Nyquist-Shannon sampling theorem applied to refinements of the atomic pair distribution function

Abstract

We have systematically studied the optimal realspace sampling of atomic pair distribution data by comparing refinement results from oversampled and resampled data. Based on nickel and a complex perovskite system, we demonstrate that the optimal sampling is bounded by the Nyquist interval described by the Nyquist-Shannon sampling theorem. Near this sampling interval, the data points in the PDF are minimally correlated, which results in more reliable uncertainty prediction. Furthermore, refinements using sparsely sampled data may run many times faster than using oversampled data. This investigation establishes a theoretically sound limit on the amount of information contained in the PDF, which has ramifications towards how PDF data are modeled.

Structural study of biotic and abiotic poorly-crystalline manganese oxides using atomic pair distribution function analysis

“…R w values were calculated between r min = 1 Å and r max = 50 Å , while fits were performed with r min = 0 Å . The r-interval used in the fits is 0.1 Å , which is smaller than the Nyquist interval (0.126 Å ) for Q max = 25 Å À1 (Farrow et al, 2011). Since the layers are stacked along the crystallographic c-axis in these materials, the morphology and internal structure are correlated and the CF formalism in Eq.…”

Structure of NanocrystallineTi3C2MXene Using Atomic Pair Distribution Function

“…Na Uncertainties on the refined parameters were determined from PDFGUI [25] on fits calculated on the Nyquist-Shannon sampling grid following [26], taking into account serial correlation in the data [27] b O=F mole ratio was determined from an energy dispersive x-ray spectroscopy (EDX) experiment and fixed in the refinements.…”

Verification of Anderson Superexchange in MnO via Magnetic Pair Distribution Function Analysis andab initioTheory

“…We then fit a parabola to the 17 (J 1 =J 2 , χ 2 ) ordered pairs for each temperature, with a common parabolic minimum refined as a global parameter to obtain the best-fit value of J 1 =J 2 . We took care to properly weight with the statistical uncertainties on the data propagated from the original PDF on the Nyquist grid [29]. Details are available in the Supplemental Material [21].…”