Magnetic Field Gradient Research Articles

Estimate of in-band eddy current effect in space gravitational wave detection

Abstract In space detection of gravitational waves (GWs), the presence of a low-frequency varying magnetic field will generate eddy currents in the test mass, resulting in a varying magnetic moment. This magnetic moment will couple with a constant magnetic field gradient, producing residual acceleration within the frequency band of GW detection, causing the test mass to deviate from the free-falling mode. However, this effect has not yet been studied clearly in the noise budget for TianQin and LISA Pathfinder since the constant DC magnetic susceptibility was applied to quantify the varying magnetic moment. This paper utilizes the parameter of alternating current (AC) magnetic susceptibility to define this process, and further analyzes and evaluates the effect. In a general magnetic field environment, the contribution of this effect to TianQin acceleration noise reaches 20\% from 0.1 mHz to 3 mHz. The contribution to LISA noise is less than 10\% from $10^{-4}$ to 0.1 Hz, and less than 10\% below $10^{-4}$ Hz. This effect does not have a potential limit on LISA low-frequency science down to 20 $\mu$Hz [Phys. Rev. Lett. 120, 061101 (2018)].

τSPECT: a spin-flip loaded magnetic ultracold neutron trap for a determination of the neutron lifetime

The confinement of ultracold neutrons (UCNs) in three-dimensional magnetic field gradient or magneto-gravitational traps allows for a measurement of the free neutron lifetime τ n with superior control over loss channels related to UCNs interacting with material surfaces. The most precise measurement τ n has been achieved using a magneto-gravitational trap, in which UCN are prevented from escaping at the top of their trap by gravity. More compact horizontal confinement geometries with variable energy acceptance ranges can be obtained by using steep magnetic field gradients in all spatial directions, generated by combinations of either permanent or (variable) superconducting magnets. In this paper, we present the first successful implementation of a pulsed spin-flip based loading scheme to fill a three-dimensional magnetic trap with externally produced UCN. The measurements with the τSPECT experiment were performed at the pulsed UCN source of the research reactor TRIGA Mainz. The extracted neutron storage time constant of τ = 859(16) s is compatible with the most precise determinations of τ n . We report on detailed, but statistically limited, investigations of major systematic effects influencing the neutron storage time. The statistical limitations are mitigated by the relocation of the experiment to a stronger UCN source.

Experimental Assessment of Magnetic Resonance Imaging Distortion for Radiation Therapy Planning

MRI is widely used in radiation therapy planning, particularly in stereotactic radiosurgery for brain metastases. The article addresses the geometric distortion of MRI images, which can lead to errors in radiation therapy planning. A series of experiments with a simple phantom were conducted on two MRI scanners with 0.5 T and 1.5 T magnetic fields. Deviations in the positions of the phantom objects from their actual locations were observed. The detected deviations can reach up to 5 mm. A theoretical dependence of the magnetic field gradient on the distance to the center of homogeneity was calculated, which agrees well with the approximation of experimental data. An assessment was made of the dependence of the image area distortion of objects located at the center of the field homogeneity on their actual sizes. It was found that the area deviation can reach up to 20%.

Ferrimagnetic Tb/Co multilayers patterned by ion bombardment as substrates for magnetophoresis

Ion bombardment with 30 keV Ga+ ions can locally change the magnetic properties of perpendicular magnetic anisotropy ferrimagnetic Tb/Co based multilayers. The induced changes in the effective magnetization create high gradients of magnetic fields in the proximity of the perimeters of the bombarded areas. Superparamagnetic, micrometer-sized beads floating in an aqueous suspension over such a patterned structure respond to the ensuing magnetostatic energy landscape. This landscape, and its associated forces on the beads, can be controlled with a time varying, external, homogeneous magnetic field. It is shown that with a 3.37 kA/m (approx. 4.2 mT) field and switching the direction at 10 Hz frequencies the beads can be driven with average forward velocities reaching 40 μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu$$\\end{document}m/s. This has the potential for use in lab-on-a-chip type assays.

