Pulsed Magnetic Field Gradients Research Articles

Wide-field Fourier magnetic imaging with electron spins in diamond

Wide-field magnetic imaging based on nitrogen-vacancy (NV) centers in diamond has been shown the applicability in material and biological science. However, the spatial resolution is limited by the optical diffraction limit (>200 nm) due to the optical real-space localization and readout of NV centers. Here, we report the wide-field Fourier magnetic imaging technique to improve spatial resolution beyond the optical diffraction limit while maintaining the large field of view. Our method relies on wide-field pulsed magnetic field gradient encoding of NV spins and Fourier transform under pixel-dependent spatial filters. We have improved spatial resolution by a factor of 20 compared to the optical resolution and demonstrated the wide-field super-resolution magnetic imaging of a gradient magnetic field. This technique paves a way for efficient magnetic imaging of large-scale fine structures at the nanoscale.

Binary Coding by Microwave Pulses on the Transverse Magnetization of the Tetracyanoethylene Radical

The possibility of coding the response of the electron–nuclear system of the tetracyanoethylene radical under microwave pulse irradiation in combination with a pulsed magnetic field gradient in the nanosecond timescale by a binary code is demonstrated. To this end, the tetracyanoethylene radical, which has a well-resolved equidistant electron paramagnetic resonance spectrum due to the interaction of the electron with equivalent magnetic nuclei, is used. The aim is to demonstrate the possibility of implementing this procedure physically rather than to encode the longest possible sequence.

A second order asymptotic model for diffusion MRI in permeable media

Starting from a reference partial differential equation model of the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses, one can use periodic homogenization theory to establish macroscopic models. A previous work introduced an asymptotic model that accounted for permeable interfaces in the imaging medium. In this paper we formulate a higher order asymptotic model to treat higher values of permeability. We explicitly solved this new asymptotic model to obtain a system of ordinary differential equations that can model the diffusion MRI signal and we present numerical results showing the improved accuracy of the new model in the regime of higher permeability.

Molecular Mobility in Aqueous Systems of Primary Aliphatic Amino Alcohols

Self-diffusion coefficients (SDCs) of 3-amino-1-propanol (3AP), monoethanolamine (MEA), and water molecules are measured via 1H NMR using a pulsed magnetic field gradient and a stimulated echo sequence throughout a range of concentrations at temperatures of 293–333 K. It is found that the studied amino alcohols and water form a nearly tetrahedral three-dimensional network of hydrogen bonds between the molecules of the system. It is shown that this network is stable in the liquid phase, despite the mobility of the molecules that constitute the network. The mechanism of molecular mobility is studied. It is shown that the SDC of the molecules of the above amino alcohols and their aqueous solutions depends linearly on temperature. It is concluded that the molecular mobility mechanism has a pattern of activation. To verify this hypothesis and interpret the mechanism, the energies of activation are calculated for the self-diffusion of all the studied molecules in water–amino alcohol systems. The determined energies of activation for the mobility of amino alcohol molecules in aqueous systems have similar values, suggesting they undergo interactions that produce either mixed water–amino alcohol spatial networks or amino alcohol–water molecule associations in the liquid phase.

Diffusion within pores fully revealed by magnetic resonance.

The diffusion propagator fully characterizes the diffusion process, which is highly sensitive to the confining boundaries and the structure within enclosed pores. While magnetic resonance has extensively been used to observe various features of the diffusion process, its full characterization has been elusive. Here, we address this challenge by employing a special sequence of magnetic field gradient pulses for measuring the diffusion propagator, which allows for "listening to the drum," mapping structural dispersity, and determining not only the pore's shape but also diffusive dynamics within it.

Features of Self-Diffusion of Tridecane Molecules in a Porous Medium of Kaolinite Used as a Model of a Chemically Inert Membrane

The present work focused on the experimental study of the specific features of self-diffusion of tridecane molecules in macroporous kaolinite, which is used as a raw material for the production of chemically inert membranes. The measurements of self-diffusion coefficients by pulsed magnetic field gradient nuclear magnetic resonance (PMFG NMR) revealed an increased translational mobility of tridecane molecules in kaolinite with incomplete filling of the pore space. This effect was accompanied by a sharp change in the slope of the Arrhenius plot of the self-diffusion coefficients of tridecane molecules in kaolinite. An analysis of the diffusion spin echo decay in the tridecane-kaolinite system revealed a discrepancy between the experimental data and the theoretical predictions, considering the effect of the geometry of porous space on molecular mobility. It was shown that the experimental results could be interpreted in terms of a model of two phases of tridecane molecules in the pores of kaolinite, in the gaseous and adsorbed state, coexisting under the fast-exchange conditions. Within the framework of the model, the activation energies of self-diffusion were calculated, which agreed satisfactorily with the experimental data. Additionally, the effects of the internal magnetic field gradients arising in a porous medium loaded with a gas or liquid on the data of the PFG NMR measurements were calculated. It was shown that the effect of magnetic field inhomogeneities on the measured self-diffusion coefficients of tridecane in kaolinite is small and could be neglected.

