Heliyon 10 (2024) e25347

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Heliyon
journal homepage: www.cell.com/heliyon

Research article

Synthesis of nano-crystallite hydroxyapatites in different media

and a comparative study for estimation of crystallite size using
Scherrer method, Halder-Wagner method size-strain plot, and
Williamson-Hall model
Md. Kawsar a, b, Md. Sahadat Hossain a, Newaz Mohammed Bahadur b,
Samina Ahmed a, c, *
a
Glass Research Division, Institute of Glass & Ceramic Research and Testing, Bangladesh Council of Scientific and Industrial Research (BCSIR),
Dhaka-1205, Bangladesh
b
Department of Applied Chemistry and Chemical Engineering, Noakhali Science and Technology University, Noakhali, Bangladesh
c
BCSIR Dhaka Laboratories, Bangladesh Council of Scientific and Industrial Research (BCSIR), Dhaka-1205, Bangladesh

A R T I C L E I N F O A B S T R A C T

Keywords: Hydroxyapatite (HAp) [Ca10(PO4)6(OH)2] is remarkably similar to the hard tissue of the human
Crystallite size body and the uses of this material in various fields in addition to the medical sector are increasing
X-ray diffraction day by day. In this research, mustered oil, soybean oil, as well as coconut oil were employed as
Peak profiling
liquid media for synthesizing nanocrystalline HAp using a wet chemical precipitation approach.
Crystallographic parameter
The X-ray diffraction (XRD) study verified the crystalline phase of the HAp in all the indicated
media and discovered similarities with the standard database. Several prominent models such as
the Scherrer’s Method (SM), Halder-Wagner Method (HWM), linear straight-line method (LSLM),
Williamson-Hall Method (W-M), Monshi Scherrer Method (MSM), Size-Strain Plot Method
(SSPM), Sahadat-Scherrer Method (S–S) were applied for the determination of crystallite size. The
stress, strain, and energy density were also computed from the above models. All the models,
without the Linear straight-line technique of Scherrer’s equation, resulted in an appropriate value
of crystallite size for synthesized products. The calculated crystallite sizes were 6.5 nm for HAp in
master oil using Halder-Wagner Method, and 143 nm for HAp in coconut oil using the Scherrer
equation which were the lowest and the largest, respectively.

1. Introduction

The word ‘crystallite size,’ has a variation from ‘particle size,’ which is very much significant for crystalline substances for pro­
ductive application [1]. And, crystallite size is particularly significant for microstructural and physical features of any crystalline
materials [2]. Crystallographically, HAp has two distinct forms of structure: (a) monoclinic and (b) hexagonal. Hexagonal HAp
contains space group of P63/m by a symmetrical axis of sixfold organized with a helix of threefold retaining a mirror plane and the
intrinsic properties were reported as (a) crystal density = 3.140 g cm− 3, (b) a = b = 9.42 Å and c = 6.88 Å, and (c) cell volume =

* Corresponding author. Glass Research Division, Institute of Glass & Ceramic Research and Testing, Bangladesh Council of Scientific and In­
dustrial Research (BCSIR), Dhaka 1205, Bangladesh.
E-mail address: shanta_samina@yahoo.com (S. Ahmed).

https://doi.org/10.1016/j.heliyon.2024.e25347
Received 22 September 2023; Received in revised form 10 January 2024; Accepted 25 January 2024
Available online 30 January 2024
2405-8440/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Md. Kawsar et al. Heliyon 10 (2024) e25347