Skyrmion dynamics in attractive and repulsive local magnetic fields

The study of the behavior of magnetic skyrmions in local magnetic fields’ nanometric length-scale has gained increasing interest in recent years due to the theoretical proposal of magnetic skyrmion–superconducting vortex pairs as potential hosts for topologically protected bound states, which hold high promise for applications in quantum computing. From a magnetic interaction point-of-view, the key interest lies in understanding the skyrmion dynamics triggered by the magnetic energy landscape generated by the superconducting vortex. Here, we present a micromagnetic study of the dynamics of nanometric skyrmions inside a Gaussian magnetic field profile, which is used as a simplified version of the vortex magnetic flux. On the one hand, our calculations show that local non-linear magnetic fields can be very effective in controlling the dynamics of magnetic skyrmions; in particular, they offer the appealing possibility to manipulate skyrmions in a two dimensional space. On the other hand, they also show that the dynamics of a skyrmion in a local magnetic field can be manipulated via a uniform external magnetic field without any change in the magnetic field gradient. An analytical expression for the skyrmion velocity is given, and the corresponding microscopic dynamics are confirmed by the micromagnetic simulations. This work is expected to motivate more theoretical and experimental studies of the behavior of magnetic skyrmions in proximity to superconducting vortices.

A Statistical Examination of Interactions Between 1‐Hz Whistler Waves and Ions in the Earth's Foreshock

AbstractThe 1 Hz whistler wave precursors attached to shock‐like structures are often observed in the foreshock. Using observations from the Magnetospheric Multiscale mission, we investigate the interactions between 1 Hz waves and ions. Incoming solar wind ions do not cyclotron‐resonate with the wave, since typically the wave is right‐handed in their frame. We demonstrate that solar wind ions commonly exhibit 180° gyro‐phase bunching from the wave magnetic field, understanding it with a reconciled linear theory and non‐linear trapping theory for non‐cyclotron‐resonant modulations. Along the longitudinal direction, solar wind ions experience Landau resonance, exhibiting either modulations at small wave potentials or trapping in phase‐space holes at large potentials. The results also improve our understanding of foreshock structure evolution and 1 Hz wave excitation. Shock‐like structures start with having incoming solar wind and remotely reflected ions from further downstream. The ion‐scale 1 Hz waves can already appear during this stage. The excitation may be due to shock‐like dispersive radiation or kinetic instabilities resonant with these remotely reflected ions. Ions reflected by local shock‐like structures occur later, so they are not always necessary for generating 1 Hz waves. The wave leads to ion reflection further upstream, which may cause reformation of shock‐like structures. In one event, locally reflected ions exhibit non‐cyclotron‐resonant modulation in the early stage, and later approach the anomalous cyclotron resonant condition with gyro‐phases ∼270°. The latter is possibly due to nonlinear trapping in regions with an upstream‐pointing magnetic field gradient, linked to reformation. Some additional special features, such as frequency dispersions, are observed, encouraging further investigations.

Case Studies on Pre-eruptive X-class Flares using R-value in the Lower Solar Atmosphere

The R-value is a measure of the strength of photospheric magnetic polarity inversion lines in active regions (ARs). This work investigates the possibility of a relation between R-value variations and the occurrence of X-class flares in ARs, not in the solar photosphere, as usual, but above it in regions closer to where flares occur. The modus operandi is to extrapolate the Solar Dynamic Observatory’s Helioseismic and Magnetic Imager magnetogram data up to a height of 3.24 Mm above the photosphere and then compute the R-value based on the extrapolated magnetic field. Recent studies have shown that certain flare-predictive parameters such as the horizontal gradient of the vertical magnetic field and magnetic helicity may improve flare prediction lead times significantly if studied at a specific height range above the photosphere, called the optimal height range (OHR). Here, we define the OHR as a collection of heights where a sudden but sustained increase in R-value is found. For the eight case studies discussed in this paper, our results indicate that it is possible for OHRs to exist in the low solar atmosphere (between 0.36 and 3.24 Mm), where R-value spikes occur 48–68 hr before the first X-class flare of an emerging AR. The temporal evolution of R-value before the first X-class flare for an emerging AR is also found to be distinct from that of nonflaring ARs. For X-class flares associated with nonemerging ARs, an OHR could not be found.

