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Analysis of the Initial Slope of the Small‐Angle Scattering Correlation Function of a Particle
Author(s) -
Gille Wilfried,
Mazzolo Alain,
Roesslinger Benoit
Publication year - 2006
Publication title -
particle and particle systems characterization
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.877
H-Index - 56
eISSN - 1521-4117
pISSN - 0934-0866
DOI - 10.1002/ppsc.200500883
Subject(s) - scattering , particle (ecology) , geometry , correlation function (quantum field theory) , physics , spheres , mathematics , mathematical analysis , optics , quantum mechanics , geology , oceanography , astronomy , dielectric
The small‐angle scattering correlation function of a particle γ( r ) results from scattering experiments. This function possesses a well‐defined slope γ′(0) at the origin. This slope is defined by the particle volume V and the whole surface area S of the particle via γ′(0) = – S /(4 V ). In this paper it is demonstrated that this slope defines the mean chord length of the particle, $ \bar l=-1/\gamma \prime (0)=4V/S $ . This theorem involves non‐convex particles, especially the case of particles with hollow parts. Consequently, for a large class of particle shapes the mean chord length is defined in terms of V and S . This extension of the Cauchy theorem is developed by closer analysis of the set covariance C ( r ), of the small‐angle scattering correlation function γ( r ), and of the so‐called linear erosion P ( r ) near the origin r →0. The cases of a single hollow sphere, of two touching spheres, and of the single hollow cylinder are discussed.