Exact solution of the slit-height correction problem in small-angle X-ray scattering. I. The general method and its accuracy in application to simulated data

A general method for obtaining the exact solution of the integral equation relating the slit height correction function g(t) and the slit transmission function f(t) is presented. This is achieved by transforming the integral equation so that it is directly solvable utilizing Laplace transforms. An expression for the corrected intensity function in terms of d[g(t)/t]/dt and the measured scattered intensity, without the necessity of differentiating the latter, is also presented. The procedure is tested in detail by applying it to simulated experimental data for several experimentally relevant examples. The key ingredients tested relate to the effects of: (1) discreteness of the data; (2) availability of data for only a restricted range of scattering angles; and (3) data beset with random noise. The procedure yields highly accurate results despite all of these difficulties.