530.301 (Å)3 [3,4]. For the assessment of particle size along with crystallite size light scattering, BET (Brunauer–Emmett–Teller)
theory, transmission electron microscopy (TEM), Atomic force microscopy (AFM), and scanning electron microscopy (SEM) are most
often utilized [5,6]. A few other less applicable techniques are selected area electron diffraction (SAED), electron back-scattered
diffraction (EBSD), and neutron diffraction (ND) [7]. The technique of X-ray diffraction (XRD) is acknowledged as an efficient and
resilient instrument for determining the size of crystals [8,9]. Mainly, the Powder X-ray diffraction (PXRD) assessment method is
widely utilized to evaluate the quantity of crystallite and deformation of the lattice. The knowledge of the quantity of crystallite and
deformation of the lattice is associated with the diffraction peak extending inducing lattice strain derived from the researched ma­
terial’s defects (stacking defects or coherency stresses) [10].
Scientists generally utilize the traditional Scherrer equation to compute the crystallite size, which was introduced in 1918 [11].
However, until now, multiple modified formulas and models, e.g., the model of straight line passing through the origin (MSLPO) of the
Scherrer equation, straight line model in Scherrer method (SLMSM), Williamson–Hall (W–H) model, Monshi–Scherrer model,
Halder-Wagner (H–W) and Size-Strain Plot (SSP) methods have been established and utilized in many studies [12–15]. The Scherer’s
approach is often used for determining crystallite size based on the broadening of the reflection. Nevertheless, the crystallite size
estimated by this approach is a tiny bit erroneous since peak broadening develops with lattice strain and crystallite dimension [16,17].
The important component of SLMSM is that rather than selecting a specific reflection peak, it examines all peaks to determine the
crystallite size. Furthermore, in the current works, it’s shown that this strategy is erroneous in the scenario of natural nanocrystalline
material [8]. In an additional technique, the Monshi–Scherrer equation was built employing the ln(log 10) edition of the Scherrer
equation to compute crystallite size. Researchers have also pointed out this model to acquire more precise estimations for crystallite
size as it keeps the relevance of lowering the deficiencies by leveraging the least squares approach and diminishing the absolute extent
of limitation [1,13]. The Williamson–Hall developed an X-ray peak profile analysis approach which is employed to estimate the
crystallite size D(hkl), stress (σ), lattice strain (ε), and energy density (u). The W–H technique is divided into three sub-methods
Uniform deformation energy-density model (expressed as UDEDM), uniform stress deformation model (expressed as USDM), and
uniform deformation model (expressed as UDM) to predict an idea of strain and stress-strain relation as a relation of energy density (u)
[18]. Furthermore, when the crystals of the materials are isotropic, it is thought that the strain inside the crystal is uniform through all
orientations and UDM is performed to quantify lattice strain [19,20]. If the sample comprises anisotropic crystals that are crystallo­
graphic planes composed of homogenous stress, the USDM can be leveraged for predicting anisotropic lattice stress [21]. Another
model, UDEDM is also employed to estimate the crystallite size together with the energy density per unit of volume inside of crystals as
a whole and is reliant on Hooke’s law [22]. Additionally, SSP interprets the size-widened component as the function of the Lorentzian
structure and the Gaussian function for the strain-broadened section of the XRD pattern. The benefit of the SSP approach is to pay extra
significance to the lower-angle XRD diffraction peak, whereby the reliability and accuracy of the XRD data are great. For this reason,
crystallite size computation from the SSP model is reported as more specific than the W–H technique [23–25]. Conversely, the H–W
Method views the peak widening as a voigt function and presumes that typical strain and size may be deduced from the XRD peak
widening [26]. In this work, we intend to find the crystal size of synthesized HAp (in water and oil media) using the aforementioned
XRD models and analyze their elastic behavior.

2. Materials and methods

2.1. Materials

Orthophosphoric acid (H3PO4), ammonium hydroxide (NH4OH), calcium hydroxide (Ca(OH)2), and nitric acid (HNO3) were
bought from E-Merck Germany. The substances employed in this research were analytical grade. The liquid organic media (Mustard
oil, Soybean oil, Coconut oil) are acquired from the local market. A double distillation method was used to prepare the deionized (DI)
water.

2.2. Synthesis method of nanocrystalline HAp

The study focuses on the use of mustard oil, soybean oil, and coconut oil as liquid media for synthesizing nano-crystalline hy­
droxyapatite (HAp) using a wet chemical precipitation method. These oils are natural, renewable, and biodegradable, making them
environmentally friendly [27–30]. They can also act as surfactants or structure-directing agents, influencing the growth and nucleation
of HAp nanoparticles [27,30]. Additionally, their chemical compositions and molecular structures can influence the nucleation and
growth of hydroxyapatite nanoparticles, resulting in variations in the crystalline structure and properties of the synthesized HAp.
Initially, an equal volume of 1.67 M Ca(OH)2 suspension and 1.0 M H3PO4 solution was prepared for the synthesis of HAp in water
solvent. The ratio of the organic medium (Mustard oil, Soybean oil, Coconut oil) and the water mixture was maintained at 50:50 (vol
%). H3PO4 was dropwise added to the calcium hydroxide suspension maintaining rate of 1 mLmin-1 fixing reaction parameters such as
(a) 10–11 of solution pH (maintained by adding dilute ammonia solution and/or nitric acid); and (b) reaction temperature: at room
temperature (25 ◦ C). A vigorous stirring (300 rpm) was applied to produce the reaction. Subsequently, a precipitate was produced that
was sorted out and then oven-dried maintaining 105 ◦ C for 6 h. The entire dry component was crushed to powder and exposed to
sintering at 900 ◦ C for 0.5 h (increment steps were 3.5 ◦ C min− 1).