Effect of colloidal magnetite (Fe3O4) nanoparticles on the electrical characteristics of the azolectin bilayer in a static inhomogeneous magnetic field

This work is devoted to the study of the combined effects of applied magnetic field and MNPs on the electrical characteristics of bilayer lipid membranes. We present results of the study of electrical parameters of azolectin membranes in a static inhomogeneous magnetic field at the one-sided addition of positively charged quasi-spherical superparamagnetic magnetite nanoparticles with a diameter of about 4 nm. The magnet was located at different distances from the membrane, and the magnetic field attracted the nanoparticles to the membrane surface with different strengths. We observed three pronounced effects that depended on the external magnetic field. Firstly, after addition of nanoparticles in a magnetic field, the conductance of the membranes increased. A smooth increase in conductance was accompanied in some cases by the appearance of current jumps, which can be associated with the formation of through pores with a radius of no more than 1 nm. The conductance increased with increasing magnetic field gradient. Secondly, at zero command voltage, a negative current through the membrane was observed. Although our experiments did not allow us to unambiguously determine which particles create this current, we believe that this current is associated with the penetration of particles through the membrane. This effect intensified with increasing magnetic field gradient. Thirdly, we observed a sharp change in the nonlinear dependence of capacitance on voltage associated both with the change in the surface potential of the azolectin membrane and with the effect of MNP binding to the membrane surface on the apparent membrane capacitance.

Use of the 5P3/2→6P3/2 electric dipole forbidden transition in Rb as a non-perturbing probe of atom dynamics in an operating magneto-optical trap

Abstract Results of spectroscopy experiments using the $5P_{3/2} \to 6P_{3/2}$ electric dipole forbidden transition in cold rubidium atoms are presented. Production of this forbidden transition is detected by observing the emission of the $420 $ nm fluorescence photons that result from the decay of the $6P_{3/2} $ state into the ground state. The experiments are performed under the steady state operation conditions of a magneto-optical trap (MOT), and thus provide non-perturbing information on the atom-light system. The hyperfine structure of the $6P_{3/2}$ level is completely resolved, and the fluorescence peaks show the expected Autler-Townes (AT) splitting of the emission lines. This hyperfine structure is used to calibrate the frequency scale of all spectra recorded. A combination of this absolute frequency scale and an expression for the AT profile allows an absolute determination of the effective Rabi frequency and detuning of the MOT. The behavior of these AT doublets is studied as a function of frequency detuning and also as a function of the trapping light intensity.
The experiments also take full advantage of the strong polarization dependence of the relative intensities of the hyperfine fluorescence components.
This study results in a sensitive probe of the relative populations of the magnetic sublevel projections relative to the trapping magnetic field gradient.
An almost isotropic population distribution was found, but small deviations from isotropy could be determined with this electric dipole forbidden probe.

Design of dual-layer heater based on genetic algorithm to optimize magnetic field gradient in vapor cell

Addressing the constraint of magnetic field gradients in the vapor cell on enhancing the sensitivity of atomic magnetometers, this paper proposed a dual-layer heater design based on genetic algorithms, effectively reduced the magnetic field gradients within the vapor cell. The study analyzed the influence of key parameters of the resistive wire, such as wire width, thickness, and spacing, on magnetic noise generation in the three-dimensional model of the heater. The parameter combinations were then optimized synchronously using genetic algorithms to reduce the magnetic field gradient in the vapor cell region and enhance the magnetic noise self-suppression capability of the heater. The simulation results confirmed that the magnetic field strength in most areas remains below 40 pT, and the magnetic field gradient was well-managed. Additionally, further magnetic field experiments demonstrated that the heater's current-generated magnetic field had a strong self-suppression effect on magnetic noise, as evidenced by the index k value of -0.05. This paper provides convincing technical support and experimental evidence for improving the performance of the SERF atomic magnetometer.

Understanding flame behaviors under gradient magnetic fields: The dynamics of non-reacting gas jets

Gradient magnetic fields impact fluid dynamics through Kelvin forces, altering flame behavior. Key complexities such as the interaction of magnetic fields with thermal gradients and chemical reactions within flames remain largely unexplored. By isolating variables such as heat release and thermal convection, this study aims to delve into fundamental mechanisms by which gradient magnetic fields affect the fluid dynamics of flames. This research adopts nitrogen, helium, argon and oxygen as jet gases, focusing exclusively on the fluid dynamics of non-reacting flow. Experimental techniques and Computational Fluid Dynamics (CFD) simulations are employed to explore the effects of varying gas species and magnetic field strengths. Results demonstrate that gas density significantly impacts jet behavior, with paramagnetic oxygen concentrating in areas of stronger magnetic fields, and other gases like helium, nitrogen, and argon showing distinct behaviors based on their molar mass. The study also establishes an optimized energy conservation equation that accurately predict jet height, considering factors such as buoyant and Kelvin forces. By studying the flow behavior of non-reacting gases under magnetic fields, the results are ultimately extended to estimate flame heights and elucidate the mechanisms of air-to-fuel diffusion.

