Inhomogeneous Magnetic Field Research Articles

Stimulus-induced rotary saturation imaging of visually evoked response: A pilot study.

Spin-lock (SL) pulses have been proposed to directly detect neuronal activity otherwise inaccessible through standard functional magnetic resonance imaging. However, the practical limits of this technique remain unexplored. Key challenges in SL-based detection include ultra-weak signal variations, sensitivity to magnetic field inhomogeneities, and potential contamination from blood oxygen level-dependent effects, all of which hinder the reliable isolation of neuronal signals. This pilot study evaluates the performance of the stimulus-induced rotary saturation (SIRS) technique to map visual stimulation response in the human cortex. A rotary echo spin-lock (RESL) preparation followed by a 2D echo planar imaging readout was used to investigate 12 healthy subjects at rest and during continuous exposure to 8Hz flickering light. The SL amplitude was fixed to the target neuroelectric oscillations at that frequency. The signal variance was used as contrast metric, and two alternative post-processing pipelines (regression-filtering-rectification and normalized subtraction) were statistically evaluated. Higher variance in the SL signal was detected in four of the 12 subjects. Although group-level analysis indicated activation in the occipital pole, analysis of variance revealed that this difference was not statistically significant, highlighting the need for comparable control measures and more robust preparations. Further optimization in sensitivity and robustness is required to noninvasively detect physiological neuroelectric activity in the human brain.

Cosmic Ray Diffusion in Magnetic Fields Amplified by Nonlinear Turbulent Dynamo

The diffusion of cosmic rays (CRs) in turbulent magnetic fields is fundamental to understanding various astrophysical processes. We explore the CR diffusion in the magnetic fluctuations amplified by the nonlinear turbulent dynamo in the absence of a strong mean magnetic field. Using test particle simulations, we identify three distinct CR diffusion regimes: mirroring, wandering, and magnetic moment scattering (MMS). With highly inhomogeneous distribution of the dynamo-amplified magnetic fields, we find that the diffusion of CRs is also spatially inhomogeneous. Our results reveal that lower-energy CRs preferentially undergo the mirror and wandering diffusion in the strong-field regions, and the MMS diffusion in the weak-field regions. The former two diffusion mechanisms play a more important role toward lower CR energies, resulting in a relatively weak energy dependence of the overall CR mean free path (MFP). In contrast, higher-energy CRs predominantly undergo the MMS diffusion, for which the incomplete particle gyration, i.e., the limit case of mirroring, in strong fields has a more significant effect than the scattering by small-scale field tangling/reversal. Compared with lower-energy CRs, they are more poorly confined in space and their MFPs have a stronger energy dependence. We stress the fundamental role of magnetic field inhomogeneity of nonlinear turbulent dynamo in causing the different diffusion behavior of CRs compared to that in sub-Alfvénic magnetohydrodynamic turbulence.

Frequency Shift Caused by Magnetic Field and Light Intensity Inhomogeneity in Optically Detected Clock

Frequency Shift Caused by Magnetic Field and Light Intensity Inhomogeneity in Optically Detected Clock

Zero-echo-time sequences in highly inhomogeneous fields.

Zero-echo-time (ZTE) sequences have proven a powerful tool for MRI of ultrashort tissues, but they fail to produce useful images in the presence of strong field inhomogeneities (14 000 ppm). Here we seek a method to correct reconstruction artifacts from non-Cartesian acquisitions in highly inhomogeneous , where the standard double-shot gradient-echo approach to field mapping fails. We present a technique based on magnetic field maps obtained from two geometric distortion-free point-wise (SPRITE) acquisitions. To this end, we employ three scanners with varying field homogeneities. These maps are used for model-based image reconstruction with iterative algebraic techniques (ART). For comparison, the same prior information is fed also to widely used Conjugate Phase (CP) algorithms. Distortions and artifacts coming from severe inhomogeneities, at the level of the encoding gradient, are largely reverted by our method, as opposed to CP reconstructions. This holds even close to the limit where intra-voxel bandwidths (determined by inhomogeneities, up to 1.2 kHz) are comparable to the encoding inter-voxel bandwidth (determined by the gradient fields, 625 Hz in this work). We have benchmarked the performance of a new method for ZTE imaging in highly inhomogeneous magnetic fields. For example, this can be exploited for dental imaging in affordable low-field MRI systems, and can be expanded for arbitrary pulse sequences and extreme magnet geometries, as in, for example, single-sided MRI.