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Md. Kawsar et al. Heliyon 10 (2024) e25347

2.3. X-ray crystallographic characterization

Phase analysis of the synthesized HAp was accomplished utilizing an X-ray diffractometer (Model: Rigaku SE). With the measuring
range at 2θ = 10◦ –60◦ , the findings were acquired in an ongoing scanning mode whilst the scanning steps were 0.01. The X-ray ra­
diation source CuKα (λ = 1.54060 Å) was operated under conditions of 40 mA current and 50 kV voltage, while the chiller temperature
was fixed at 22–23 ◦ C. A similar experiment was taken out employing all the HAp samples derived from mustard oil, soybean oil, and
coconut oil along with calcining at 900 ◦ C. All the discovered reflected peaks were recognized by comparing them with the standard
ICDD database files. Before evaluating the manufactured hydroxyapatite, the equipment was calibrated using a standard silicon
reference sample which was also used to anticipate the instrumental broadening.

3. Results and discussion

3.1. XRD data analysis

The XRD patterns of the synthesized HAp employing organic media such as mustard oil, soybean oil, as well as coconut oil are
exhibited in Fig. 1, and the crystallographic parameters were examined from the developed patterns. The 2θ (degree) diffracted po­
sitions of the HAp phases were visualized at 25.93 (002), 31.83 (211), 32.24 (112), 32.96 (300), 34.12 (202), 39.88 (130), 46.75 (222),
and 49.53 (213), which were matched with the standard ICDD database of the card no: #01-074-0565 for hydroxyapatite and a
hexagonal structure was predicted. A very similar form of data was observed for all the synthesized HAp.
The crystallographic study assesses crystalline features such as lattice parameters, cell volume, crystallinity index, dislocation
density, crystallite size, degree of crystallinity, and microstrain utilizing equations (1)–(7), as mentioned as well as explained else­
where [31–33].
( )2 ( )
1 4 h2 + hk + k2 l2
Lattice parameter equation, = + 2 (1)
dhkl 3 a2 c
√̅̅̅
3 2
Cell volume, V = ac (2)
2

Fig. 1. X-ray diffractogram of calcined hydroxyapatites using different organic media.

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Md. Kawsar et al. Heliyon 10 (2024) e25347

Kλ
Crystallite size, Dc = (3)
β cos θ

Microstrain, ε = β /4 tan θ (4)

( )3 ( )3
Ka 0.24
Degree of crystallinity, Xc = = (5)
β β

1
Dislocation density, δ = (6)
(Dc )2

∑ H(202) + H(300) + H(112)

Crystallinity index, CIXRD = (7)
H(211)

In the aforementioned equations, the unit cell is denoted by plane (h,k,l) and a,b,c represents lattice parameters, Dc = size of repeating
unit, β = FWHM (full width at half maximum) in radian; θ = angle of diffraction (in degree), Xc = crystallinity degree, K = shape factor
(arbitrary constant)/Scherrer’s constant = 0.94 [34], δ = dislocation density, H(hkl) = peak height of the respective plane, Ka = 0.24,
for HAp, and CIXRD = crystallinity index. By utilizing eqn (8), the specific surface area of the synthesized HAp was calculated, where
crystallite size and density of HAp were represented by Dc and ρ (3.16 g cm− 3) [35].
6
Specific surface area, s = g-1 m2 (8)
ρ × Dc
Crystallite sizes in ordered materials are significant in diverse applications, since small crystallites are characterized by large
surface areas, and vice versa [36].
The physical arrangement of atoms or molecules in any well-aligned material in a three-dimensional frame is revealed by the
degree of crystallinity. The crystallinity level considerably determines the features of materials, however properly managing crys­
tallinity is quite challenging. From the study, it’s evident that HAp demonstrates diverse magnitudes in the degree of crystallinity. The
data was calculated using equation (5).
In crystalline solids, microstrain corresponds to the intrinsic stress of crystal planes, which might emerge as either compressive or
tensile forces. As a consequence of microstrain, crystallite deformation occurs, giving conception to changes in the properties of
substances, notably in suitability. A constant variation in microstrain has emerged from the estimated data produced by applying
equation (4).
Imperfection in crystalline materials is displayed because of several flaws such as point dislocation, line dislocation, and area
dislocation, frequently known as dislocation. Dislocation density analyzes the number of dislocation lines per given surface area and is
directly linked to the crystal size [37]. However, the amount of line dislocation is computed using equation (6). The data is registered
in Table 1. Crystallite sizes in ordered materials are significant in diverse applications, since small crystallites are characterized by
large surface areas, and vice versa. The crystallinity index (CIXRD) is discussed for assessing the numerical quantification of crystal
structure. In this specific section, the X-ray diffraction (XRD) data were evaluated to compute the crystallinity using equation (7), as
well as the results obtained are displayed in Table 1.

