Extracting the Pair Distribution Function of Liquids and Liquid-Vapor Surfaces by Grazing Incidence X-ray Diffraction Mode

We show that the structure factor $S(q)$ of water can be obtained from X-ray synchrotron experiments at grazing angle of incidence (in reflection mode) by using a liquid surface diffractometer. The corrections used to obtain $S(q)$ self-consistently are described. Applying these corrections to scans at different incident beam angles (above the critical angle) collapses the measured intensities into a single master curve, without fitting parameters, which within a scale factor yields $S(q)$. Performing the measurements below the critical angle for total reflectivity yields the structure factor of the top most layers of the water/vapor interface. Our results indicate water restructuring at the vapor/water interface. We also introduce a new approach to extract $g(r)$, the pair distribution function (PDF), by expressing the PDF as a linear sum of Error functions whose parameters are refined by applying a non-linear least square fit method. This approach enables a straightforward determination of the inherent uncertainties in the PDF. Implications of our results to previously measured and theoretical predictions of the PDF are also discussed.