Accounting for instrument resolution in the pair distribution functions obtained from total scattering data using Hermite functions.

The use of Hermite functions to describe pair distribution functions (PDFs) from total scattering data was previously proposed by Krylov & Vvedenskii [J. Non-Cryst. Solids (1995), 192-193, 683-687]. Hermite functions have a suitable form for describing both the total scattering data and the PDF, and have the useful feature that they are eigenfunctions of the Fourier transform operation. We demonstrate that, by fitting Hermite functions to total scattering data, it is possible to take into account the effects of experimental resolution when deriving the PDF from the scattering data. This is particularly advantageous for neutron time-of-flight data, where different banks of detectors have different resolution functions and the resolution widths vary with the size of the scattering vector. A number of technical points are discussed and illustrated using examples of synthetic data, including both amorphous and crystalline materials. These include a solution to the problem of handling the sharp Bragg peaks, and how to scale the scattering function and PDF to match the scale of the Hermite functions. A number of examples using real scattering data, both synchrotron X-ray and spallation neutron data, are also shown. To account for uncertainties in the levels of the scattering functions, we have modified a method of Billinge & Farrow [J. Phys. Condens. Matter (2013), 25, 454202] to remove backgrounds by fitting with low-order orthogonal (Chebyshev) functions.