A general, weighted least-squares method for the evaluation of small-angle X-ray data without desmearing

In order to avoid some of the disadvantages associated with the desmearing methods, a procedure has been developed where the smeared, primary, intensity data can be evaluated directly without desmearing. The procedure consists of the following: first, a model depending on a vector of unknowns, x = (x₁, ...., x_n), is constructed; then, an iterative search is made for the vector x, and a scale factor s, which corresponds to a local minimum in the error square sum based on the primary, slit-smeared, intensity data. The main advantages with the present method are that the comparison between theory and experiment is made directly with the experimental quantity; thus the experimental errors can be considered in this comparison. Furthermore, some of the disadvantages associated with the desmearing methods are avoided; the method is numerically stable and no extrapolations outside the measured angular range are necessary. Several data sets measured at different concentrations and with different, completely arbitrary, primary-beam weighting functions can be considered in the same refinement. The interparticle scattering effect may also be included in the least-squares refinement. The method is general, so that different models can be tested simply by changing only one subroutine of the computer program. It may also be used to evaluate data impaired by other types of resolution errors; for example, effects due to polychromatic radiation or resolution errors in neutron scattering. Two constructed examples of the application of the method are given: (1) the calculation of the dimensions and the molecular weight of particles with a shape which can be approximated with an ellipsoid of revolution; (2) the calculation of the dimensions and electron-density distribution for spherical particles.