Effects of permeable boundaries on the diffusion-attenuated MR signal: insights from a one-dimensional model

The local magnetization distribution M(x,t) and the net MR signal S arising from a one-dimensional periodic structure with permeable barriers in a Tanner-Stejskal pulsed-field gradient experiment are considered. In the framework of the narrow pulse approximation, the general expressions for M(x,t) and S as functions of diffusion time and the bipolar field gradient strength are obtained and analyzed. In contrast to a system with impermeable boundaries, the signal S as a function of the b-value is modeled well as a bi-exponential decay not only in the short-time regime but also in the long-time regime. At short diffusion times, the local magnetization M(x,t) is strongly spatially inhomogeneous and the two exponential components describing S have a clear physical interpretation as two "population fractions" of the slow- and fast-diffusing quasi-compartments (pools). In the long-diffusion time regime, the two exponential components do not have clear physical meaning but rather serve to approximate a more complex functional signal form. The average diffusion propagator, obtained by means of standard q-space analysis procedures in the long-diffusion time regime is explored; its structure creates the deceiving appearance of a system with multiple compartments of different sizes, while in reality, it reflects the permeable nature of boundaries in a system with multiple compartments all of the same size.     

Relaxation of nuclear magnetization in a nonuniform magnetic field gradient and in a restricted geometry

We study the influence of restriction on Carr-Purcell-Meiboom-Gill spin echo response of magnetization of spins diffusing in a bounded region in the presence of a nonuniform magnetic field gradient. We consider two fields in detail-a parabolic field which, like the uniform-gradient field, scales with the system size, and a cosine field which remains bounded. Corresponding to three main length scales, the pore size, L(S), the dephasing length, L(G), and the diffusion length during half-echo time, L(D), we identify three main regimes of decay of the total magnetization: motionally averaged, localization, and short-time. In the short-time regime (L(D) < L(S), L(G)), we confirm that the leading order behavior is controlled by the average of the square of the gradient, (nablaB(z))(2), and in the motionally averaged regime (MAv), where L(S) < L(D), L(G), by (integral dxB(z))(2). We verify numerically that two different fields for which those two averages are identical result in very similar decay profiles not only in the limits of short and long times but also in the intermediate times, with important practical implications. In the motionally averaged regime we found that previous estimates of the decay exponent for the parabolic field, based on a soft-boundary condition, are significantly altered in the presence of a more realistic, hard wall. We find the scaling of the decay exponent in the MAv regime with pore size to be L(2)(S) for the cosine field and L(6)(S) for the parabolic field, as contrasted with the linear gradient scaling of L(4)(S). In the localization regime, for both the cosine and the parabolic fields, the decay exponent depends on a fractional power of the gradient, implying a breakdown of the second cumulant or the Gaussian phase approximation. We also examined the validity of time-evolving the total magnetization according to a distribution of effective local gradients and found that such approximation works well only in the short-time regime and breaks down strongly for long times. Copyright 2000 Academic Press.     

“Aharonov-Bohm antiferromagnetism” and compensation points in the lattice of quantum rings

We investigate the magnetic properties of the lattice of non-interacting quantum rings using the 2D rotator model. The exact analytic expressions for the free energy as well as for the magnetization and magnetic susceptibility are found and analyzed. It is shown that such a system can be considered as a system with antiferromagnetic-like properties. We have shown also that all observable quantities in this case (free energy, entropy, magnetization) are periodic functions of the magnetic flux through the ring's area (as well known, such a behavior is typical for the Aharonov-Bohm effect). For the lattice of quantum rings with two different geometric parameters we investigate the ordinary compensation points (“temperature compensation points”, i.e. points at which the magnetization vanishes at fixed values of the magnetic field strength). It is shown that the positions of compensation points in the temperature scale are very sensitive to small changes in the magnetic field strength.     

A Curie-Weiss Model with Dissipation

We consider stochastic dynamics for a spin system with mean field interaction, in which the interaction potential is subject to noisy and dissipative stochastic evolution. We show that, in the thermodynamic limit and at sufficiently low temperature, the magnetization of the system has a time periodic behavior, despite of the fact that no periodic force is applied.     

Ordered states and structural transitions in a system of Abrikosov vortices with periodic pinning

A system of Abrikosov vortices in a quasi-two-dimensional HTSC plate is considered for various periodic lattices of pinning centers. The magnetization and equilibrium configurations of the vortex density for various values of external magnetic field and temperature are calculated using the Monte Carlo method. It is found that the interaction of the vortex system with the periodic lattice of pinning centers leads to the formation of various ordered vortex states through which the vortex system passes upon an increase or a decrease in the magnetic field. It is shown that ordered vortex states, as well as magnetic field screening processes, are responsible for the emergence of clearly manifested peaks on the magnetization curves. Extended pinning centers and the effect of multiple trapping of vortices on the behavior of magnetization are considered. Melting and crystallization of the vortex system under the periodic pinning conditions are investigated. It is found that the vortex system can crystallize upon heating in the case of periodic pinning.     

Strain-Induced Orbital Magnetization in a Weyl Semimetal

Strains in a crystal lattice give rise to gauge pseudofields. In this work, the magnetization of a distorted Weyl semimetal induced by the pseudomagnetic field is studied. It is shown that such magnetization is nonzero because an additional electric field (gradient of the deformation potential) is generated simultaneously with the pseudomagnetic field. It is also shown that the usual applied electric field controls the magnetization, providing an opportunity of decreasing it to zero. The system under study can also exhibit the coexistence of two types of Weyl fermions, types I and II.     