Silk Fibroin Film Decorated with Ultralow FeCo Content by Sputtering Deposition Results in a Flexible and Robust Biomaterial for Magnetic Actuation.

Magnetically responsive soft biomaterials are at the forefront of bioengineering and biorobotics. We have created a magnetic hybrid material by coupling silk fibroin─i.e., a natural biopolymer with an optimal combination of biocompatibility and mechanical robustness─with the FeCo alloy, the ferromagnetic material with the highest saturation magnetization. The material is in the form of a 6 μm-thick silk fibroin film, coated with a FeCo layer (nominal thickness: 10 nm) grown by magnetron sputtering deposition. The sputtering deposition technique is versatile and eco-friendly and proves effective for growing the magnetic layer on the biopolymer substrate, also allowing one to select the area to be decorated. The hybrid material is biocompatible, lightweight, flexible, robust, and water-resistant. Electrical, structural, mechanical, and magnetic characterization of the material, both as-prepared and after being soaked in water, have provided information on the adhesion between the silk fibroin substrate and the FeCo layer and on the state of internal mechanical stresses. The hybrid film exhibits a high magnetic bending response under a magnetic field gradient, thanks to an ultralow fraction of the FeCo component (less than 0.1 vol %, i.e., well below 1 wt %). This reduces the risk of adverse health effects and makes the material suitable for bioactuation applications.

The added value of diffusion tensor imaging with systematic bias correction for the assessment of liver morphology and physiology.

Diffusion-weighted images of the human liver are prone to artifacts from bulk motions, poor SNR, non-uniformity of magnetic field gradients, and non-optimal choice of diffusion weightings. These factors markedly affect diffusion tensor imaging (DTI) metrics such as mean diffusivity (MD) and fractional anisotropy (FA). This work presents a simple preprocessing pipeline for enhanced magnetic field gradient non-uniformity calibration and analysis of the systematic bias removal attained in each correction step. Liver DTI scans were conducted in two isotropic phantoms and one healthy volunteer. Diffusion tensor was calculated for the original data and after denoising, B1 correction, rigid body registration, and magnetic field gradient non-uniformity correction applying the B-matrix spatial distribution (BSD) method and then, compared with the standard approach (sDTI). MD and FA were determined in three segments of the right lobe from DTI using four different combinations of b-values from the set 0, 400, and 800 s/mm2. Results showed that the proposed preprocessing and BSD methods have a significant impact on MD and FA values in off- and iso-centered isotropic phantoms. The applied corrections applied to the human liver resulted in a 11% change in MD and a - 64% change in FA. By manipulating the b-values used in the diffusion tensor calculation, DTI metrics that reflect only morphology or additional information about liver tissue physiology can be obtained. Accurate quantification of the human liver by diffusion requires appropriate preprocessing and carefully chosen b-value. Noise, B1 inhomogeneity, mis-registration, and non-uniform magnetic field gradients significantly change distributions of DTI metrics in isotropic phantoms and the human liver. Basic preprocessing and the B-matrix spatial distribution (BSD) method perform differently for off-center and isocenter locations. In the human liver, they removed systematic bias of FA and MD by up to -63% and 11%, respectively. Visible variability of FA and MD among b-value sets indicates the possibility of DTI sensitization to different liver compartments.

Superconducting surface trap chips for microwave-driven trapped ions

Microwave-driven trapped ion logic gates offer a promising avenue for advancing beyond laser-based logic operations. In future microwave-based operations, however, the joule heat produced by large microwave currents flowing through narrow microwave electrodes would potentially hinder improvements in gate speed and fidelity. Moreover, scalability, particularly in cryogenic trapped ion systems, is impeded by the excessive joule heat. To address these challenges, we present a novel approach: superconducting surface trap chips that integrate high-Q microwave resonators with large current capacities. Utilizing sub-ampere microwave currents in superconducting Nb resonators, we generate substantial magnetic field gradients with significantly reduced losses compared to conventional metal chips. By harnessing the high Q factors of superconducting resonators, we propose a power-efficient two-qubit gate scheme capable of achieving a sub-milliwatt external microwave input power at a gate Rabi frequency of 1 kHz.