3.2. Crystallite size calculation using various models

Precise crystallite size estimation for each application is a crucial criterion. Yet, for determining the dimension of the crystallite of
HAp samples, multiple approaches and mathematical equations have been used including Scherrer’s Method (SM), linear straight-line
method (LSLM), Size-Strain Plot Method (SSPM), Williamson-Hall Method (WHM), Monshi Scherrer Method (MSM), Sahadat-Scherrer
Method (S–S), Halder-Wagner Method (HWM). The Williamson-Hall Method was further diversified emphasizing the UDSM, UDM,
and UDEDM models.

Table 1
Crystallographic parameters of the prepared HAp samples by using different organic media.
Parameter Pure HAp Mustard oil Soybean oil Coconut oil

Lattice parameter a = b = 9.39 a = b = 9.40 a = b = 9.40 a = b = 9.41

c = 6.82 c = 6.87 c = 6.872 c = 6.87
Crystallite size, nm 81.97 65.75 51.88 54.93
Degree of crystallinity 13.41 6.96 3.40 4.063
Microstrain, ε 1.54 × 10− 3 1.923 × 10− 3 2.4407 × 10− 3
2.30 × 10− 3
Dislocation density, (1015 lines per m2) 0.1488 0.2313 0.3715 0.3313
Crystallinity index, CIXRD 1.46 1.31 1.52 1.48
Specific surface area, S (g− 1m2) 0.02316 0.02887 0.03659 0.03456
Volume of unit cell(Å3) 521.416 526.188 525.859 527.598

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Md. Kawsar et al. Heliyon 10 (2024) e25347

3.2.1. Scherrer’s method (SM)

Scherer’s formula for ideal conditions for diffraction on monodisperse particles of crystalline materials with homogeneous orga­
nized domains was developed for parallel, monochromatic, and infinitely thin X-ray beams [38]. The broadening of XRD spectra in
nanocrystals is linked to non-intrinsic strain effects and crystallite size, involving instrumental and physical expanding components
[15]. To minimize this inaccuracy of the instrument, equation (9) may be deployed:
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
βd = β2m − β2i (9)

In equation (9), βm is the determined broadening, βi is the broadening by the instrumental, and βd is established as the modified
broadened accountable for crystal dimension. However, the physical and instrumental broadening of the sample was measured
through the full-width at half maxima (FWHM). So, we could calculate the average crystal size and overlook the influence of the strain
by using the Scherrer method with the following equation (10) [39].
Kλ
Crystallite size, Dc = (10)
βCos θ
Here, Dc indicates the crystallite size, K is the shape constant (K is equal to 0.9, for cubic crystal), wavelength (λ) of the Cu-radiation
was 1.54056 Å for CuKα1 radiation, β is the full width of the reflection at half of the maximum intensity, and the diffraction angle is θ.
The crystal sizes obtained from this model were 83.23 nm for Pure HAp, 119.33 nm for Mustard oil, 105.68 nm for Soybean oil, and
143.41 nm for Coconut oil.

3.2.2. Liner straight-line method of Scherrer’s equation (LSLMSE)

The fundamental requirement of the LSLMSE method was to evaluate all peaks in synthesized samples (shown in Fig. 2(a–d)),
rather than focusing on a specific scattering peak [8]. The Scherrer equation can be reformed in the new equation (11). This resultant
equation can be used to estimate crystallite size (DL), widely articulated in various studies [39,40]. The mathematical Equation for this
model can be written as follows:
Kλ 1 Kλ 1
Cos θ = × = × (11)
Dc β DL β
The crystallite size, denoted by DL, was calculated using the LSLMSE method. A graph was constructed by plotting Cosθ as well as
1/β on the Y-axis and X-axis respectively. The crystal size was measured using the gradient m = Kλ/(DL), were 2 × 10− 4, 1 × 10− 5, 2 ×

Fig. 2. Determination of crystallite size using liner straight line model of Scherrer equation for (a) Pure HAp (b) Mustard oil (c) Soybean oil and (d)
Coconut oil.

