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

We have systematically studied the optimal real-space sampling of atomic pair distribution (PDF) data by comparing refinement results from oversampled and resampled data. Based on nickel and a complex perovskite system, we show that not only is the optimal sampling bounded by the Nyquist interval described by the Nyquist-Shannon (NS) sampling theorem as expected, but near this sampling interval, the data points in the PDF are minimally correlated, which results in more reliable uncertainty estimates in the modeling. Surprisingly, we find that PDF refinements quickly become unstable for data on coarser grids. Although the Nyquist-Shannon sampling theorem is well known, it has not been applied to PDF refinements, despite the growing popularity of the PDF method and its adoption in a growing number of communities. Here, we give explicit expressions for the application of NS sampling theorem to the PDF case, and establish through modeling that it is working in practice, which lays the groundwork for this to become more widely adopted. This has implications for the speed and complexity of possible refinements that can be carried out many times faster than currently with no loss of information, and it establishes a theoretically sound limit on the amount of information contained in the PDF that will prevent over-parametrization during modeling.

Figure 1

Demonstration of aliasing in $F\left(Q\right)$. (Top) Experimental nickel $F\left(Q\right)$ with $Q_{\max }=29.9{\text{Å}}^{-1}$ featuring regions above and below $Q^{\prime }=20.9{\text{Å}}^{-1}$. (Center) Experimental nickel $F\left(Q\right)$ with the region above $Q^{\prime }$ “folded” over to lower $Q$. (Bottom) Aliased $F\left(Q\right)$ obtained by sampling the PDF from the experimental $F\left(Q\right)$ on a grid with interval $dr=0.15\text{Å}$ and Fourier transforming back to $F\left(Q\right)$ (solid line). This sampling interval is larger than the Nyquist interval ($d{r}_N=0.105\text{Å}$) and corresponds to $Q^{\prime }=\pi /dr=20.9{\text{Å}}^{-1}$. Overlaid is the $F\left(Q\right)$ obtained by adding the unfolded and folded segments of the experimental $F\left(Q\right)$ (dashed line). Note that the $Q$ axis starts at $10{\text{Å}}^{-1}$.

Figure 2

Fits to sampled Ni PDFs. (a) Unaltered data with $dr=0.01\text{Å}$. (b) Sampled data with $dr=0.1\text{Å}$. (c) Sampled data with $dr=0.3\text{Å}$. The data are shown as circles, the fits are the lines through the data, and the difference is shown offset below. All fits are of similar quality, despite the poor visual quality of the data in panels (b) and (c). The data shown in panel (c) are undersampled and produced unacceptably uncertain results, though this is not apparent from the difference curve.

Figure 3

Fits to sampled LaMnO${}_3$ PDFs. (a) Unaltered data with $dr=0.01\text{Å}$. (b) Sampled data with $dr=0.1\text{Å}$. (c) Sampled data with $dr=0.3\text{Å}$. The data are shown as circles, the fits are the lines through the data, and the difference is shown offset below. All fits are of similar quality, despite the poor visual quality of the data in panels (b) and (c). The data shown in panel (b) and (c) are undersampled, and the data in panel (c) produced unacceptably uncertain results. Note that in panel (c) several peaks are not resolved.

Figure 4

Refined parameter quality (open symbols) and refinement times (solid circles) measured using sampled Ni (top) and LaMnO${}_3$ (bottom) data. The dotted horizontal line shows the cutoff between acceptable and unacceptable parameter quality. The dashed vertical line shows the value of $d{r}_N$ predicted by the sampling theorem. For $dr$ values larger than this, the quality of some parameters transition into the unacceptable region. The time values demonstrate the decrease in refinement time with increasing $dr$ with more than a seven-fold speedup near $d{r}_N$. The solid curve through the time values is fit to the form $a+b/dr$.