# Simplified Polydispersion Analysis of Small-Angle Scattering Data
## 7 Conclusions  
The SPA was used to perform a simplified analysis of the SAS data of polydisperse samples using simple interpolations employing commercial software, and it does not require a high level of computational skills. To summarize, the SPA can be applied when the following conditions are met:  
* Two-phase system;  
Figure 10: Ludox silica particles HS30 with a 0.3% volume fraction: the Weibull SDF (blue line) obtained with the fitting procedure, with corrections for the wavelength spread (\(\Delta\lambda/\lambda\) = 18%) and for the multiple scattering, and the lognormal SDF (red line) calculated with the SPA.  
Figure 9: Analysis of experimental SCS of Ludox silica particles HS30 with a 0.3% volume fraction: (**a**) The SPA with the fits of the Guinier (red line), as well as of the Porod (blue line) approximations along with the values of the optimized parameters and their uncertainties. The points used for both fits are shown with black symbols, while the others are red. (**b**) A fitting procedure with a Weibull SDF, where corrections for the wavelength spread (\(\Delta\lambda/\lambda\) = 18%) and for the multiple scattering were applied [29].  
* Isotropic scattering;
* Sharp interfaces of the inhomogeneities;
* The Q-range includes both the Guinier and Porod approximations;
* Presence of one family of inhomogeneities, described by a lognormal SDF;
* The presence formalism has been developed for spherical inhomogeneities; however, it can be extended to other shapes.  
The SPA provides the log-normal SDF of the investigated samples by performing the interpolation of the two approximated trends at small and large \(Q\) values, where the three parameters needed to define the SDF are calculated numerically from the optimized parameters. The check-list below summarizes the workflow of the determination of the SDF once the experimental SCSs are available:  
* Guinier approximation available in the experimental \(Q\) range (Equation (5));
* Porod approximation available in the experimental \(Q\) range (Equation (14));
* The three parameters \(R_{g*}\), \(\frac{d\Sigma}{d\Omega}(0)\), and \(A_{p}\) are calculated;
* The three parameters A, B, and C are calculated (Equations (24)-(26));
* The three quantities describing a lognormal SDF are calculated (Equations (27)-(29)).  
The SPA was applied to both simulated (with and without an additional incoherent background) and experimental SCSs, and in all cases, the validity of the proposed method was demonstrated, where the values of the physical quantities were clearly reproduced and their uncertainties were larger than those estimated with the traditional fitting method. Last but not least, in many large-scale facilities, the simple SPA approach can be applied to real-time analysis during experiments for the efficient tuning of experimental details.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.  
[PERSON] is warmely acknowledged for his critical reading of the manuscript.  
The following abbreviations are used in this manuscript:  
ED electron density  
RID refraction index density  
SAS small-angle scattering  
SCS scattering cross-section  
SDF size distribution function  
SLD scattering length density  
SPA simplified polydispersion analysis