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10− 5, and 2 × 10− 5 were obtained for pure HAp, mustard oil, soybean oil, and coconut oil correspondingly. The estimated crystal size
was found to be 693.27 nm for pure HAp, 13865.4 nm for mustard oil, 6932.7 nm for soybean oil, and 6932.7 nm for coconut oil.

3.2.3. Monshi–Scherrer method (MSM)

The Scherrer equation reveals higher nanocrystalline size when d-spacing as well as 2θ values drop. Modifying the equation may
eliminate shortcomings or Σ(±Δln β)2 to offer a more accurate evaluation of crystal size from all or a part of unique peaks, enhancing
the stability of β. Cosθ [13]. The Monshi–Scherrer model is shown in equation (12), with the dimension of the crystallite being
indicated by DM–S [41].
1 Kλ
ln β = ln + ln (12)
Cosθ DM− s
This model was examined by plotting lnβ and ln 1/cosθ on the Y-axis, and X-axis (visualized in Fig. 3(a–d)). The straight-line
equation (y = mx + c) and equation (12) were compared with each other to find out the slope, which gives a checkpoint for veri­
fying the correctness of the findings.
The plot showed that the values of slope as 2.105 for pure HAp, 14.889 for mustard oil, 5.8653 for soybean oil, and 7.4216 for
coconut oil. The crystal size values for pure HAp, mustard oil, soybean oil, and coconut oil were computed, with a total of 89.67,
103.57, 111.19, and 123.91 nm, correspondingly.

3.2.4. Williamson–Hall method (WHM)

The modified Scherrer’s Equation addresses crystallite size impact through XRD reflection, neglecting inherent strain in nano­
crystals. However, intrinsic strain is crucial due to grain boundaries, point defects, dislocations, and stacking in nanocrystals [42].
Consequently, the Williamson-Hall approach analyzes strain from XRD peak widening, following estimating intrinsic strain from
crystallite size values. This approach is more accurate for calculating crystallite size with several reflected peaks [43]. Eventually, the
overall broadening may be described as equation (13) [15].
βTotal = βsize + βstrain (13)

where βsize is the broadening due to its size and βstrain is connected to the strain broadening effect. The modified form of the Williamson-
Hall considers a UDM, USDM, UDEDM, and the sizes-strain plot method (SSP) will be discussed in this context [44].

3.2.4.1. Uniform deformation model (UDM). The estimation of strain obtained by crystalline defects and distortion in the synthesized
powder can be mathematically expressed as equation (14) [45]:
βhkl
ϵ= (14)
4 tan θ

Fig. 3. Determination of crystallite size using Monshi–Scherrer method equation for (a) pure HAp (b) mustard oil (c) soybean oil (d) coconut oil.

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Md. Kawsar et al. Heliyon 10 (2024) e25347

The UDM is founded on the concept that the strain is considered uniform in all orientations. The lattice strain is consequently
perceived as an isotropic property that is independent of the extent of direction [46]. The peak broadening occurred by lattice strain
can be presented as equation (15):
βstrain = 4ϵ tan θ (15)
The overall broadening, βhkl reflecting the FWHM of a reflected peak which is related to the influence of the strain of crystal lattice
(βstrain) and the value of the size of the crystals (βsize) in a specific peak that may be stated as equations (16)–(18).
β hkl = βsize + βstrain (16)

Kβ λ
βhkl= + 4ϵ tan θ (17)
DW .cosθ
Equation (9) can be written as:
Kβ λ
βhkl cosθ = + 4ϵsinθ (18)
DW
The linear equation, incorporating 4sinθ (X-axis) and βhkl*cosθ (Y-axis), enables for the estimation of slope (ε) and crystallite size
(Dw) in a graph, as seen in Fig. 4(a–d), and given in Table 2.
The slope of the UDM curve reveals the existence of intrinsic strain, a phenomenon involving the lattice expansion of nanocrystals
[47]. Pure HAp as well as coconut oil had a slope of 4 × 10− 4, whereas mustard oil and soybean oil had slopes of 2 × 10− 4 and 3 × 10− 4
respectively. The estimated crystal size of pure HAp was 106.66 nm, whereas mustard oil had 63.02 nm, soybean oil had 92.44 nm, and
coconut oil had 138.65 nm.

