## 2 Protocols for Synchrotron X-Ray Experiments
### Pdf
#### 2.5.2 PDF Analysis Procedure  
The scattering data are corrected for background contributions using custom software developed and maintained by the BL04B2 staff specifically designed for PDF data processing. The corrected scattering intensity \(I_{\text{corrected}}(\theta)\) is calculated using the equation:  
\[I_{\text{corrected}}(\theta)=\frac{I_{\text{obs}}(\theta)-\epsilon I_{\text{BG} }-\delta}{P(\theta)A(\theta)} \tag{8}\]  
Here, \(\epsilon\) represents a scaling factor applied to account for differences between the scattering intensities of the sample and the background, while \(\delta\) is an offset term used to correct systematic baseline variations. These corrections ensure that the scattering signal accurately reflects the structural features of the sample. Additional corrections for polarization \(P(\theta)\) and absorption \(A(\theta)\) are applied to further enhance data quality. In this experiment, the polarization correction \(P(\theta)\) is expressed by the following formula:  
\[P(\theta)=\cos^{2}(2\theta)+p\sin^{2}(2\theta) \tag{9}\]  
where \(p\) is a fitting parameter, typically ranging from 0.05 to 0.1. The absorption correction \(A(\theta)\) was also applied to account for the attenuation of the X-ray beam in the sample and capillary. Together, these corrections enhance the accuracy of the scattering data and help ensure that the results are representative of the sample's intrinsic properties. The next step involves the calculation of the structure factor \(S(Q)\), which represents the normalized scattering intensity as a function of the scattering vector \(Q\). The structure factor is computed using the equation (Faber-Ziman structure factor [30]):  
\[S(Q)=\frac{I_{\text{norm}}(Q)-I_{\text{inc}}^{*}(Q)-\left(\left\langle f(Q)^{2} \right\rangle-\left\langle f(Q)\right\rangle^{2}\right)}{\left\langle f(Q) \right\rangle^{2}} \tag{10}\]  
where \(I_{\text{inc}}^{*}(Q)\) denotes the incoherent scattering intensity, and \(f(Q)\) is the atomic scattering factor, describing the scattering amplitude of individual atoms as a function of \(Q\). This structure factor forms the foundation for deriving real-space structural information. Figure 11 shows representative plots of \(I_{\text{corrected}}(\theta)\) and \(S(Q)\). The reduced PDF \(G(r)\), which captures local atomic correlations in real space, is obtained by applying Fourier transformation to \(S(Q)\):  
\[G(r)=\frac{2}{\pi}\int_{Q_{\text{min}}}^{Q_{\text{max}}}Q[S(Q)-1]\text{sin}(Qr )\text{d}Q \tag{11}\]  
The total radial distribution function \(T(r)\), representing the cumulative atomic density at a given distance \(r\), is calculated from \(G(r)\) using the equation:  
\[T(r)=G(r)+4\pi r\rho_{0} \tag{12}\]  
Here, \(\rho_{0}\) denotes the average atomic number density of the sample, a parameter derived from the sample's composition and bulk density. Representative plots of the reduced \(G(r)\) and the total radial distribution function \(T(r)\), obtained through the above equations, are shown in Figure 12. The areas of the first to fifth peaks in the \(T(r)\) distribution are valuable metadata for structural analysis. When the composition and density of the sample are well-characterized, such as in platinum-based materials, these peak areas can determine relative coordination numbers in absolute terms, normalized to known structural parameters. Even when the exact composition or density is unknown, comparative analysis  
Figure 11: **(a)** Intensity as a function of \(2\theta\) for the hydrogen-reduced TEC10V30E sample, supplied by TANAKA Precious Metals. The red line (\(I_{\text{obs}}\)) represents raw data, and the blue line (\(I_{\text{BG}}\)) shows background data from a vacant quartz capillary, obtained by combining measurements from seven detectors positioned at different angles. The black line (\(I_{\text{corrected}}\)) indicates the corrected data, obtained after applying background subtraction and corrections for absorption and polarization. **(b)** The structure factor \(S(Q)\), emphasizing high \(Q\) oscillations.