Effective phase diffusion for spin phase evolution under random nonlinear magnetic field.

The general theoretical description of spin self-diffusion under a nonlinear gradient magnetic field is proposed, which extends the effective phase diffusion method for a linear gradient field. Based on the phase diffusion, the proposed method reveals the general features of phase evolutions in nonlinear gradient fields. There are three types of phase evolutions: phase diffusion, float phase evolution, and shift evolution based on the starting position. For spin diffusion near the origin of the nonlinear field, these three phase evolutions significantly affect the nuclear magnetic resonance (NMR) signal. The traditional methods have difficulties in handling these three-phase evolutions. Notably, the phase from float phase evolution is missed or misplaced in traditional methods, which leads to incorrect NMR signal attenuation or phase shift. The method here shows that the diffusing and float phase evolutions come from the first and second derivatives of the gradient field. Based on these three phase evolutions, the phase variance and corresponding NMR signal attenuation are obtained, as demonstrated by calculating the phase diffusions under both parabolic and cubic fields. The results indicate that signal attenuation obeys Gaussian attenuation for a short time, then changes to follow Lorentzian or Mittag-Leffler function attenuations as time increases, significantly different from Gaussian attenuation. For spins starting diffusion far away from the origin of the field gradient, the signal attenuation is Gaussian, but the float phase still has an important effect on the total phase shift of even-order gradient fields, which could be used to measure the diffusion coefficient directly. Random walk simulations were performed, which support the obtained theoretical results. General theoretical expressions are obtained, which can handle random order nonlinear gradient fields. The results could help develop advanced experimental techniques based on a nonlinear gradient field in NMR and magnetic resonance imaging.

Fabrication of Magnetic Ferrofluid Microrobot for Tissue Mechanical Measurement

Ferrofluids are a uniform mixture of magnetic nanoparticles and a carrier liquid, such as water or oil. They have become increasingly popular in the medical field due to their biocompatibility, manipulability, and their wide range of possible medical applications, one of them being mechanical measurement in human tissues. Our goal is to fabricate a bio-compatible ferrofluid droplet to perform these mechanical measurements. To fabricate the ferrofluid, magnetite nanoparticles (Fe3O4) were chosen along with vegetable oil as the carrier liquid due to their biocompatibility. The magnetic nanoparticles were also coated with oleic acid to serve as a lubricant layer to prevent them from agglomerating. We first coated the magnetic particles (Nanoshel) with oleic acid (Sigma). The magnetic particles were added to distilled water that was heated up to 60oC and mechanically stirred at 500 rpm. For each gram of magnetic nanoparticles used, an equal amount of oleic acid was added into the mixture and stirred for 15 minutes. The coated particles were then rinsed with distilled water followed by ethyl alcohol and dried. To fabricate the ferrofluid, we mixed the vegetable oil with the nanoparticles overnight to ensure uniform mixing with weight rations including 30%wt, 40%wt, and 50%wt, to find the optimal ratio for magnetic control and stability. To validate the technology using ferrofluid to perform tissue mechanical measurement, a droplet of ferrofluid was injected into the agar gel mimicking the stiffness of tissue and then actuated with a magnet. The ferrofluid showed observable deformation under a microscope after magnetic actuation. Our next steps will include repeating this process under a controlled gradient magnetic field while using a camera to track its deformation and movement and from there the hope is to perform on human tissue.

Hydromagnetic Non-Newtonian Nanofluid Flow Past Linealy Stretching Convergent-Divergent Conduit with Chemical Reaction