3.2.4.2. Uniform stress deformation model (USDM). The Uniform Stress Deformation Model (USDM) was validated by incorporating
the anisotropic character into lattice strain analysis. This modified model focuses on the lattice deformation stress for crystal plane
directions with low microstrains uniformly, addressing the issue of sample uniformity and the potential anisotropic nature of actual
crystals [42]. Hooke’s law, expressed as equation (19), relates stress (σ) and strain (ε), with higher accuracy for low-stress values [48].
σ = Yhkl ε (19)

where Yhkl is Young’s modulus or modulus of elasticity, the mathematical expression is just an approximation that is reliable for a

Fig. 4. Determination of crystallite size using Uniform deformation model for (a) Pure HAp (b) Mustard oil (c) Soybean oil (d) Coconut oil.

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Md. Kawsar et al. Heliyon 10 (2024) e25347

Table 2
Microstructural characteristics of hydroxyapatite utilizing various models in this study.
Model Name Crystal size (in nm), strain (in N/m2), energy density (in J/m3)

Pure HAp Mustard Oil Soybean Oil Coconut oil

Scherrer’s Model 83.23 119.33 105.68 143.41

The linear straight-line method of Scherrer’s equation 693.27 13865.4 6932.7 6932.7
Monshi- Scherrer’s 89.68 103.57 111.19 123.91
Method
Williamson-Hall Method UDM ε = 0.0004 ε =0.0002 ε = 0.0003 ε =0.0004
Dw = 106.66 Dw = 63.02 Dw = 92.44 Dw = 138.65
USDM σ = 1015.2 σ = 1877.9 σ = 1497.4 σ = 414.66
D(hkl)= 99.04 D(hkl)= 99.04 D(hkl)= 99.04 D(hkl)= 99.04
3
UDEDM u = 4.52 × 10− u = 1 × 10− 14 u = 2.116 × 10− 3
u = 70.11 × 10− 3

Dw = 98.32 Dw = 77.03 Dw = 77.0 Dw = 99.04

Size-strain plot 53.33 38.52 37.45 26.16
Halder-Wagner Method 47.61 6.5 43.48 34.48
Sahadat-Scherrer’s Model 86.66 106.66 126.05 138.65

minimal strain. Furthermore, raising the strain produces a variation of particles from being linear in nature, respectively [49]. In this
context, 6 × 109 N/m2 is considered the value for Young’s modulus [50]. By rearranging and replacing equation (19) with equation
(18), we have the following relation (equation (20)):
KB λ sinθ
βhkl cosθ = +4σ (20)
Dhkl Yhkl
A linear graph was generated by plotting βtotal*cosθ and 4*Sinθ/Y(hkl) along the Y-axis as well as X-axis. The gradient of this straight-
line measured stress (σ) and crystallite size D(hkl) of HAp nanocrystals. The crystal size was estimated from the uniform stress defor­
mation model, with 99.04 nm for pure HAp, 99.04 nm for mustard oil, 99.04 nm for soybean oil, and 99.04 nm for coconut oil. The
stress calculated was 1015.2 (N/m2) for pure HAp, 1877.9 (N/m2) for mustard oil, 1497.4 (N/m2) for soybean oil, and 414.66 (N/m2)
for coconut oil. The plots are illustrated in Fig. 5(a–d), and the computed σ and D(hkl) values are registered in Table 2.

3.2.4.3. Uniform deformation energy density model (UDEDM). The orientational arrangement in UDM necessitates adjustment of the
W–H relationship from anisotropic crystal, since isotropy and homogeneity are not essential for efficient arrangement [51]. The

Fig. 5. Determination of crystallite size using Uniform stress deformation model for (a) pure HAp (b) mustard oil (c) soybean oil (d) coconut oil.

