Nuclear spin relaxation in zero- to ultralow-field magnetic resonance spectroscopy

Abstract

Nuclear-magnetic-resonance experiments can interrogate a broad spectrum of molecular-tumbling regimes and can accurately measure interatomic distances in solution with sub-nanometer resolution. In the zero- to ultralow-field (ZULF) regime, population and coherence decay reveal nontrivial behavior due to strong coupling between nuclear spins. We note, in particular, the surprising effects that different resonances show different relaxation rates, depending on (i) the (pre)polarizing magnet field and the shuttling trajectory to the detection region at nano- and microtesla fields, (ii) the strength of the measurement field, (iii) the detection method (single-channel or quadrature), and even (iv) the nutation angle induced by the excitation pulse. We describe herein experimental data of relaxation rates measured for a 13C-labeled formic acid sample, with an atomic-magnetometer-based ZULF setup, and develop a theoretical framework to explain the detected effects and extract molecular properties. The observed effects could be used for spectral assignment, for the establishment of specific motional regimes, for image contrast, and for the characterization of relaxation processes at nano- to microtesla magnetic fields.

Figures (3)

• Figure 1: Breit-Rabi diagram showing the shifts of energy levels as a function of magnetic field. The populations have been adiabatically transferred from high to low field. Filled and empty circles represent overpopulated and underpopulated states, respectively. The diagram was calculated for the [$^{13}$C-$^{1}$H] pair in formic acid ($J_{IS} = 222$ Hz).
• Figure 2: a) Energy-level diagram for the [$^{13}$C-$^{1}$H] system of $^{13}$C-labeled formic acid at ultralow field ($B_{0}=2.7\,\upmu$T and $J_{CH}=222$ Hz). Observed transitions $\nu_{ij}$ connect nuclear-spin eigenstates, satisfying the selection rule $\Delta m_{f} = \pm 1$. b) Example of the simulated ULF spectrum corresponding to either single-channel detection or dual-channel quadrature detection. Reduced spectral crowding, absolute determination of peak frequencies, and increased signal amplitude are achieved using quadrature detection c) Field dependence of the observable peak frequencies. d) Simulated nutation profile (Eqs. \ref{['eq:nut13']}--\ref{['eq:nut24']}) of the observable coherences $\hat{\rho}_{ij}=\ket{\psi_{i}}\bra{\psi_{j}}$ under a DC pulse along the $y$-axis, characterized by a magnetic field $B_{y}=30\,\upmu$T and total duration $t_p=1.6$ ms; Theoretical relaxation rates of populations (e) and observable coherences (f) under dipolar interactions (DD) as a function of mixing angle $\theta$ for a [$^{13}$C-$^{1}$H] system in the fast-tumbling regime. The internuclear distance is $r_{\text{CH}}=1.1$ Å and the rotational correlation time is $\tau_{c}^{\text{DD}}=3.24$ ps. Theoretical relaxation rates of populations (g) and observable coherences (h) under random field fluctuations (RFF), as a function of the mixing angle $\theta$ for a [$^{13}$C-$^{1}$H] system in the fast-tumbling regime. Parameters used for the simulations are $\omega_{\text{rms}}^{\text{C}}=85.32\cdot 10^3\,$rad$;\omega_{\text{rms}}^{\text{H}}=49.63 \cdot10^{3}\,$rad$; \kappa_{\text{CH}}=0.49$, and $\tau_{c}^{\text{RFF}}=19.34$ ps.
• Figure 3: a) Schematic representation of the experimental setup (lateral view in the laboratory frame): the sample is shuttled from a prepolarizing 1 T Halbach magnet array through a distance of around 36 cm in a piercing solenoid into the magnetically shielded region for storage, pulse and detection. b) Top view of gradiometric quadrature detection scheme, consisting of two optical atomic magnetometers (dual-axis vector detectors) with four acquisition channels. c) Pulse sequences used for nutation (indicated by pulse time $t_p$) and polarization-lifetime (indicated by storage time $t_s$) experiments in nT--$\upmu$T fields. Spin polarization in the Halbach array lasted 10 s. d) Nutation curves of the three observed peaks for the [$^{13}$C] formic-acid sample, using a detection pulse with a constant background field amplitude of $B_y = -30\,\upmu$T for storage time $t_{s} = 1 s$. Equations \ref{['eq:nut13']}--\ref{['eq:nut24']} were used to fit individual nutation profiles. e) Signal decay for $+J$ and $-J$ peaks as a function of storage time $t_s$ using two different pulse lengths $t_p$ prior to detection. Decay profiles were fitted with a mono-exponential function $S(t_s)=A\exp(-R_1\,t_s)$, with characteristic rates shown as inset. All experiments were performed at a measurement field of $B_0=0.1\,\upmu$T. f) Nutation curves of the four observed peaks nZF1 ($\nu_{34}$), nZF2 ($\nu_{13}$), $+J$ ($\nu_{24}$), and $-J$ ($\nu_{12}$) and g) Signal decay for $+J$ and $-J$ peaks as a function of storage time $t_s$ for similar [$^{13}$C] formic-acid experiments performed at a measurement field of $B_0=5\,\upmu$T.