The Effect of Technical Operational Parameter on the Performance Stability of RF-Power Transfer for Cyclotron DECY-13 in Yogyakarta

Abstract Radio Frequency (RF) power transfer needs special condition in order to achieve high transmission coefficient, the matching condition between source and load must be met. In this case there are 3 important technical operation parameters, namely conductivity of cooling water, vacuum pressure, and magnetic field. They have influence on the impedance of the RF-generator load. An RF-generator of the frequency of 77.76 MHz having power up to 10 kW has been used in the construction of self-built cyclotron DECY-13 by BRIN in Yogyakarta. The experiment showed the increasing of power reflectivity up to 61% by rising of water conductivity 16%, i.e. 0.97 µS/cm, that used for cooling of the entire system inside the cyclotron chamber. The results also showed that the homogeneity of magnetic field from 1,5 to 2,0 ppm has changed the detected of impedance to be ~10 Ω at its real part and ~5 Ω at its imaginary part, by using network analyser based on Schmidt Chart analysis. On the other hand, the effect of vacuum pressure starting at 10−3 mbar to high vacuum have no significant effect on its real and imaginary parts of impedance.

Implementing a two-stage, shim field-calibrated superconducting shimming method on a 7 T cryogen-free small animal MRI magnet

Ultrahigh field systems (≥ 7 T) can increase the signal-to-noise ratio of magnetic resonance imaging (MRI), improving imaging performance compared to systems with lower fields. However, these enhancements heavily rely on a high B0 magnetic field homogeneity level, which can be achieved through superconducting shimming. This paper presents a novel two-stage superconducting shimming method designed to achieve precise shimming for a 7 T MRI superconducting magnet. In the initial stage, detailed measurements and fittings were conducted to determine the current polarity and the axial or circumferential positions of the shim fields. Subsequently, an optimization strategy was implemented to determine the optimal shim currents with a flexible target field. The second stage involves an iterative process to fine-tune the current of a specific shim coil, identified as having the most significant impact on field homogeneity. The overall fitness of 99.5% underscores the precision in determining the current polarity and position of the shim fields. Significantly, the calibrated shim system substantially improves the peak-to-peak and Root Mean Square Error (RMSE) field homogeneities from 107.42 ppm and 37.00 ppm to 11.12 ppm and 3.26 ppm, respectively, representing improvements of 89.65% and 91.19%. Furthermore, the simulation results of the fine-tuning stage demonstrate additional enhancements in peak-to-peak field homogeneity, to 9.9 ppm by reducing the current of the Z2 shim coil by 51.3 mA. Additionally, the shimmed magnetic field exhibited high time stability, with a maximum variation of only 27 µT observed within 48 h. Thus, the proposed two-stage superconducting shimming framework effectively addresses the challenge of imperfect B0 magnetic fields, enhancing peak-to-peak and RMSE field homogeneity. The stepwise optimized approach also mitigates deviations caused by shim-to-shim coupling, demonstrating its efficacy in achieving precise shimming in ultrahigh-field MRI systems.

Uniform microwave field formation for control of ensembles of negatively charged nitrogen vacancy in diamond.

The homogeneity of the microwave magnetic field is essential in controlling a large volume of ensemble spins, for example, in the case of sensitive magnetometry with nitrogen-vacancy (NV) centers in diamond. This is particularly important for pulsed measurement, where the fidelity of control pulses plays a crucial role in its sensitivity. So far, several magnetic field-forming systems have been proposed, but no detailed comparison has been made. Here, we numerically study the homogeneity of five different systems, including a planar antenna, a dielectric resonator, a cylindrical inductor, a barrel-shaped coil, and a nested barrel-shaped coil. The results of the simulation allowed us to optimize the design parameters of the barrel-shaped field-forming system, which led to significantly improved magnetic field uniformity. To measure this effect, we experimentally compared the homogeneity of a field-forming system having a barrel shape with that of a planar field-forming system by measuring Rabi oscillations of an ensemble of NV centers with them. Significant improvements in inhomogeneity were confirmed in the barrel-shaped coil.

