Benford distributions in NMR

TL;DR

This paper investigates the prevalence of Benford's Law in NMR data, revealing that NMR signals generally follow Benford distribution across various conditions and can distinguish real spectra from simulations.

Contribution

It is the first comprehensive study showing NMR signals follow Benford's Law in multiple domains and parameters, with applications in data validation and analysis.

Findings

NMR signals follow Benford's Law in time and frequency domains.

Benford analysis can differentiate genuine spectra from simulated ones.

Chemical-shift data and RF pulse amplitudes also adhere to Benford's Law.

Abstract

Benford's Law is an empirical law which predicts the frequency of significant digits in databases corresponding to various phenomena, natural or artificial. Although counter intuitive at the first sight, it predicts a higher occurrence of digit 1, and decreasing occurrences to other larger digits. Here we report the Benford analysis of various NMR databases and draw several interesting inferences. We observe that, in general, NMR signals follow Benford distribution in time-domain as well as in frequency domain. Our survey included NMR signals of various nuclear species in a wide variety of molecules in different phases, namely liquid, liquid-crystalline, and solid. We also studied the dependence of Benford distribution on NMR parameters such as signal to noise ratio, number of scans, pulse angles, and apodization. In this process we also find that, under certain circumstances, the Benford analysis can distinguish a genuine spectrum from a visually identical simulated spectrum. Further we find that chemical-shift databases and amplitudes of certain radio frequency pulses generated using optimal control techniques also satisfy Benford's law to a good extent.