Self-Diffusion of Ferulic and Sinapic Acids in the Binary Carbon Tetrachloride–Acetone-d6 System

1H NMR with a pulsed magnetic field gradient method are used to measure the coefficients of self-diffusion of ferulic and sinapic acids in binary carbon tetrachloride–acetone-d6 solvent at temperatures of 278 and 298 K. Data show that the acids’ coefficients of self-diffusion grow along with the concentration of acetone-d6 and temperature. It is shown that the effective hydrodynamic radii of acids do not depend on the composition of the binary solvent within the experimental error. In light of molecular association, this behavior can be explained by the competition between two processes: acid–acetone heteroassociation and acid–acid self-association.

Quadratic spacing of the effective gradient area for spatially encoded diffusion NMR

Diffusion NMR experiments rely on the measurement of signal attenuation as a function of the area of diffusion-encoding pulsed magnetic-field gradients. In conventional experiments, arbitrary series of gradient values can be used, and different gradient spacing strategies have different advantages. Ultrafast diffusion NMR relies on the spatial parallelisation of effective gradient area values to collect full 2D diffusion data sets in a single scan. Until recently, only linear spacing was available. We have shown that quadratic spacing can be achieved using a tailored frequency swept pulse. Here we describe the design of the pulse and validate it with numerical spin simulations, that make it possible to check the effect of the quadratic spacing pulse at different stages of the pulse sequence. We also show that quadratic spacing makes it possible to use a recently reported analysis method for diffusion NMR, the Matrix Pencil Method. We describe the results obtained with the MPM and those obtained with the direct exponential curve resolution algorithm (DECRA), which also requires quadratic gradient spacing. Overall, these developments open new opportunities for applications of spatially encoded diffusion experiments, such as ultrafast DOSY NMR and ultrafast Laplace NMR.

Spinnoise gradient echoes.

Nuclear spinnoise spectroscopy in the absence of radio frequency pulses was studied under the influence of pulsed field gradients (PFGs) on pure and mixed liquids. Under conditions where the radiation-damping-induced line broadening is smaller than the gradient-dependent inhomogeneous broadening, echo responses can be observed in difference spectra between experiments employing pulsed field gradient pairs of the same and opposite signs. These observed spinnoise gradient echoes (SNGEs) were analyzed through a simple model to describe the effects of transient phenomena. Experiments performed on high-resolution nuclear magnetic resonance (NMR) probes demonstrate how refocused spin noise behaves and how it can be exploited to determine sample properties. In bulk liquids and their mixtures, transverse relaxation times and translational diffusion constants can be determined from SNGE spectra recorded following tailored sequences of magnetic field gradient pulses.

Practical computation of the diffusion MRI signal based on Laplace eigenfunctions: permeable interfaces.

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal, called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the tissue geometry and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. In a previous publication, we presented a simulation framework that we implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for biological cells subject to impermeable membrane boundary conditions. In this work, we extend our simulation framework to include geometries that contain permeable cell membranes. We describe the new computational techniques that allowed this generalization and we analyze the effects of the magnitude of the permeability coefficient on the eigendecomposition of the diffusion and Bloch-Torrey operators. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

Frequency-swept pulses for ultrafast spatially encoded NMR

Ultrafast NMR based on spatial encoding yields arbitrary multidimensional spectra in a single scan. The dramatic acceleration afforded by spatial parallelisation makes it possible to capture transient species and processes, and has notably been applied to the monitoring of reactions and the analysis of hyperpolarised species. At the heart of ultrafast NMR lies the spatially sequential manipulation of nuclear spins. This is virtually always achieved by combining a swept radio-frequency pulse with a magnetic field gradient pulse. The dynamics of nuclear spins during these pulse sequence elements is key to understand and design ultrafast NMR experiments, and can often be described by surprisingly simple models. This article describes the spatial encoding of relaxation, chemical shift and diffusion in a common framework and discusses directions for future developments.

