*3.1. Positive and Negative Slope Values of Unit Cell and Super Cells (2* × *2* × *2)*

The difference of Young's modulus values in the unit cell and super cells (2 × 2 × 2) is attributed to the locations of atoms. The calculated slope is a negative value in super cells (2 × 2 × 2). In the first aspect, it is clear that the slope is dependent on the planar density of diffracted planes, so the fitting based on the planar density of super cells with a matrix of eight unit cells could submit a better result of intercept [35]. It is because eight cells besides each other are completed and have more symmetry than two cells (Figure S8) [35]. In the second aspect, the reason for the positive slope in the unit cell and the negative slope in the super cells (2 × 2 × 2) is related to the defects (imperfections) including point defects (vacancies, substitutional and interstitials), line defects (screw and edge dislocation), surface defects (grain boundaries) and volume defects (lack of order of atoms due to amorphous region in a very tiny area). The effect of these imperfections is more effective in super cells (2 × 2 × 2), while the unit cell is more ideal and less affected by these imperfections. This means that when the density of atoms in planes is increased, lower forces for dislocation motion are required and the strength will be decreased and, consequently, in super cells (2 × 2 × 2) the slope is negative. In the case of super cells, when the number of atoms is increased, by the increase of planar density, the effect of dislocation motion is increased; the strength and Young's modulus will be decreased so the slope is negative. The intercept of the fitting line is a value of Young's modulus, which can show the Young's modulus of the plane with zero planar density as a plane without any specific atom. Therefore, a discussion of defects and imperfections cannot be considered for such a plane without an atom at the origin and so, in this case, the Young's modulus value of the unit cell is less than that of the super cells (2 × 2 × 2) and consequently the plane without an atom is more realistic in a smaller area of a unit cell than in a wider area of the super cells. This means that the intercept in a unit cell is closer to the real values of the Young's modulus. As mentioned above, the planar density depends on the array and position of atoms into the plane and the situation of the planes. For illustration, according to Figure 4b, the planar density of (210) > (510) > (102) > (023) > (043), because sequencing of the planes is not related to the diffraction (XRD), but it depends on the planar density values. With this method, it can be possible to consider two or more planes with similar planar density values; for example, in super cells (2 × 2 × 2), planar density values of (040) and (331) are equal to 0.250. The control of the displacement and deformation process of

the atoms in the planes are associated with the dislocation networks. The work or energy (W) for the movement of the atoms in each plane corresponds to Equation (20) [44,45].

$$\mathbf{W} = \mathbf{G} \mathbf{b}^2 \mathbf{l} \tag{20}$$

In this equation, G, b and l are shear modulus, Burgers vector and dislocation length, respectively. In addition, by merging Equations (20) and (21), Equation (22) can be registered. In Equations (21) and (22), E and ν are Young's modulus and Poisson's ratio tandemly.

$$\mathbf{G} = \frac{\mathbf{E}}{2(1+\mathbf{v})} \tag{21}$$

$$\mathbf{W} = \frac{\mathbf{E}}{2(1+\mathbf{v})} \mathbf{b}^2 \mathbf{l} \tag{22}$$

There are several studies on the properties of hydroxyapatite, especially with regard to biocompatibility and bioactivity, and these properties are dependent on the identification and recognition of hydroxyapatite structures [46,47] For example, when atoms are widely spaced, such as corner atoms (Ca) in (111) super cells (2 × 2 × 2), atoms require higher values of applied force for approaching, so knowledge on the planes of hydroxyapatite structure can be helpful for doping metals, polymers and other ceramics, for aims such as the controlled release of protein and also the fabrication of bioactive monolithic fragments in the biomaterials industry [48,49]. In addition, the hydroxyl ion is in the center of each unit cell. The hydroxyl group in the center of the hydroxyapatite lattice is surrounded by three calcium ions per hexagon, forming a ring (six calcium ions). A chord is formed by the structure, as these rings are responsible for many properties of hydroxyapatite, especially the biocompatibility and the position of the hydroxyl group in the planes have considerable importance for doping mechanisms [15,50].

### *3.2. Williamson-Hall Method in USDM Model*

The W-H method has been used for determining different elastic properties. The best procedure is to mathematically reduce the errors and obtain the values of the Young's modulus by all the diffracted peaks, using the least squares method. The W-H method is a simplified integral expanse and, taking into account the peak width, strain-induced broadening is specified [51]. Taking into account the W-H method in the USDM model, Young's modulus values are examined. It is clear that in Equation (23) and Figure 5, the term of 4sin<sup>θ</sup> Ehkl is along the X-axis and the term of <sup>β</sup>hkl.cos <sup>θ</sup> is along the Y-axis individually.

$$
\beta\_{\rm hkl.} \cos \theta = (\frac{\rm K\lambda}{\rm L}) + 4\sigma \frac{\sin \theta}{\rm E\_{\rm hkl}} \tag{2.3}
$$

