## 3 Results and Discussion
### Characterization of Morphology and Chemical States of Polycrystalline Diamond Films
#### 3.1.2 X-ray Diffraction (XRD) Analysis  
XRD analysis was used to determine the presence of the diamond crystals in the PCD films as well as to determine the grain size (less than 20 nm) for samples A and B. Figure 2 shows results from X-ray diffraction (XRD) analysis of PCD films A to D (a) and E to G (b).  
The XRD results from Figure 2 show a typical diffraction pattern [19] for PCD films with polycrystalline diamond growth. The broad peak around 2\(\theta\)-55\({}^{\circ}\) is related to the substrate and growing conditions.  
Table 2 shows the measured 2\(\theta\) XRD diffraction angle for each of the allowed reflections for a faced centered cubic structure. The values for an unbounded single crystal diamond (SCD) are also shown in the table. The interplanar \(d\)-spacing for each Miller index was calculated using Braggs law (Equation (1)).  
\[d~{}=~{}\frac{n~{}\lambda}{2\sin~{}\theta} \tag{1}\]  
where \(\lambda\) = 1.5406 A (Cu K\(\alpha\) radiation) is the X-ray wavelength, \(n\) = 1, and \(\theta\) is the scattering angle.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Miller Indices (hkl)** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **SCD** \\ \hline \multirow{3}{*}{**(111)**} & 2 \(\theta\) & 43.75 & 43.9 & 43.94 & 43.95 & 43.94 & 43.95 & 43.95 & 43.93 \\ \cline{2-10}  & d (Å) & 2.07 & 2.06 & 2.06 & 2.06 & 2.06 & 2.06 & 2.06 & 2.06 \\ \hline \multirow{3}{*}{**(220)**} & 2 \(\theta\) & 75.5 & 75.33 & 75.37 & 75.32 & 75.33 & 75.31 & 75.34 & 75.3 \\ \cline{2-10}  & d (Å) & 1.26 & 1.26 & 1.26 & 1.26 & 1.26 & 1.26 & 1.26 & 1.26 \\ \hline \multirow{3}{*}{**(311)**} & 2 \(\theta\) & 90.75 & 91.41 & 91.71 & 91.51 & 91.48 & 91.5 & 91.49 & 91.5 \\ \cline{2-10}  & d (Å) & 1.08 & 1.08 & 1.08 & 1.08 & 1.08 & 1.08 & 1.08 & 1.08 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calculation of d-spacing for (hkl) planes for samples A–G compared with a single crystal diamond (SCD).  
Figure 2: X-ray diffraction (XRD) spectra from PCD films grown with different grain sizes. (**a**) Samples A–D grown for 2 h with different Ar:H gas mixtures. (**b**) Samples E–G grown with H (no Ar) gas flow for different deposition time (Table 1).  
It can be inferred from the table that the diamond crystal in the PCD is not strained due to the packing of the grains in the PCD films or due to the presence of defects. The difference in unit cell sizes does not exceed 0.3% of the expected value from an SCD.  
The grain size was obtained for samples A and B from the peak (111), which is the only peak with sufficient resolution to calculate the FWHM. The FWHM was calculated using OriginPro 9.0 software and by applying the Scherrer equation (Equation (2)):  
\[T=\frac{K\,\lambda}{\beta\,\text{Cos}\,\theta} \tag{2}\]  
where \(K\) is a shape factor that varies from 0.89 to 0.94. Usually, the shape factor is set to 0.9 for particles of unknown size, \(\lambda\) is the X-ray wavelength, and \(\beta\) is the line broadening at half the maximum intensity (FWHM). The Scherrer equation is used to determine crystallite size (\(T\)) from XRD spectra for small particles where the FWHM is greater than the instrument limitations. After subtracting the instrumental line broadening in radians (also sometimes denoted as \(\Delta(2\theta)\)), we obtained a grain size of 5 nm for sample A and 7 nm for sample B. These values are within the range of previously measured grain sizes using high resolution transmission electron microscopy (HRTEM) for similar growth conditions. The variation in size for a particular sample is about 30%. The combined measurements for grain size are shown in Figure 3.