Asymptotic behavior and Lorentz factor for small-angle elastic scattering profiles from preferentially oriented asymmetric bodies

Abstract

The small-angle scattered intensity functions I(h) were analyzed for asymmetric bodies of thin disks (TD; R≫2L), narrow rods (NR; 2L≫R), and prolate (PE; va≫a) and oblate ellipsoids of revolution (OE; va≪a). h, R, and 2L, 2a, and 2av are the magnitude of scattering vectors defined as h=(4π/λ)sin θ, with 2θ and λ being the scattering angle and wavelength of the radiation in the medium, respectively, the radius and length of the disk (or rod), and the lengths of two major axes of the ellipsoid of revolution, respectively. The functions I(h) scale as h−γ (γ=0,1,2,4) and show crossovers between different exponents γ, which depend on the aspect ratio (L/R or v) and the orientation distribution function W(α). When the aspect ratio is much larger or much smaller than unity, the relationship between the intensities at a given direction Ir(h) and that perpendicular to the longest axis of the particle Ip(h) was found to be related by Ir(h)∼h−γL Ip(h), for h≫1/max(R,L) or 1/max (va,v), where h−γL is the so-called Lorentz factor or Lorentz correction factor. The criteria for this relationship to be valid were discussed in terms of the aspect ratio and W(α) of the particle.