Regulating mechanisms of external axial magnetic field on flow and heat transfer behavior for process optimization in plasma beam polishing

Plasma beam polishing has emerged as an innovative alternative to conventional metal finishing techniques. However, the precision machining ability of plasma arc encounters challenges due to the formation of surface undulations, while previous research has predominantly focused on parameter adjustments. In this paper, the feasibility of optimizing the plasma beam polishing process was validated through magnetic field regulation, as the surface topography and subsurface structure were obviously flattened. Given this, a micro plasma arc model under inhomogeneous axial magnetic field was established to explore the regulating mechanisms of the external axial magnetic field on the flow and heat transfer behavior of the plasma arc. Through this model, the plasma characteristics (magnetic induction intensity, electric potential, arc current, Lorentz force, plasma velocity and electron temperature) were comprehensively discussed. The results indicate that the external shape of the plasma arc is determined by the self-induced magnetic field, while the externally applied axial magnetic field significantly influences the internal flow of the plasma arc. The axial and radial magnetic fields generated by the excitation coil can induce circumferential rotation by dragging the plasma arc with Lorentz force, creating a more suitable energy beam for the polishing process. To verify the model, the arc temperature was measured using spectroscopic diagnostics, and the arc shape was captured using a high-speed camera, both of which were basically consistent with the simulation results.

Formation and Dynamics of Droplets in a Magnetic Fluid in Microchannels in an Inhomogeneous Magnetic Field of a Ring Magnet

Formation and Dynamics of Droplets in a Magnetic Fluid in Microchannels in an Inhomogeneous Magnetic Field of a Ring Magnet

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.

Spatially-resolved spectroscopic investigation of the inhomogeneous magnetic field effects on a low-pressure capacitively-coupled nitrogen plasma

Although magnetized plasmas have been frequently used to enhance the process rate or improve the film quality via the control of ion flux as well as energy and plasma density in semiconductor processes, the inhomogeneous magnetic field—which leads to plasma non-uniformity—remains as a problem to be solved. To address this problem, it is essential to conduct a comprehensive assessment of the magnetic effect throughout the entire discharge. Therefore, in the present study, we investigated the magnetic field effects (B < 100 G) on a capacitively-coupled nitrogen plasma based on spectroscopic analyses. The spatially-resolved emission spectra were measured along the radial direction at various vertical positions under the pressures of 10 mTorr and 250 mTorr both with and without magnetic field. By analyzing emission spectra such as N2 FPS, N2 SPS, N2+ FNS, and N I, we were able to obtain the radial distributions of reactive species density, vibrational temperature, and excitation temperature. In low-pressure plasma, with the application of a magnetic field, maximum increases in vibrational temperature and excitation temperature of 462 K and 491 K, respectively, were observed within the bulk region beneath the magnet. This magnetic effect resulted in a significant increase in reactive species density along the radial direction. It was also found that the local enhancement of ion density by magnetic field was strongly related to the increase in excitation temperature and the density of the N2+(B) state. From this result, it is suggested that introducing an asymmetric magnetic field could modulate the spatial distributions of the physical and chemical properties of the plasma.

Compact Electron Paramagnetic Resonance on a Chip Spectrometer Using a Single Sided Permanent Magnet.

Electron paramagnetic resonance (EPR) spectroscopy provides information about the physical and chemical properties of materials by detecting paramagnetic states. Conventional EPR measurements are performed in high Q resonator using large electromagnets which limits the available space for operando experiments. Here we present a solution toward a portable EPR sensor based on the combination of the EPR-on-a-Chip (EPRoC) and a single-sided permanent magnet. This device can be placed directly into the sample environment (i.e., catalytic reaction vessels, ultrahigh vacuum deposition chambers, aqueous environments, etc.) to conduct in situ and operando measurements. The EPRoC reported herein is comprised of an array of 14 voltage-controlled oscillator (VCO) coils oscillating at 7 GHz. By using a single grain of crystalline BDPA, EPR measurements at different positions of the magnet with respect to the VCO array were performed. It was possible to create a 2D spatial map of a 1.5 mm × 5 mm region of the magnetic field with 50 μm resolution. This allowed for the determination of the magnetic field intensity and homogeneity, which are found to be 254.69 mT and 700 ppm, respectively. The magnetic field was mapped also along the vertical direction using a thin film a-Si layer. The EPRoC and permanent magnet were combined to form a miniaturized EPR spectrometer to perform experiments on tempol (4-hydroxy-2,2,6,6-teramethylpiperidin-1-oxyl) dissolved in an 80% glycerol and 20% water solution. It was possible to determine the molecular tumbling correlation time and to establish a calibration procedure to quantify the number of spins within the sample.

