Separate and detailed characterization of signal and noise at low resonance frequencies

When developing or deploying a Nuclear Magnetic Resonance (NMR) spectrometer, especially for Overhauser Dynamic Nuclear Polarization (ODNP) or other experiments that require low-volume low-field measurements, the ability to mitigate noise and to quantitatively predict signal amplitude prove crucial. A quantitative treatment allows separate analysis of signal and noise and independent optimization of each. In particular, the results here emphasize that clarity and insight come from (1) characterizing the spectral distribution of the noise, and (2) integrating elements of theory and notation originally developed for Electron Spin Resonance (ESR) spectroscopy. Specifically, the spectral noise density “fingerprint spectrum” identifies sources of electromagnetic interference (EMI) and definitively confirms which actions do and do not mitigate the EMI, while the quantitative ratio ($\Lambda$) of $B_1$ to the square root of the power on the transmission line provides a useful focal point that simplifies the prediction of signal intensity and that decomposes into a few simple but exact factors. Thus, this article provides a relatively comprehensive overview of signal and noise in low-field NMR instruments. The protocol/toolkit introduced here should apply to a wide range of instruments, and give most spectroscopists the freedom to systematically design sensitive NMR hardware even in cases where it must be integrated with multiple other hardware modules (e.g., an existing ESR system), or where other requirements constrain the design of the NMR hardware. It enables a systematic approach to instrument design and optimization. For the specific X-band ODNP design demonstrated here (and utilized in other laboratories), it facilitates a reduction of the noise power by more than an order of magnitude, and accurately predicts the signal amplitude from measurements of the nutation frequency. Finally, it introduces reasoning to exactly determine the field distribution factor (${\eta }^{\prime }$, essentially, a more specific definition of the filling factor) experimentally from $\Lambda$ and thus identifies the inefficient distribution of fields in the hairpin loop probe as the main remaining bottleneck for the improvement of low-field, low-volume ODNP signal-to-noise ratio (SNR).