Resolution function and flux at the sample for small-angle X-ray scattering calculated in position-angle-wavelength space

A method for calculating the flux at the sample position and the resolution function based on the general description of synchrotron X-ray beamlines in position-angle-wavelength space has been developed. A mathematical formulation is presented, in which source, transmission function for the slits and acceptance windows of the monochromator crystals are approximated by Gaussian functions. These approximations allow all of the algebra in connection with combining the different contributions to be done analytically. When the beam passes the various components of the beamline, such as flight paths, monochromators and mirrors, it results in coordinate transformations in parameter space. By inserting the transformations in the intensity distribution of the source, transmission functions of the slits and acceptance windows of the monochromators, these are transformed to the position of the sample. Their product gives the intensity distribution at this position and from this the resolution function in reciprocal space is calculated. The resolution of the detector is easily included by a convolution. The transmissions of the slits and the acceptance windows of the monochromator crystals have been normalized to give the same integrated transmission and reflectivities as the functions they approximate. Therefore the flux at the sample position can be calculated by multiplication of the brilliance of the source and the result obtained from an integration of the distribution over the two position parameters, the two angle parameters and the wavelength. The theory has been applied to an example and gives reasonable results for sample intensity and resolution function.