This paper investigates hydromagnetic non-Newtonian nanofluid flow past linearly stretching convergent-divergent conduit with chemical reaction using spectral ralaxation method. The fluid considered here is electrically conducting and is subjected to a constant pressure gradient and variable magnetic field. The two non-parallel walls are assumed not to intersect and the angle between the inclined walls is <i>θ</i>. The governing equations are continuity equation, momentum equation, species concentration, induction equation and energy equation. On modelling, the resulting partial differential equations are non-linear and are first transformed into system of ordinary differential equations through similarity transformation. The resulting boundary value problem is solved numerically using Spectral Relaxation Method. The results obtained after varying Hartman number, Unsteadiness parameter, Reynolds number, Solutal and Thermal Grashof number on velocity, concentration, temperature and induction profiles are represented in form of graphs. Some of the application of this study are, when extracting the energy from earth crust that varies in length between five to ten kilometres and temperature in between 500° and 1000°, nano-fluids are employed to cool the machinery and equipment working under high friction and high temperature. This present study considers nanofluid acting as a coolant of such equipment as well as acting as a lubricant thus reducing the rate of wear and tear of the equipment. The copper-water increases the thermophysical properties thus increasing heat transfer coefficient and hence increasing cooling rate.

Dynamically Reconfigurable Micro-Patterned Hydrogels Based on Magnetic Pickering Emulsion Droplets.

Reconfigurability within hydrogels has emerged as an attractive functionality that can be used in information encryption, cargo/delivery, environmental remediation, soft robotics, and medicine. Here micro-patterned polymer hydrogels capable of temperature-dependent reconfigurability are fabricated. For this, the hydrogels are provided with micron-sized Pickering emulsion droplets stabilized by magnetic particles, which are capable of harnessing energy from external force fields. The droplets can both migrate under magnetic field gradients and heat the environment when laser irradiated. These functions not only affect a single compartment but have higher-order effects on the mesoscale, thanks to the temperature-responsiveness of the polymeric network. This double responsiveness is exploited to control the spatial organization of hundreds of droplets within the hydrogel matrix and form predesigned and sophisticated patterns. Furthermore, pattern self-reconfiguration driven by the droplets themselves upon laser irradiation is induced. Finally, we show that due to their internal liquid phase, the droplets can be used as reservoirs of hydrophobic nutrients for living cells (i.e., Yarrowia lipolytica yeast) in the solid-like environment of the polymeric network, and demonstrate communication between the droplets and the cells to facilitate nutrient uptake. Altogether, the results provide opportunities for the development of stimuli-sensitive polymer hydrogels with post-synthesis reprogrammable response using micro-compartments as building blocks.

Dynamic fringe field effect of the rapid ramping quadrupole magnets at a Rapid Cycling Synchrotron

For a DC quadrupole magnet, the normalized magnetic field distribution along the beam trajectory is mainly determined by the mechanical structure of the magnet and remains almost independent of the exciting current. However, the magnetic field measurement results of the Rapid Cycling Synchrotron (RCS) at the China Spallation Neutron Source (CSNS) show that the normalized magnetic field distribution along the beam trajectory varies during the exciting current ramping process for rapid ramping magnets. Consequently, for rapid ramping quadrupole magnets, there are significant differences between the dynamic and static magnetic field gradient distribution. And the changes in the distribution of the magnetic field gradient can cause time dependent tune-shifts during the exciting current ramping process. For RCS, the tune is crucial, and the shifts of tune may lead to the beam approaching the resonance line, thereby causing beam loss. The impact of the fringe field effect of the rapid ramping quadrupole magnets on beam dynamics at CSNS-RCS was detailly studied. A new method for correcting the time-dependent tune shifts caused by the dynamic field distribution, based on waveform compensation at the quadrupole magnets, has also been proposed for the RCS of CSNS. Related simulation and experiment have been conducted, and the simulation results and experimental results are consistent with theoretical study results.

Nonparametric distributions of tensor-valued Lorentzian diffusion spectra for model-free data inversion in multidimensional diffusion MRI.

Magnetic resonance imaging (MRI) is the method of choice for noninvasive studies of micrometer-scale structures in biological tissues via their effects on the time- and frequency-dependent (restricted) and anisotropic self-diffusion of water. While new designs of time-dependent magnetic field gradient waveforms have enabled disambiguation between different aspects of translational motion that are convolved in traditional MRI methods relying on single pairs of field gradient pulses, data analysis for complex heterogeneous materials remains a challenge. Here, we propose and demonstrate nonparametric distributions of tensor-valued Lorentzian diffusion spectra, or "D(ω) distributions," as a general representation with sufficient flexibility to describe the MRI signal response from a wide range of model systems and biological tissues investigated with modulated gradient waveforms separating and correlating the effects of restricted and anisotropic diffusion.