Simulation of the response of a magnetic polymersome in an inhomogeneous magnetic field

Purpose. Identification of the behavior of a magnetic polymersome placed in an inhomogeneous field of a point dipole using coarse-grained molecular dynamics simulations.Methods. The system under the study is represented as a set of interacting particles of two types: polymeric particles simulating a double layer of an amphiphilic membrane, and magnetic nanoparticles located in the membrane layer. Polymeric particles interact through elastic potentials designed to maintain an equilibrium spherical vesicular geometry. Magnetic nanoparticles interact with each other and external fields as point dipoles. The excluded volume and steric interaction of magnetic particles with polymeric walls are modeled in the form of soft repulsion. The entire system is under isothermal conditions, and the model parameters are chosen according to typical interaction energies relative to thermal fluctuations. A model polymersome with a diameter of about 100 nm in an aqueous solution at 25°C is considered. An inhomogeneous magnetic field is created by a dipole of constant direction located at a fixed distance. The dynamics of the system is monitored by numerical solution of the equations of particles motion with introduced interactions and imposed conditions.Results. In numerical experiments, the response of the polymersome to a magnetic field was obtained for three values of the parameter describing the inhomogeneity of the field. The problem with a gradual increase in the strength (and its gradient) of the magnetic field near the polymersome is considered. Changes in the magnetization of the system are shown, and the redistribution of the concentration of nanoparticles is analyzed. A simulation of the situation when the center of the polymersome at the initial moment was placed at a point with a fixed value of the magnetic field for three cases of inhomogeneity of the field was carried out. A significant restructuring of the magnetic layer of the vesicle combined with movement of the entire capsule was detected. Estimates are made about the average velocity based on the displacement of the object.Conclusion. The model allows to evaluate the features of the combined magnetic, structural and mechanical response of the polymersome to an inhomogeneous field in the context of potential applications for controlled delivery of contents into cells.

Mechanical properties of silicone polymer with magnetic filler in magnetic field

Purpose. To investigate the change in the mechanical properties of a magnetorheological silicone elastomer consisting of a polymer filled with magnetite nanoparticles under the influence of an inhomogeneous magnetic field of an electromagnet. Methods. The experiments were carried out on a magnetic response research facility developed and manufactured independently based on known methods. The value of the deflection angle of the magnetically active receiver was determined by the optical method. Two-component silicone rubbers filled with magnetite particles were studied as samples. The manufactured samples differed in geometric dimensions, magnetic phase concentrations of 1%, 5%, 10% and 20%, and polymerization mechanism. The source of the magnetic field was electromagnets of various sizes connected to power sources. The images were captured using a Micmed 5.0 digital USB microscope.Results. The analysis of the structure of the manufactured magnetorheological silicone elastomers was carried out, as well as studies of the effect of the magnetic field strength and sample parameters on the deflection angle of the magnetically active cantilever. A theoretical interpretation of the obtained results is proposed.Conclusion. The experimental dependence of the tilt angle of a magnetically active polymer cantilever on the equilibrium position relative to the magnitude of the magnetic field strength of the electromagnet is determined. The obtained research results can be used to develop actuators and magnetoactive sensors.