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Md. Kawsar et al. Heliyon 10 (2024) e25347

relationship between stress (σ) and strain (ε) in USDM is linear, based on Hooke’s law. However, in actual crystalline materials, defects
like agglomerations and dislocations cause imperfections. UDEDM examines crystal imperfection, anisotropic deformation, and
distortion as expressions of energy density (u), ensuring that constants related to stress and strain remain independent [52]. The u
(energy per unit volume) as an expression of ε is determined by Hooke’s expression as equation (21).
Yhkl
u = ε2 (21)
2
The UDEDM equation can be obtained by reorganizing equation (13) to ε and replacing it with equation (9), resulting in equation
(22).

KB λ √2
βhkl Cosθ = + 4Sinθ√u (22)
Dw √Yhkl

The graph plotted (shown in Fig. 6(a–d)), between β(hkl) cos θ (Y-axis) as well as 4Sinθ √2
√Yhkl
(X-axis) to estimate anisotropic energy
density (u) and crystallite size (Dw). The energy density was reported as 4.52929 × 10 (J/m3) for pure HAp, 1 × 10− 14 (J/m3) for
− 3

mustard oil, 2.116 × 10− 3 (J/m3) for soybean oil, and 70.11 × 10− 3 (J/m3) for coconut oil, whereas Crystal size was measured 98.32
nm for pure HAp, 77.03 nm for mustard oil, 77.03 nm for soybean oil, and 99.04 nm for coconut oil, correspondingly.

3.2.5. Size-strain plot method (SSP)

The Size-strain plot is a method used to determine XRD peak broadening, incorporating both Gaussian as well as Lorentz functions.
The widening is regulated by strain, with the crystallite size of synthesized materials contributing to the broadening, indicated as a
Lorentz function [42]. This model employs data at decreased diffraction angles to compensate for the lack of accuracy and significant
peak overlaps caused to higher-degree influence planes [53]. So, the total SSP peak widening may thus be represented as the sum­
mation of broadening generated by Lorentzian (βL) and Gaussian (βG) functions (equation (23)).
βtotal = βL + βG (23)
The SSP is mathematically expressed using equation (24) [54,55].

Fig. 6. Determination of crystallite size using Uniform deformation energy density model for (a) pure HAp (b) mustard oil (c) soybean oil (d)
coconut oil.

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KB λ ( ) ε2
(dhkl βhkl cosθ)2 = dhkl 2 βhkl cosθ + (24)
Dw 4

where dhkl is the lattice distance between the (hkl) planes or d-spacing. Which was measured by the following equation (25).
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
a2
dhkl = (25)
h2 + k2 + l2

A linear fit is produced by plotting the term (dhkl 2 βhkl cos θ) on the X-axis and (dhkl βhkl cos θ)2 on the Y-axis as a function of all XRD
reflections (Fig. 7(a–d)). The linear fit slope and Y-intercept determine the average size of crystallite (Dw) and intrinsic strain (ϵ). The
slop values equate to KDBwλ, with values given as 0.0026 for pure HAp, 0.0036 for mustard oil, 0.0037 for soybean oil, and 0.0053 for
coconut oil. The intercept causes the intrinsic strain, with a value equal to ε4 , making the computation of intrinsic strain impossible for
2

the synthesized samples. The crystal sizes found from this model are 53.33 nm for pure HAp, 38.515 nm for mustard oil, 37.47 nm for
soybean oil, and 26.16 nm for coconut oil.

3.2.6. Halder-Wagner method (HWM)

The SSP approach employs the tension expansion as the Gaussian function and size widening as the Lorentzian function in the XRD
pattern. However, the XRD peak area matches the Gaussian function, with significant tail collapse. The bottoms of the profile fit the
Lorentz function, but not the XRD peak [56,57]. That is why the Halder-Wagner technique approach demands the symmetrical Voigt
function, a convolution that integrates Lorentzian and Gaussian functions [57,58]. Thus, the Voigt function may be expressed as
equation (26), which represents the FWHM of the physical profile.

βhkl 2 = βL βhkl + βG 2 (26)

Where, βL = FWHM for Lorentzian function. βG = FWHM for Gaussian function. This technique offers a higher weight to the Bragg
peaks in the small and intermediate angle, and the overlapping of the reflection peaks was minimal, and the correlation between the
crystallite size and the lattice ε relates to the H–W technique expressed by equations (27)–(29) [15].
( )2 ( ). ( )
∗βhkl 1 ∗βhkl ε 2
= 2
+ (27)
∗dhkl Dw ∗dhkl 2

Fig. 7. Determination of crystallite size using Stress-Strain plot for (a) Pure HAp (b) Mustard oil (c) Soybean oil (d) Coconut oil.