Radiofrequency encoded Only Parahydrogen SpectroscopY

A new pulse sequence aimed to filter out NMR signals coming from thermally polarized protons in PHIP experiments based on the OPSY pulse sequence (Only Parahydrogen SpectroscopY) is presented. In analogy to OPSY, which removes thermal polarization by using a pair of magnetic field gradient pulses with an intensity ratio 1:2 and equal duration, the same effect can be achieved using inhomogeneous radiofrequency fields. The spatial dependence of the radiofrequency field is used to control the Hamiltonian, which results in an effective suppression of thermal contributions in the NMR signal, while PHIP originated signals remain unmodified. A theoretical model for the radiofrequency encoded only parahydrogen (REOPSY) sequence is presented along with an experimental implementation on a birdcage coil in a 7 T magnetic field. The control level achieved by this strategy allows the inclusion of a long train of refocusing pulses. Therefore, the new sequence can be combined with the parahydrogen discriminated PHIP (PhD-PHIP) pulse sequence as a detection block to improve sensitivity and resolution in a single–scan experiment. Experiments with REOPSY and REOPSY+PhD-PHIP are presented in thermally and hyperpolarized samples.

Diffusion Spectrum of Polymer Melt Measured by Varying Magnetic Field Gradient Pulse Width in PGSE NMR.

The translational motion of polymers is a complex process and has a big impact on polymer structure and chemical reactivity. The process can be described by the segment velocity autocorrelation function or its diffusion spectrum, which exhibit several characteristic features depending on the observational time scale—from the Brownian delta function on a large time scale, to complex details in a very short range. Several stepwise, more-complex models of translational dynamics thus exist—from the Rouse regime over reptation motion to a combination of reptation and tube-Rouse motion. Accordingly, different methods of measurement are applicable, from neutron scattering for very short times to optical methods for very long times. In the intermediate regime, nuclear magnetic resonance (NMR) is applicable—for microseconds, relaxometry, and for milliseconds, diffusometry. We used a variation of the established diffusometric method of pulsed gradient spin-echo NMR to measure the diffusion spectrum of a linear polyethylene melt by varying the gradient pulse width. We were able to determine the characteristic relaxation time of the first mode of the tube-Rouse motion. This result is a deviation from a Rouse model of polymer chain displacement at the crossover from a square-root to linear time dependence, indicating a new long-term diffusion regime in which the dynamics of the tube are also described by the Rouse model.

Superresolution localization of nitrogen-vacancy centers in diamond with quantum-controlled photoswitching

We demonstrate the super-resolution localization of the nitrogen vacancy centers in diamond by a novel fluorescence photoswitching technique based on coherent quantum control. The photoswitching is realized by the quantum phase encoding based on pulsed magnetic field gradient. Then we perform super-resolution imaging and achieve a localizing accuracy better than 1.4 nm under a scanning confocal microscope. Finally, we show that the quantum phase encoding plays a dominant role on the resolution, and a resolution of 0.15 nm is achievable under our current experimental condition. This method can be applied in subnanometer scale addressing and control of qubits based on multiple coupled defect spins.

Gradient-based pulse sequences for benchtop NMR spectroscopy

Benchtop NMR spectroscopy has been on the rise for the last decade, by bringing high-resolution NMR in environments that are not easily compatible with high-field NMR. Benchtop spectrometers are accessible, low cost and show an impressive performance in terms of sensitivity with respect to the relatively low associated magnetic field (40–100 MHz). However, their application is limited by the strong and ubiquitous peak overlaps arising from the complex mixtures which are often targeted, often characterized by a great diversity of concentrations and by strong signals from non-deuterated solvents. Such limitations can be addressed by pulse sequences making clever use of magnetic field gradient pulses, capable of performing efficient coherence selection or encoding chemical shift or diffusion information. Gradients pulses are well-known ingredients of high-field pulse sequence recipes, but were only recently made available on benchtop spectrometers, thanks to the introduction of gradient coils in 2015. This article reviews the recent methodological advances making use of gradient pulses on benchtop spectrometers and the applications stemming from these developments. Particular focus is made on solvent suppression schemes, diffusion-encoded, and spatially-encoded experiments, while discussing both methodological advances and subsequent applications. We eventually discuss the exciting development and application perspectives that result from such advances.

Practical computation of the diffusion MRI signal of realistic neurons based on Laplace eigenfunctions.

