Small-Angle Scattering from Anisotropic Systems in the Guinier Region

A general formula for scattering in the Guinier region is derived for uniaxially symmetric system in terms of the inertia of ellipsoid. It is shown that the azimuthal angle dependence of the scattering pattern is independent of the shape of the particle and depends only on the axial and lateral dimensions. A method involving double-tilt is proposed and discussed to quantitatively estimate the size when the Guinier region is difficult to access for the long axis parallel to the detector plane (e.g., voids scattering from highly drawn fibers) or in cases when the axis of symmetry cannot coincide with the detector plane (e.g., single chain scattering from planar oriented films). A family of “particle” shapes ranging from convex to concave surface is considered to quantitatively estimate the error between the actual and measured aspect ratio.