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Md. Kawsar et al. Heliyon 10 (2024) e25347

βhkl cos(θ)
Here, ∗βhkl . = (28)
λ

2dhkl sin(θ)
∗dhkl . = (29)
λ
A plot of (* β hkl ∕ *dhkl)2 on the Y-axis and* β hkl ∕(*dhkl)2 on X-axis yields a straight line (presented in Fig. 8(a–d)), with a slope
equal to 1/Dw, calculating microstrain. The values of the slope were reported as 0.0021 for pure HAp, 0.0152 for mustard oil, 0.0023
for soybean oil, and 0.0029 for coconut oil, corresponding crystallite diameters of 47.61, 6.57, 43.48, and 34.48 nm, respectively.

3.2.7. Sahadat–Scherrer model

The Sahadat-Scherrer model offers an accurate approximation of the crystallite size, despite various limitations connected to
previously stated models possibly leading to enormous crystallite sizes [1]. The technique involved studying each peak like previous
models, but with a distinct linear route passing through the origin, resulting in a more precise model due to the straight line crossing
through the origin [59]. The mathematical expression of this model can be written as equation (30):
Kλ 1
Cosθ = × (30)
DS− S FWHM

In this model 1/ β and cosθ are applied in horizontal as well as vertical axis to build a straight line across the origin (shown in Fig. 9
(a–d)). An intercept model was developed using excel, matching the y = mx equation. The crystallite size was determined by
comparing the slope with Kλ/D(S–S). The model reported the magnitude of slopes as 0.0016 for pure HAp, 0.0013 for mustard oil,
0.0011 for soybean oil, and 0.0010 for coconut oil, additionally the crystal size as 86.66 nm for pure HAp, 106.66 nm for mustard oil,
126.05 nm for soybean oil, and 138.65 nm for coconut oil, correspondingly.

4. Conclusion

The synthesis of HAp by the wet chemical precipitation technique is further explored with the use of X-ray diffraction (XRD)
analysis for their structural and crystalline characteristics. To estimate various elastic parameters for example energy density, stress,
and inherent strain, several techniques e.g. Scherer’s method, Stress-Strain plot method, Halder-Wagner method, and different models
of Williamson-Hall such as UDM, UDEDM, and USDM commonly used. Although the value of crystallite dimension for each of the
samples are almost similar except for mustard oil (6.57 nm) in the Halder- Wagner model, after comparing all these models it is
unequivocally, stated that the Williamson-hall model provides the more precise crystallite size along with accurate stress, strain, as

Fig. 8. Determination of crystallite size using Halder- Wagner model for (a) Pure HAp (b) Mustard oil (c) Soybean oil (d) Coconut oil.

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Md. Kawsar et al. Heliyon 10 (2024) e25347

Fig. 9. Determination of crystallite size using Sahadat-Scherre’s plot for (a) Pure HAp (b) Mustard oil (c) Soybean oil, and (d) Coconut oil.

well as energy density value. Finally, the models evaluated crystallite sizes below 150 nm for the synthesized HAp nanocrystal utilizing
different oil mediums, except the linear straight-line method (LSLM) that ended up resulting in a crystallite size of 693.27 nm for Pure
HAp along with Soybean and Coconut oil, and 13865.4 nm for Mustard oil.

Data availability statement

Data will be made available on request and no data was stored in any publicly available repository.

CRediT authorship contribution statement

Md. Kawsar: Writing – original draft, Formal analysis, Data curation. Md. Sahadat Hossain: Writing – review & editing, Su­
pervision, Methodology, Formal analysis, Conceptualization. Newaz Mohammed Bahadur: Supervision. Samina Ahmed: Supervi­
sion, Resources, Project administration, Funding acquisition.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing in­
terests:There is nothing to declare If there are other authors, they declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors are grateful to the Bangladesh Council of Scientific and Industrial Research (BCSIR) authority for financial support
through the R&D project (ref. no. February 39, 0000.011.14.134.2021/900; Date: December 30, 2021) and Ministry of Science and
Technology for approving special allocation project (ref. no. 39.00.0000.009.99.023.23–363; Date: December 18, 2023). Md. Kawsar
wishes to thank the Department of Applied Chemistry and Chemical Engineering, Noakhali Science and Technology University,
Noakhali, Bangladesh for approving the M.S. Thesis program.

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Md. Kawsar et al. Heliyon 10 (2024) e25347

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