Paths from weak to strong coupling in NMR

Abstract

$J$-coupled nuclear magnetic resonance (NMR) spectroscopy in the strong coupling regime at low magnetic field ($10{}^{\ensuremath{-}7}$ T $<B<{10}^{\ensuremath{-}3}$ T) is more complex than at high field ($B>{10}^{\ensuremath{-}3}$ T) and at ultralow field ($B<{10}^{\ensuremath{-}7}$ T). We show that several upper and lower boundaries ${B}_{i}^{\mathrm{up}}$ and ${B}_{i}^{\mathrm{low}}$ of the magnetic field $B$ exist, where the complexity of $J$-coupled NMR spectra changes in terms of the number of lines. The index $i=1,2,\dots{}$ for ${B}_{i}^{\mathrm{up}}$ at high field specifies the perturbation order of the dominating Zeeman interaction and for ${B}_{i}^{\mathrm{low}}$ at ultralow field the perturbation order of the dominating $J$-coupling interaction. Mathematical expressions for these boundaries are derived for the case of a $J$-coupled $S$-$I$${}_{N}$ group where $S$ and $I$ are rare and abundant spins $\frac{1}{2}$ and $N$ counts the abundant spins $I$. The entire $B$-field range can further be delineated into two weak coupling regimes, one at high field with ${B}_{2}^{\mathrm{up}}<B<{B}_{1}^{\mathrm{up}}$ ($10{}^{\ensuremath{-}3}$ T $<B<{10}^{2}$ T), one at low field with ${B}_{1}^{\mathrm{low}}<B<{B}_{2}^{\mathrm{low}}({10}^{\ensuremath{-}8}$ T $<B<{10}^{\ensuremath{-}7}$ T), and a strong coupling regime with ${B}_{2}^{\mathrm{low}}<B<{B}_{2}^{\mathrm{up}}$ ($10{}^{\ensuremath{-}7}$ T $<B<{10}^{\ensuremath{-}3}$ T). The corresponding NMR spectra for the $S\ensuremath{-}{I}_{N}$ group are investigated by experiment and by simulation. In the strong coupling regime, the maximum number of lines is $(N+1){}^{2}$. In the weak coupling regime ${B}_{1}^{\mathrm{low}}<B<{B}_{2}^{\mathrm{low}}$ at low field, symmetric multiplet structures group around the frequencies $0$, $J$, ($3/2$)$J$, $2J$, ($5/2$)$J$, etc. These spectra determine the structure of the $S\ensuremath{-}{I}_{N}$ group unambiguously and are in dual correspondence to the weakly coupled spectra at high field. High-resolution NMR spectroscopy at ultralow field may open up new ways for chemical analysis by small and mobile instruments with many applications in science and technology.