Influence of inhomogeneous magnetic field source location on the intensity of thermomagnetic convection in a closed loop

Purpose. Obtaining information on the influence of the location of the inhomogeneous magnetic field source relative to the heated section of a vertical hydrodynamic loop filled with magnetic fluid on the intensity of convective heat transfer along the loop.Methods. Experiments were carried out using a hydrodynamic loop made of a thin tube of circular cross-section and placed in a vertical plane. Heat source was carried out by a heater on a short vertical section of the loop, and heat removal was implement out by blowing the entire surface of the tube with thermostated air. The source of the magnetic field was the flat pole tips of the ferrite magnetic core, in the gap between which the heater was located. The position of the pole tips relative to the heater varied vertically in the experiments. In the control experiments, the source of the magnetic field was deleted. The circuit was filled with medium concentrated magnetic liquid of the type "magnetite - kerosene - oleic acid". The intensity of steady-state convective heat flow along the tube was calculated from the results of measuring the tube surface temperature by copper-constantane thermocouples. The measurement results were presented in dimensionless form - the relationship between the Nusselt number and Rayleigh number.Results. The unfluctuating mixed, thermomagnetic and gravitational, convection of the magnetic fluid in the loop was observed at any location of the pole tips of the magnetic core relative to the heater. At location of pole tips above the heater, competition of gravitational and thermomagnetic convection was observed, and the heat flux was weak. When the pole tips were placed below the heater, the Nusselt number was 2 - 4 times higher than in the control tests (only gravitational convection) with equal Rayleigh numbers. The highest Nusselt numbers were obtained when the field source was placed in center of the heater.Conclusion. Information on the influence of the relative location of the magnetic field source and the heater on convective heat transfer by magnetic fluid in a hydrodynamic loop is obtained experimentally. The optimal concerning of heat transfer intensity position of the field source was found.

Metabolic imaging in living plants: A promising field for chemical exchange saturation transfer (CEST) MRI.

Magnetic resonance imaging (MRI) is a versatile technique in the biomedical field, but its application to the study of plant metabolism in vivo remains challenging because of magnetic susceptibility problems. In this study, we report the establishment of chemical exchange saturation transfer (CEST) for plant MRI. This method enables noninvasive access to the metabolism of sugars and amino acids in complex sink organs (seeds, fruits, taproots, and tubers) of major crops (maize, barley, pea, potato, sugar beet, and sugarcane). Because of its high signal detection sensitivity and low susceptibility to magnetic field inhomogeneities, CEST analyzes heterogeneous botanical samples inaccessible to conventional magnetic resonance spectroscopy. The approach provides unprecedented insight into the dynamics and distribution of sugars and amino acids in intact, living plant tissue. The method is validated by chemical shift imaging, infrared microscopy, chromatography, and mass spectrometry. CEST is a versatile and promising tool for studying plant metabolism in vivo, with many applications in plant science and crop improvement.

Development and Testing of a Helicon Plasma Thruster Based on a Magnetically Enhanced Inductively Coupled Plasma Reactor Operating in a Multi-Mode Regime

A disruptive Electric Propulsion system is proposed for next-generation Low-Earth-Orbit (LEO) small satellite constellations, utilizing an RF-powered Helicon Plasma Thruster (HPT). This system is built around a Magnetically Enhanced Inductively Coupled Plasma (MEICP) reactor, which enables acceleration of quasi-neutral plasma through a magnetic nozzle. The MEICP reactor features an innovative design with a multi-dipole magnetic confinement system, generated by neodymium iron boron (NdFeB) permanent magnets, combined with an azimuthally asymmetric half-wavelength right (HWRH) antenna and a variable-section ionization chamber. The plasma reactor is followed by a solenoid-free magnetic nozzle (MN), which facilitates the formation of an ambipolar potential drop, enabling the conversion of electron thermal energy into ion beam energy. This study explores the impact of an inhomogeneous magnetic field on the heating mechanism of the HPT and highlights its multi-mode operation within a pulsed power range of 200 to 500 W of RF. The discharge state, characterized by high-energy electron-excited ions and low-energy excited neutral particles in the plasma plume, was analyzed using optical emission spectroscopy (OES). The experimental testing campaign, conducted under pulsed power excitation, reveals that, as RF input power increases, the MEICP reactor transitions from inductive (H-mode) to wave coupling (W-mode) discharge modes. Spectrograms, electron temperature, and plasma density measurements were obtained for the Helicon Plasma Thruster within its operational envelope. Based on OES data, the ideal specific impulse was estimated to exceed 1000 s, highlighting the significant potential of this technology for future LEO/VLEO space missions.

Repeatability of alkaline inorganic phosphate quantification in the skeletal muscle using 31P-magnetic resonance spectroscopy at 3T.

