Nuclear magnetic resonance diffraction with subangstrom precision

We have combined ultrasensitive force-based spin detection with high-fidelity spin control to achieve NMR diffraction (NMRd) measurement of ~2 million 31P spins in a (50 nm)3 volume of an indium-phosphide (InP) nanowire. NMRd is a technique originally proposed for studying the structure of periodic arrangements of spins, with complete access to the spectroscopic capabilities of NMR. We describe two experiments that realize NMRd detection with subangstrom precision. In the first experiment, we encode a nanometer-scale spatial modulation of the z-axis magnetization of 31P spins and detect the period and position of the modulation with a precision of <0.8 Å. In the second experiment, we demonstrate an interferometric technique, utilizing NMRd, to detect an angstrom-scale displacement of the InP sample with a precision of 0.07 Å. The diffraction-based techniques developed in this work extend the Fourier-encoding capabilities of NMR to the angstrom scale and demonstrate the potential of NMRd as a tool for probing the structure and dynamics of nanocrystalline materials.

Scattering techniques that employ coherent sources, such as X-rays, neutrons, and electrons, are universal tools in many branches of natural science for exploring the structure of matter. In crystalline materials, these approaches provide a direct and efficient means of characterizing the periodicity of charge and magnetic order. MRI, like other scattering approaches, is a reciprocal space technique, in which the measured signal is proportional to the Fourier transform of the spin density. This similarity between MRI and scattering was recognized very early in the development of MRI and led Mansfield and Grannell in 1973 to propose NMR “diffraction” (NMRd) as a method for determining the lattice structure of crystalline solids (1–3), taking advantage of the chemical specificity of NMR.The main challenge to achieving atomic-scale NMRd lies in the difficulty of generating a sufficiently large wavenumber k, capable of encoding a relative phase difference as large as 2π between adjacent spins on a lattice, separated by angstrom-scale distances. For example, the largest encoding wavenumbers achieved in clinical high-resolution MRI scanners are of order k/(2π)~104 m−1, more than a factor of 10⁵ smaller than what is needed to measure typical atomic spacings in condensed-matter systems (4). Consequently, while MRI has become a transformative technique in medical science, earning Sir Peter Mansfield and Paul Lauterbur the Nobel Prize in Physiology and Medicine, the original vision of NMRd as a method for exploring material structure has not yet been realized.The realization of atomic-scale NMRd would be a powerful tool for characterizing periodic nuclear spin structures, combining the spectroscopic capabilities of NMR with spatial encoding at condensed matter’s fundamental length scale. NMRd is a phase-sensitive technique that permits real-space reconstruction of the spin density, without the loss of phase information common to scattering techniques, such as X-rays, that measure the scattered-field intensity (5). Being nondestructive and particularly sensitive to hydrogen, NMRd could be of great importance in the study of ordered biological systems, such as protein nanocrystals that are of great interest in structural biology (6, 7). Furthermore, the combination of scattering with NMR’s rich repertoire of spectroscopic tools opens additional avenues for spatially resolved studies of nuclear-spin Hamiltonians (e.g., chemical shifts or spin–spin interactions), which are currently achieved only through increasingly complex and indirect methods (8). Finally, NMRd could be used to study quantum many-body dynamics on the atomic scale. NMR scattering experiments have previously been used in the direct measurement of spin diffusion in CaF₂ on the micrometer scale (9). Experiments on many-body dynamics have also been conducted in engineered quantum simulators, such as ultracold atoms (10–12), trapped ions (13–15), superconducting circuits (16–18), and quantum dots (19). However, these measurements have thus far been limited to small-scale quantum systems that are at most hundreds of qubits. Angstrom-scale NMRd measurements would permit studying the dynamics of complex large-scale spin networks in condensed-matter systems on length scales as short as the lattice spacing.Over the past two decades the principal technologies needed to encode nuclear spin states with wavenumbers of order 1 Å−1 have been developed in the context of force-detected nanoMRI (20–26). In this work, we report two experiments that utilize key advances in nanoMRI technology—namely the ability to generate large time-dependent magnetic-field gradients and the ability to detect and coherently control nanoscale ensembles of nuclear spins (27–31)—to generate encoding wavenumbers as large as k/(2π)~0.48 Å−1.Our first experiment demonstrates the use of spatial spin-state modulation that encodes position information the way it was envisioned in the initial NMRd proposal. Phase-sensitive NMRd detection enables us to determine the position and period of a “diffraction grating” with a precision of <0.8 Å. The diffraction grating itself is a z-axis 31P spin magnetization modulation, the mean period of which is 4.5 nm in our (50 nm)3 detection volume. Our second experiment utilizes spatially modulated spin phase in an alternative way—as a label for the physical displacement of the spins. Our interferometric technique detects an angstrom-scale displacement of the indium-phosphide (InP) sample with a precision of 0.07 Å.