# Simplified Polydispersion Analysis of Small-Angle Scattering Data
## 5 Simulations
The fit is shown with a red line. Figure 3: Guinier approximation of the simulated SCS by using the SDF with \(\sigma=0.2\). The points used for the Guinier approximation are shown with black symbols, while the others are red. The fit is shown with a red line. \(R_{g*}\), and \(A_{p}\), and from them, the SDF parameters. In principle, it is also possible to use a non-linear fit of the SCSs to calculate the three physical quantities to optimize, but the conversion from the linear parameters to the physical quantities is straightforward. The values of the SDF parameters used to simulate the SCSs, as well as those calculated by using the SPA are shown in Table 1. The fit was performed by using commercial software with the linear interpolation option in the \(ln(\frac{d\overline{z}}{d\Omega})-Q^{2}\) plane for the Guinier and in the \(log(\frac{d\overline{z}}{d\Omega})-logQ\) plane for the Porod approximations, respectively. The analysis of polydisperse SAS data by SPA then becomes a very easy process. The SDFs' parameters are in excellent agreement with the nominal values, and their estimated uncertainties are acceptable. The main physical quantities of the inhomogeneities, such as number density \(N\), average radius \(\langle R\rangle\), total surface \(S\) and volume fraction \(C_{v}\), were calculated and the results shown in Table 2. The values calculated with the nominal parameters of the SDF and those calculated by the SPA are in excellent agreement. The agreement between the values of the physical quantities calculated using the nominal SDF parameters and those calculated using SPA is excellent, and the estimated uncertainties are acceptable. The SDFs reconstructed with the SPA are shown in Figure 5 for the four different cases. The agreement between the original and reconstructed SDFs is excellent. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter** & \(\sigma=0.1\) & \(\sigma=0.2\) & \(\sigma=0.3\) & \(\sigma=0.4\) \\ \hline nominal values & & & & \\ \(N\) (\(10^{17}\) cm\({}^{-3}\)) & 2.77 & 2.21 & 1.52 & 0.90 \\ \(\langle R\rangle\) (Å) & 20.3 & 21.2 & 22.9 & 25.4 \\ \(\langle S\rangle\) (\(10^{5}\) cm\({}^{-1}\)) & 1.45 & 1.30 & 1.09 & 0.86 \\ \(C_{v}(\%)\) & 1.00 & 1.00 & 1.00 & 1.00 \\ \hline calculated values & & & & \\ \(N\) (\(10^{17}\) cm\({}^{-3}\)) & \(2.75\pm 0.62\) & \(2.15\pm 0.19\) & \(1.52\pm 0.11\) & \(0.78\pm 0.05\) \\ \(\langle R\rangle\) (Å) & \(20.4\pm 6.6\) & \(21.6\pm 2.8\) & \(22.9\pm 2.6\) & \(27.7\pm 2.4\) \\ \(\langle S\rangle\) (\(10^{5}\) cm\({}^{-1}\)) & \(1.45\pm 0.35\) & \(1.31\pm 0.14\) & \(1.10\pm 0.11\) & \(0.86\pm 0.07\) \\ \(C_{v}(\%)\) & \(1.00\pm 0.25\) & \(1.01\pm 0.13\) & \(1.00\pm 0.12\) & \(1.04\pm 0.09\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Nominal and calculated values of the main physical quantities, the number density \(N\), the average radius \(\langle R\rangle\), the average surface \(\langle S\rangle\), and the volume fraction \(C_{v}\), of spherical inhomogeneities obtained by applying the SPA on the simulated SCS. Figure 5: SDFs reconstructed with the parameters (\(N_{o}\), \(R_{o}\) and \(\sigma\)) calculated with the SPA for the four cases (\(\sigma=0.1\), \(0.2\), \(0.3\) and \(0.4\)). The original SDFs are shown in Figure 1.