# Simplified Polydispersion Analysis of Small-Angle Scattering Data
## 5 Simulations  
The SCSs of polydisperse systems of spherical inhomogeneities were calculated by using a lognormal SDF defined between 1 and 250 A, with \(R_{o}\) of 20 A, four values of \(\sigma\) (0.1, 0.2, 0.3, and 0.4), and normalized to a volume fraction \(C_{v}\) of 1.0%. The SDF functions used to simulate the SCSs are shown in Figure 1. The nominal values of the three parameters \(N_{o}\), \(R_{o}\), and \(\sigma\) calculated with the SDFs are reported in the upper part of Table 1.  
A contrast factor \((\Delta\rho)^{2}\) of \(1.0\times 10^{-12}\)A\({}^{-4}\) was used to simulate the SCSs in absolute units (cm\({}^{-1}\)) in the ideal Q range between 0.00005 and 10 A\({}^{-1}\) to ensure the presence of the Guinier and of the Porod approximations. An experimental error of 5% was applied to mimic the experimental conditions. The simulated SCSs are shown in Figure 2.  
For \(\sigma=0.1\), the simulated system shows reduced polydispersity, and therefore, the oscillations typical of the monodisperse systems are not smeared out. In this respect, it is \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 & & & & \\ \(N_{o}\) (10\({}^{24}\) cm\({}^{-4}\)) & 5.49 & 2.16 & 0.97 & 0.41 \\ \(R_{o}\) (Å) & 20.0 & 20.0 & 20.0 & 20.0 \\ \(\sigma\) & 0.1 & 0.2 & 0.3 & 0.4 \\ \hline calculated values & & & & \\ \(N_{o}\) (10\({}^{24}\) cm\({}^{-4}\)) & \(6.13\pm 1.03\) & \(2.22\pm 0.17\) & \(0.97\pm 0.07\) & \(0.35\pm 0.02\) \\ \(R_{o}\) (Å) & \(20.2\pm 0.4\) & \(20.5\pm 0.4\) & \(20.1\pm 0.4\) & \(22.7\pm 0.4\) \\ \(\sigma\) & \(0.09\pm 0.01\) & \(0.19\pm 0.01\) & \(0.30\pm 0.01\) & \(0.37\pm 0.01\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nominal and calculated values of the three parameters of the lognormal SDF of spherical inhomogeneities obtained by applying the SPA to the simulated SCSs. Figure 1: SDFs used to simulate the SCSs of polydisperse inhomogeneities; \(N_{o}\) are normalized to ensure a volume fraction of 1%; \(R_{o}=20\) Å; four values of \(\sigma\) are considered (0.1, 0.2, 0.3, and 0.4). Figure 2: Simulated SCSs of a lognormal SDF of spherical inhomogeneities; \(N_{o}\) are normalized to ensure a volume fraction of 1%; \(R_{o}=20\) Å; four values of \(\sigma\) are considered (0.1, 0.2, 0.3, and 0.4). The Guinier and the Porod approximations are present in the investigated Q range. worth bearing in mind that the form factor of a sphere depends on the spherical Bessel function \(J_{1}\). The Guinier and the Porod approximations were fit to the simulated SCSs, each on the proper Q-range. As an example, the linear fits for the simulated SCS with \(\sigma=0.2\) are shown in Figures 3 and 4 for the Guinier and the Porod approximations, respectively. In both cases, the points used for the fit are shown with black symbols with their error bars, while the other simulated points are shown in red. With the SPA approach, the SCSs were fit by using two linear fits in the two plots described above, i.e., \(In\frac{d\Sigma}{d\Omega}-Q^{2}\) at low \(Q\) values for the Guinier and \(log\frac{d\Sigma}{d\Omega}-logQ\) at large \(Q\) values for the Porod approximation, respectively. In the case shown in Figure 3, \(Q_{max}R_{g}\) was found to be 1.13, beyond the theoretical limit of validity of the Guinier approximation (\(Q_{max}R_{g}\lesssim 1.0\)) in order to obtain a more precise result. There is a two-fold argument for this exception: (1) the data are effectively reproduced by a straight line even at higher Q values than the theoretical limit, and moreover, (2) the difference in the results is smaller than the calculated uncertainty. From the optimized parameters of the linear interpolations and their estimated errors, it is possible to calculate the physical quantities needed to apply the SPA, i.e., \(\frac{d\Sigma}{d\Omega}(0)\)  
Figure 4: Porod approximation of the simulated SCS by using the SDF with \(\sigma=0.2\). The points used for the fit are shown with black symbols, while the others are red.