energy density model (UDEDM)

The intercept of the linear fit of the plot of β h k l ⋅ cos θ as a function of 4 sin θ ⋅ 2 Y h k l (Figure 7), given by K∙λ/D = 0.0008933, yielded an average crystallite size of approximately 155.21 nm. The slope u = 0.00575 was used to estimate the anisotropic strain energy density u = 3.306 × 10⁻⁵ GPa. By rearranging Equation (8), the microstrain value ε was determined to be approximately 9.24 × 10⁻⁴, corresponding to a stress of σ = 0.0712 GPa.

3.3.3. Size-Strain Plot Method

The linear fit of the experimental data of the plot of ( d h k l 2 ⋅ β h k l ⋅ cos θ ) versus ( d h k l ⋅ β h k l ⋅ cos θ ) 2 , presented in Figure 8, enabled the crystallite size and lattice strain to be estimated from the slope (K∙λ/D = 0.0012) and the intercept (ε²/4 = 9.80678.10⁻⁶), respectively. The calculated crystallite size and lattice strain are 115.54 nm and ε = 6.26 × 10⁻³, respectively.

Figure 5. UDM.

Figure 6. USDM.

Figure 7. UDEDM.

3.3.4. Halder-Wagner Method

The linear fitting of the plot of ( β h k l 2 tan θ ) 2 as a function of ( β h k l 4 tan θ ⋅ sin θ ) derived from the experimental data (Figure 9) made it possible to estimate the crystallite size and lattice strain from the slope (Kλ/D = 0.0012) and the intercept (4ε² = 4.12991 × 10⁻⁶). The calculated crystallite size and strain are D = 115.542 nm and ε = 1.02 × 10⁻³, respectively.

Figure 8. Size-Strain plot.

Figure 9. Halder-Wagner.

3.3.5. Modified Warren-Averbach Method

The slope λ/D = 0.0012 and the intercept ( 25 〈 ε 2 〉 = 1.65196 × 10 − 4 ), obtained from the linear fit of the plot of β h k l 2 tan θ versus β h k l tan θ sin θ , as shown in Figure 10, enabled the estimation of crystallite size and lattice strain. The calculated crystallite size and lattice strain are 128.383 nm and 8.13 × 10⁻⁴, respectively.

Figure 10.Modified warren-averbach method.

Analysis of the diffraction peak broadening data (Figure 11) revealed a pronounced dependence of the crystallite size and lattice microstrain values on the analytical method employed. The estimated crystallite sizes ranged from 75.47 to 158.54 nm, whereas the corresponding microstrain values were on the order of 10⁻⁴ - 10⁻³. Similar discrepancies have been widely reported in the literature, demonstrating that these analytical approaches may yield significantly different results for the same material [21][23][43]. This variability primarily reflects the distinct theoretical assumptions underlying each method rather than intrinsic differences in the material itself. The Scherrer and Monshi-Scherrer methods produced the smallest crystallite sizes, namely 75.47 and 98.71 nm, respectively, which is consistent with the fact that these approaches neglect the contribution of lattice microstrain [21]. In contrast, the Williamson-Hall models, namely UDM, USDM, UDEDM, yielded the lagest crystallite sizes of 158.54, 149.46, and 155.21 nm, respectively, owing to the explicit separation of size and strain-induced broadening. However, these models rely on assumptions of isotropic deformation (UDM) or simplified elastic approximations (USDM ans UDEDM), with may substantially influence the calculed values. Consequently, these results should be regarded as model-dependent estimates rather than absolute physical quantities. The Size-strain plot and Halder-Wagner methods produced similar crystallite sizes of approximately 115.54 nm, indicating good internal consistency in the treatment of diffraction peak broadening. This agreement suggest that these approches may provide a more realistic estimate of the average coherent domain size [31][44] in nanocrystalline materials such zeolite. The modified Warren-Averbach method yielded an intermediate crystallite size of 128.38 nm, lying between the estimates obtained using the Scherrer-type and Williamson-Hall approaches. For analcime, these results are consistent with a highly crystalline framework while also indicating the presence of local lattice distortions and point defects [45][46]. The calculated microstrain values were relatively low, on the order of 10⁻³ - 10⁻⁴, and exhibited good consistency in the terms of magnitude. These low indicate the pressence of moderate internal lattice strain while suggesting that the associated structural imperfections remain limited and do not significantly disrupt the long-range crystalline order, in agreement with the overall high crystallinity of the synthesized analcime. Overall, the differences observed among the various analytical methods primarily arise from their underlying modeling assumptions and their intrinsic sensitivity to the respective contributions of crystallite size and lattice strain to diffraction peak broadening. Within this context, the size-strain plot and Halder-Wagner methods appear to provide the most reliable estimates of the average coherent domain size, whereas the Scherrer and Williamson-Hall methods may reasonably be regarded as providing the lower and upper bounds of the crystallite size estimation, respectively. Finally, the crystallite sizes determined in this study should be interpreted as the diffraction coherence domain size, corresponding to the crystal regions that diffract coherently with the synthesized analcime. (Figure 11)

Figure 11.Correlation crystallite size - strain.

Figure 12 and Figure 13 present the SEM micrographs and the results of the EDS analysis of the synthesized product. Figure 12(a) and Figure 12(b) show homogeneous particles with spherical shapes. The observation of Figure 12(c) and Figure 12(d) shows a trapezohedral or deltoidal icositetrahedral morphology, characteristic of analcime [47]-[49], whose particle size (Figure 14) ranges from 6.97 to 14.79 µm, with an average particle size of 11.7 µm (Figures 12-14).

Figure 15 shows the infrared spectrum of the synthesized product, and the observed bands are consistent with those reported in the literature. The band at 619 cm⁻¹ is attributed to the T-O-T (T = Si or Al) bending vibration [50][51]. The

Figure 12. SEM of synthetized Analcime; scale: (a) 50 µm, (b) 20 µm, (c); 10 µm, (d) 5 µm.

(a)

(b) (c)

Figure 13. SEM-EDS of synthetized compound.

Figure 14. Particle size of the synthesized compound.

band at 730 cm⁻¹ corresponds to the external symmetric vibration of T-O-T linkages. The broad band around 965 cm⁻¹ is assigned to the asymmetric stretching vibration of T-O bonds [50][52]. The band at 1633 cm⁻¹ is related to the bending vibration of OH groups from water molecules located within the zeolite micropores. In contrast, the absorption band around 3599 cm⁻¹ is associated with the O-H stretching vibration of silanol groups and adsorbed water molecules [50][52][53].

Figure 15. Infrared spectrum of the synthesized compound.

Figure 16 presents the chemical composition of the synthesized product. The data indicate that the material is predominantly an aluminosilicate, with a high silicon content or silica (Si: 61.9% or SiO₂: 58.9%). The proportions of aluminum or alumina (Al: 22.2% or Al₂O₃: 24.2%) and sodium or sodium oxide (15.3% or Na₂O: 16.7%) are also significant. The Si/Al ratio of 2.8 suggests the formation of an analcime-type zeolite [54].