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch-Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b-values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch-Torrey operator and compute the Bloch-Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

Electron spin resonance microfluidics with subnanoliter liquid samples

Microfluidics is a well-established technique to synthesize, process, and analyze small amounts of materials for chemical, biological, medical, and environmental applications. Typically, it involves the use of reagents with a volume smaller than ~ 1 micro-l—ideally even nano- or picoliters. When the sample of interest contains paramagnetic species, it can in principle be quantified and analyzed by electron spin resonance (ESR) spectroscopy. However, conventional ESR is typically carried out with a sample volume of ~ 1 ml, thereby making it incompatible with most microfluidics applications. Here we show that by using a new class of miniature surface resonators combined with photolithography to prepare microfluidic patterns, ESR can be applied to measure small liquid samples, down to picoliter volumes, without considerably sacrificing concentration sensitivity. Our experiments, carried out with resonators whose mode volumes range from ~1 to 3.6 nL, showed that with a sample volume of ~0.25 nL good signals could be obtained from solutions with spin concentrations of less than 0.1 μM. The advantage of using microfluidics ESR is evident in our work, not only because it facilitates the use of a very small sample volume, but also because it makes it possible to apply huge ( ~ 1000 T/m) and fast ( ~ 1 μs) pulsed magnetic field gradients to the sample. This is a key capability to measuring unique properties such as nanoscale real-space diffusion and quantum spin diffusion. All our experiments are performed at room temperature, making our technique compatible with future microfluidics applications that might employ a complete system of compact resonators, microfluidic chips, miniature magnets, and a compact ESR-on-a-chip spectrometer. This could result in a completely new approach to processing and measuring paramagnetic liquid samples for use in a variety of chemical, biological, medical, and environmental applications.

High Field MicroMRI Velocimetric Measurement of Quantitative Local Flow Curves.

Performing rheo-microMRI velocimetry at a high magnetic field with strong pulsed field gradients has clear advantages in terms of (chemical) sensitivity and resolution in velocities, time, and space. To benefit from these advantages, some artifacts need to be minimized. Significant sources of such artifacts are chemical shift dispersion due to the high magnetic field, eddy currents caused by the pulsed magnetic field gradients, and possible mechanical instabilities in concentric cylinder (CC) rheo-cells. These, in particular, hamper quantitative assessment of spatially resolved velocity profiles needed to construct local flow curves (LFCs) in CC geometries with millimeter gap sizes. A major improvement was achieved by chemical shift selective suppression of signals that are spectroscopically different from the signal of interest. By also accounting for imperfections in pulsed field gradients, LFCs were obtained that were virtually free of artifacts. The approach to obtain quantitative LFCs in millimeter gap CC rheo-MRI cells was validated for Newtonian and simple yield stress fluids, which both showed quantitative agreement between local and global flow curves. No systematic effects of gap size and rotational velocity on the viscosity of a Newtonian fluid and yield stress of a complex fluid could be observed. The acquisition of LFCs during heterogeneous and transient flow of fat crystal dispersion demonstrated that local constitutive laws can be assessed by rheo-microMRI at a high magnetic field in a noninvasive, quantitative, and real-time manner.

SpinDoctor: A MATLAB toolbox for diffusion MRI simulation

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment Bloch-Torrey partial differential equation. Under the assumption of negligible water exchange between compartments, the time-dependent apparent diffusion coefficient can be directly computed from the solution of a diffusion equation subject to a time-dependent Neumann boundary condition.This paper describes a publicly available MATLAB toolbox called SpinDoctor that can be used 1) to solve the Bloch-Torrey partial differential equation in order to simulate the diffusion magnetic resonance imaging signal; 2) to solve a diffusion partial differential equation to obtain directly the apparent diffusion coefficient; 3) to compare the simulated apparent diffusion coefficient with a short-time approximation formula.The partial differential equations are solved by P1 finite elements combined with built-in MATLAB routines for solving ordinary differential equations. The finite element mesh generation is performed using an external package called Tetgen.SpinDoctor provides built-in options of including 1) spherical cells with a nucleus; 2) cylindrical cells with a myelin layer; 3) an extra-cellular space enclosed either a) in a box or b) in a tight wrapping around the cells; 4) deformation of canonical cells by bending and twisting; 5) permeable membranes; Built-in diffusion-encoding pulse sequences include the Pulsed Gradient Spin Echo and the Oscillating Gradient Spin Echo.We describe in detail how to use the SpinDoctor toolbox. We validate SpinDoctor simulations using reference signals computed by the Matrix Formalism method. We compare the accuracy and computational time of SpinDoctor simulations with Monte-Carlo simulations and show significant speed-up of SpinDoctor over Monte-Carlo simulations in complex geometries. We also illustrate several extensions of SpinDoctor functionalities, including the incorporation of T2 relaxation, the simulation of non-standard diffusion-encoding sequences, as well as the use of externally generated geometrical meshes.

T^{3} Stern-Gerlach Matter-Wave Interferometer.

We present a unique matter-wave interferometer whose phase scales with the cube of the time the atom spends in the interferometer. Our scheme is based on a full-loop Stern-Gerlach interferometer incorporating four magnetic field gradient pulses to create a state-dependent force. In contrast to typical atom interferometers that make use of laser light for the splitting and recombination of the wave packets, this realization uses no light and can therefore serve as a high-precision surface probe at very close distances.