We present a method to calculate solution X-ray scattering form-factors of various geometries with generalized electron density and polydispersity profiles. We create arbitrary and physically relevant electron density profiles using a set of smooth hyperbolic tangent functions. To numerically calculate arbitrary electron density profiles, we formulate an algorithm that adaptively transforms the functions to a series of uniform discrete steps. We solve the models both numerically and analytically for the case of multiple spherical shells and compare the results to show the consistency of the algorithm. Other geometries are solved numerically. Various form-factors are analysed and compared with earlier results. We then compare polydispersity probability density functions (uniform, normal, and Cauchy distributions) of concentric hollow cylinder thicknesses. The relationship of the shape of arbitrary electron density profiles to the features of the scattering form-factor is discussed.
Bibliographical noteFunding Information:
We thank Daniel Harries for helpful discussions and Slava Kler, Orly Ben-Shaul, and Areilla Oppenheim for experimental help with the SV40 VLPs. The ESRF synchrotron, beamline ID02 and Elettra, 5.2L SAXS beamline are acknowledged, as the presented data were acquired there. This project was supported by the Israel Science Foundation (grant number 1372/13), the US-Israel Binational Science Foundation (grant number 2009271), Kamin program of the chief scientist of Israel Ministry of Economy, and the FTA-Hybrid Nanomaterials program of the Planning and Budgeting Committee of the Israel Council of Higher Education. We thank the Safra, Wolfson, and Rudin Foundations for supporting our laboratory.
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- density functional calculations
- generalized electron density
- solution X-ray scattering