**Figure 5.** W-H in USDM model and plot of unit cell and super cells (2 × 2 × 2) of hydroxyapatite.

Here, βhkl is the broadening peak from (hkl) plane and, in this study, the instrumental broadening is taken as 0.02 degree for each diffracted peak and is subtracted from βhkl values before multiplying to <sup>π</sup> <sup>180</sup> to convert the degree to radian. In this model, the condition is performed to calculate the strain and the average Young's modulus. The average E value has been calculated in the research and studies on using the W-H method, but it is subject to errors, because if the average values of the Young's modulus are considered (through Equation (19)), the final value would be far from the standard value of Young's modulus in each peak extracted by X-ray diffraction. In addition, in some studies the Young's modulus value is considered a value that is listed in the literature but is not associated with the prepared materials. As an illustration, based on the study in Reference [10], which refers to the use of the W-H method to calculate the crystal size and Scherrer analysis of hydroxyapatite, the average value of Young's modulus has a larger deviation than the actual value of the Young's modulus of hydroxyapatite. The line broadening of the diffracted peak is extracted with Crystal Diffract 6.7.2.300 software. According to Figure 5, the slope values are associated with the stress (σ). The values obtained were positive, and the positive values of intrinsic strain and stress can prove the tensile stress and strain, and if the values were negative, they are associated with compressive stress and strain. Additionally, the resulting values of stress (σ) and strain (ε) by utilizing the W-H method in the USDM model are shown in Table 6. The value of σ is in good agreement with the values obtained from Reference [52].

**Table 6.** Mechanical properties values of unit cell and super cells (2 × 2 × 2) of hydroxyapatite.


a: Experiments of Young's modulus values (This Study).

### **4. Conclusions**

In this study, crystal of hydroxyapatite was successfully synthesized with the thermal treatment process. Moreover, the position of atoms and extracted planar density values of each diffracted plane (32 planes) of hydroxyapatite were determined and calculated for the first time. In addition, a new method based on the relationship between the measurement of elastic modulus and atomic planar density of crystalline hydroxyapatite, consisting of a unit cell and super cells (2 × 2 × 2), was fully presented. According to the method of this study, the slope of modules of elasticity against planar density in super cells (2 × 2 × 2) was negative; the reason was related to imperfections, and this means that when the density of atoms in the planes is increased, lower forces are required for the dislocation motion. As a result, a plane without an atom is more realistic in a smaller area of the unit cell than a larger area of the super cells (2 × 2 × 2), and this means that the intercept of the unit cell is closer to the real values of the Young's modulus in the hydroxyapatite lattice. Moreover, a comparison of theoretical and experimental data of the Young's modulus of hydroxyapatite showed that there is a small difference between the values for both unit cell and super cells (2 × 2 × 2), namely 10.47 GPa and 10.57 GPa, respectively; this means that the theoretical calculation is valid and, by decreasing by about 10 GPa, the experimental value can be obtained. Furthermore, the Young's modulus values of hydroxyapatite in the unit cell and super cells (2 × 2 × 2) were achieved at 108.15 and 121.17 GPa tandemly. Finally, one of the applications of the presented method was carried out in this study and the Williamson-Hall method in the USDM model can be used to minimize the errors in the least squares method and to obtain the correct elastic modulus of hydroxyapatite, which is much more accurate than the average value.

**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/10 .3390/ma14112949/s1, Figure S1: Images of production route of hydroxyapatite obtained from cow bones, Table S1: Crystallographic parameters of XRD pattern related to the natural hydroxyapatite obtained from cow bones, Figure S2: EDX spectrum and stoichiometric composition of hydroxyapatite obtained from cow bones, Figure S3: (a) Raw hexagonal structure, (b) fundamental axis, (c) principal crystallographic directions and (d) between the orthogonal axes and crystallographic directions, Figure S4: Transfer of hydroxyapatite unite cell in P63/m to P63 space group, Figure S5: Schematic of a design of cutted samples, Table S2: The values of Longitudinal and Transvers velocity of samples, Figure S6: Geometry and the situation of involved atoms in diffracted planes related to the unit-cell of hydroxyapatite (a) 210, (b) 211, (c) 112, (d) 300, (e) 202, (f) (301), (g) (130), (h) (310), (i) (032), (j) (040), (k) (023), (l) (222), (m) (320), (n) (230), (o) (213), (p) (321), (q) (042), (r) (033), (s) (004), (t) (050), (u) (501), (v) (331), (w) (043), (x) (124), (y) (510), (z) (511), (a1) (332) and (b1) (143), Figure S7: Geometry and the situation of involved atoms in diffracted planes related to the super-cell of hydroxyapatite (a) 210, (b) 211, (c) 112, (d) 300, (e) 202, (f) (301), (g) (130), (h) (310), (i) (032), (j) (040), (k) (023), (l) (222), (m) (320), (n) (230), (o) (213), (p) (321), (q) (042), (r) (033), (s) (004), (t) (050), (u) (501), (v) (331), (w) (043), (x) (124), (y) (510), (z) (511), (a1) (332) and (b1) (143), Table S3: Young's modulus values of hydroxyapatite related to the literatures and present study, Figure S8: (a) The un-symmetry of two unit cells and (b) symmetry of eight unit cells of hydroxyapatite.

**Author Contributions:** M.R., A.P., G.J., A.V.: Investigation, Writing—original draft and Validation; S.N.: Writing—review and editing; A.D. (Amir Dashti), A.D. (Akram Doustmohammadi): Formal analysis and Software; A.M.: Methodology. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by a grant No. S-MIP-19-43 from the Research Council of Lithuania.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing is not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

