# Simplified Polydispersion Analysis of Small-Angle Scattering Data
## 4 Simplified Polydispersion Analysis  
Usually, the analysis for the interpretation of the SAS data of polydisperse inhomogeneities is performed by fitting the experimental SCSs with theoretical models, and this sometimes represents a barrier for many scientists to use the SAS technique. This simplified polydispersion analysis (SPA) allows the polydispersion analysis of the SAS data to take place using a simplified approach that does not require a high level of computational skills. The SPA opens up the SAS technique with polydispersion analysis to the fraction of the scientific community with no experience in fitting experimental data and has the ambitious goal to recruit more scientists interested in obtaining their results via this powerful technique. The calculation of the SDF using global scattering functions was introduced by [22], and the present SPA approach represents a further simplification in the determination of the SDFs described below. By considering the lognormal SDF and its momenta given in Equation (19), Equations (15)-(17) can be written as follows:  
\[R_{g^{*}}^{2}=\frac{3}{5}exp(16\sigma^{2})R_{o}^{2} \tag{21}\]  
\[\frac{d\Sigma}{d\Omega}(0)=(\frac{4}{3}\pi)^{2}(\Delta\rho)^{2}N_{o}R_{o}^{7} exp(\frac{49}{2}\sigma^{2})\sigma\sqrt{2\pi} \tag{22}\]  
\[A_{p}=8\pi^{2}(\Delta\rho)^{2}N_{o}R_{o}^{3}exp(\frac{9}{2}\sigma^{2})\sigma \sqrt{2\pi} \tag{23}\]  
By defining the following quantities:  
\[A=\frac{\frac{d\Sigma}{d\Omega}(0)}{(\frac{4}{3}\pi)^{2}(\Delta\rho)^{2}}=N_{ o}R_{o}^{7}exp(\frac{49}{2}\sigma^{2})\sigma\sqrt{2\pi} \tag{24}\]  
\[B=R_{g^{*}}^{2}=\frac{\langle R^{8}\rangle}{\langle R^{6}\rangle}=R_{o}^{2} exp(16.0\sigma^{2}) \tag{25}\]  
\[C=\frac{A_{p}}{8\pi^{2}(\Delta\rho)^{2}}=N_{o}R_{o}^{3}exp(\frac{9}{2}\sigma^{ 2})\sigma\sqrt{2\pi} \tag{26}\]  
the three equations above relate the three main physical quantities obtained from the experimental SCS to the three independent parameters of the lognormal SDF, which are found to be  
\[\sigma=\frac{1}{12}\ ln\frac{B^{2}C}{A} \tag{27}\]  
\[R_{o}=[B\ exp(-16\sigma^{2})]^{1/2} \tag{28}\]  
\[N_{o}=\frac{C}{R_{o}^{3}\ exp(\frac{9}{2}\sigma^{2})\ \sigma} \tag{29}\]  
A complete analytical solution of the system with the three equations above is possible. However, for the sake of simplicity, I suggest first calculating \(\sigma\) by using Equation (27) and then calculating the other two quantities. The uncertainties \(\sigma_{*}\) (different from \(\sigma\), the parameter of the lognormal SDF) of the estimated parameters of the lognormal SDF are given by  
\[\sigma_{\sigma}=\frac{1}{24\sigma}[(\frac{\sigma_{A}}{A})^{2}+(2\frac{\sigma_ {B}}{B})^{2}+(\frac{\sigma_{C}}{C})^{2}]^{(\frac{1}{2})} \tag{30}\]  
\[\frac{d\Sigma}{d\Omega}(0)=(\frac{4}{3}\pi)^{2}(\Delta\rho)^{2}N_{o}R_{o}^{7} exp(\frac{49}{2}\sigma^{2})\sigma\sqrt{2\pi} \tag{22}\]0  
\[\frac{d\Sigma}{d\Omega}(0)=(\frac{4}{3}\pi)^{2}(\Delta\rho)^{2}N_{o}R_{o}^{7} exp(\frac{49}{2}\sigma^{2})\sigma\sqrt{2\pi} \tag{22}\]1  
where the estimated uncertainties of the quantities \(A\), \(B\), and \(C\) are given by  
\[\frac{d\Sigma}{d\Omega}(0)=(\frac{4}{3}\pi)^{2}(\Delta\rho)^{2}N_{o}R_{o}^{7} exp(\frac{49}{2}\sigma^{2})\sigma\sqrt{2\pi} \tag{22}\]2\[\frac{d\Sigma}{d\Omega}(0)=(\frac{4}{3}\pi)^{2}(\Delta\rho)^{2}N_{o}R_{o}^{7} exp(\frac{49}{2}\sigma^{2})\sigma\sqrt{2\pi} \tag{22}\]3 \[\frac{d\Sigma}{d\Omega}(0)=(\frac{4}{3}\pi)^{2}(\Delta\rho)^{2}N_{o}R_{o}^{7} exp(\frac{49}{2}\sigma^{2})\sigma\sqrt{2\pi} \tag{22}\]4  
where \(\sigma_{\frac{\sigma\Sigma}{2B}(0)}\), \(\sigma_{R_{g}}\), and \(\sigma_{A_{p}}\) are the estimated uncertainties of \(\frac{d\Sigma}{d\Omega}(0)\), the effective radius of gyration \(R_{g*}\), and the \(A_{p}\) constant, respectively.