The detection of a secondary inorganic phosphate (Pi) resonance, a possible marker of mitochondrial content in vivo, using phosphorus magnetic resonance spectroscopy (31P-MRS), poses technical challenges at 3Tesla (T). Overcoming these challenges is imperative for the integration of this biomarker into clinical research. To evaluate the repeatability and reliability of measuring resting skeletal muscle alkaline Pi (Pialk) using with 31P-MRS at 3 T. After an initial set of experiments on five subjects to optimize the sequence, resting 31P-MRS of the quadriceps muscles were acquired on two visits (~4days apart) using an intra-subjects design, from 13 sedentary to moderately active young male and female adults (22 ± 3years old) within a whole-body 3T MR system. Measurement variability attributed to changes in coil position, shimming procedure, and spectral analysis were quantified. 31P-MRS data were acquired with a 31P/-proton (1H) dual-tuned surface coil positioned on the quadriceps using a pulse-acquire sequence. Test-retest absolute and relative repeatability was analyzed using the coefficient of variation (CV) and intra-class correlation coefficients (ICC), respectively. After sequence parameter optimization, Pialk demonstrated high intra-subject repeatability (CV: 10.6 ± 5.4%, ICC: 0.80). Proximo-distal change in coil position along the length of the quadriceps introduced Pialk quantitation variability (CV: 28 ± 5%), due to magnetic field inhomogeneity with more distal coil locations. In contrast, Pialk measurement variability due to repeated shims from the same muscle volume (0.40 ± 0.09mM; CV: 6.6%), and automated spectral processing (0.37 ± 0.01mM; CV: 2.3%), was minor. The quantification of Pialk in skeletal muscle via surface coil 31P-MRS at 3T demonstrated excellent reproducibility. However, caution is advised against placing the coil at the distal part of the quadriceps to mitigate shimming inhomogeneity.

Total Magnetic Field Perturbations at Martian Low Altitudes (≤500 km): MAVEN Observations in 2014–2023

AbstractMagnetic perturbations in the Martian ionosphere are often considered as representing plasma irregularities, but thorough statistical comparisons between the two independent phenomena have been rarely conducted. In this study, we investigate the statistical relationship between total magnetic field perturbations and those of O2+ density as observed by the Mars Atmosphere and Volatile EvolutioN (MAVEN) spacecraft at altitudes below 500 km in 2014–2023. From the 1 Hz time series of magnetic field strength, perturbation intensity is estimated by subtracting a signal smoothed with a Savitzky‐Golay filter (window size ∼36 km). Statistically, the magnetic perturbations are stronger on the dayside than at night, in the local summer hemisphere than in winter, and away from crustal magnetic anomalies than nearby. Also, day‐to‐day variability of the magnetic perturbations exhibits a weak but significant correlation with that of solar wind electron density. The statistical behavior of total magnetic field perturbations is not the same as that of O2+ density perturbations: that is, the former is not a perfect proxy for the latter. The seemingly counter‐intuitive discrepancy can be explained by effects of inhomogeneous background magnetic field strength (participating in total pressure balance across plasma perturbations) and localized ionospheric currents at altitudes unreachable by MAVEN.

A hybrid Eulerian-Lagrangian Vlasov method for nonlinear wave-particle interaction in weakly inhomogeneous magnetic field

We present a hybrid Eulerian-Lagrangian (HEL) Vlasov method for nonlinear resonant wave-particle interactions in weakly inhomogeneous magnetic field. The governing Vlasov equation is derived from a recently proposed resonance tracking Hamiltonian theory. It gives the evolution of the distribution function with a scale-separated Hamiltonian that contains the fast-varying coherent wave-particle interaction and slowly-varying motion about the resonance frame of reference. The hybrid scheme solves the fast-varying phase space evolution on Eulerian grid with an adaptive time step and then advances the slowly-varying dynamics by Lagrangian method along the resonance trajectory. We apply the HEL method to study the frequency chirping of whistler-mode chorus wave in the magnetosphere and the self-consistent simulations reproduce the chirping chorus wave and give high-resolution phase space dynamics of energetic particles at low computational cost. The scale-separated HEL approach could provide additional insights of the wave instabilities and wave-particle nonlinear coherence compared to the conventional Vlasov and particle-in-cell methods.