Geometric properties of particle ensembles in terms of their set covariance

Abstract

The set covariance C(r) is a basic function in stochastic geometry,frequently applied in the eld of image analysis. On the other hand, C(r) is interrelated to the real space structure functions dened in the eld of small-angle-scattering (SAS). Fundamental results of stochastic geometry can be transformed to the eld of SAS. By use of integral transformation, scattering data of sample materials lead to the real space structure functions. These functions reflect specic geometric properties of the sample material. Let the order range of a sample be denoted by L. Let   =  (r), 0 r L,  (r) 0 if L < r, be the SAS correlation function of an isotropic two-phase ensemble of homogeneous, hard particles of volume fraction c. Then,  (r) = [C(r)=c c]=(1 c). It is shown that the rst zero point  (r1) = 0, 0 < r1 L, can be traced back to four terms: To c of the particles, to a term  0(r1) involving the correlation function  0(r) of the isolated single particle, to a term PAB (second order particle shape specic probability) and to the particle to particle pair correlation function g(r). The obtained formula (a quasi-diluted particle ensemble represents a special case) is exemplied for the model correlation function of a Dead Leaves model and for selected experimental cases.