Diffusion and relaxation effects in general stray field NMR experiments

We analyze the evolution of magnetization following any series of radiofrequency pulses in strongly inhomogeneous fields, with particular attention to diffusion and relaxation effects. When the inhomogeneity of the static magnetic field approaches or exceeds the strength of the RF field, the magnetization has contributions from different coherence pathways. The diffusion or relaxation induced decay of the signal amplitude is in general nonexponential, even if the sample has single relaxation times T(1), T(2) and a single diffusion coefficient D. In addition, the shape of the echo depends on diffusion and relaxation. It is possible to separate contributions from different coherence pathways by phase cycling of the RF pulses. The general analysis is tested on stray field measurements using two different pulse sequences. We find excellent agreement between measurements and calculations. The inversion recovery sequence is used to study the relaxation effects. We demonstrate two different approaches of data analysis to extract the relaxation time T(1). Finite pulse width effects on the timing of the echo formation are also studied. Diffusion effects are analyzed using the Carr--Purcell--Meiboom--Gill sequence. In a stray field of a constant gradient g, we find that unrestricted diffusion leads to nonexponential signal decay versus echo number N, but within experimental error the diffusion attenuation is still only a function of g(2)Dt(3)(E)N, where t(E) is the echo spacing.     

Measurement of One-Dimensional Spatially Resolved Self-Diffusion Coefficients and Longitudinal Relaxation Times with a Single B1 Gradient

A very simple experimental setup, involving a single coil for generating a radiofrequency field gradient, enables one to determine the self-diffusion coefficient and the longitudinal relaxation time along the gradient axis. This is accomplished by a two-dimensional experiment involving three gradient pulses. The first part of the sequence includes two pulses of identical duration separated by an evolution interval which, by proper phase cycling, encodes longitudinal magnetization according to translational diffusion. The last pulse is incremented for purposes of spatial encoding.     

Suppression of spin diffusion near a micron-size ferromagnet

We have used the large gradients generated near the ferromagnetic tip of a magnetic resonance force microscope to locally suppress spin diffusion in a silica sample containing paramagnetic electron spins. By controlling the slice location with respect to the tip, the magnetic field gradient was varied from 0.01 to 36 mT/microm, resulting in a fourfold decrease in T-11 and a similar decrease in T(-1)(1 rho). The observed dependence of the relaxation rates on field gradient is consistent with the quenching of flip-flop interactions that mediate the transport of magnetization between slow and fast relaxing spins.     

Inverse crystallization of a system of Abrikosov vortices with periodic pinning

A system of vortices in a quasi-two-dimensional HTSC plate with periodic pinning is considered. The magnetization curves are calculated by the Monte Carlo method for different values of an external magnetic field and different temperatures. It is shown that the vortex system with periodic pinning may crystallize with an increase in temperature.     

Gate-Voltage-Induced Magnetization Reversal and Tunneling Anisotropic Magnetoresistance in a Single Molecular Magnet with Temperature Gradient

We study the control of gate voltage over the magnetization of a single-molecule magnet(SMM) weakly coupled to a ferromagnetic and a normal metal electrode in the presence of the temperature gradient between two electrodes.It is demonstrated that the SMM's magnetization can change periodically with periodic gate voltage due to the driving of the temperature gradient.Under an appropriate matching of the electrode polarization,the temperature difference and the pulse width of gate voltage,the SMM's magnetization can be completely reversed in a period of gate voltage.The corresponding flipping time can be controlled by the system parameters.In addition,we also investigate the tunneling anisotropic magnetoresistance(TAMR) of the device in the steady state when the ferromagnetic electrode is noncollinear with the easy axis of the SMM,and show the jump characteristic of the TAMR.     

Influential Factors of Internal Magnetic Field Gradient in Reservoir Rock and Its Effects on NMR Response

Nuclear magnetic resonance (NMR) plays a significant role in porous media analysis and petroleum exploration, but its response is significantly influenced by the internal magnetic field gradient in fluid saturated porous medium, which obviously limits the accuracy of rock core analysis and logging interpretation. The influential factors of the internal magnetic field gradient in formation and its influences on NMR response are studied in this paper, based on NMR mechanism through one- and two-dimensional core NMR experiments. The results indicate that the internal magnetic field gradient is positively correlated with the static magnetic field strength and the magnetic susceptibility difference between pore fluid and solid grains, while it presents negative correlation with pore radius. The internal magnetic field gradient produces an additional diffusion relaxation in hydrogen relaxation system and accelerates the attenuation of magnetization vector. As a result, T₂ spectrum shifts to the left and NMR porosity and diffusion coefficient of the fluid could be inaccurate. This research sets a foundation for the NMR porosity correction and fluid distribution on T₂-G maps based on the internal magnetic field gradient correction.     

Out-of-Plane Torque Influence on Magnetization Switching and Susceptibility in Magnetic Multilayers

Based on both the spin diffusion equation and the Landau-LlTshitz-Gilbert （LLG） equation, we demonstrate the influence of out-of-plane spin torque on magnetization switching and susceptibility in a magnetic multilayer system. The variation of spin accumulation and local magnetization with respect to time are studied in the magnetization reversal induced by spin torque. We also research the susceptibility subject to a microwave magnetic field, which is compared with the results obtained without out-of-plane torque.     

Diffusion Imaging with HyperpolarizedHe Gas

We used MRI of hyperpolarized³He to demonstrate some novel aspects of gas diffusion. Two different techniques were used. First, a slice was burned into a one-dimensional image by inverting the spins in the slice and diffusion was studied by measuring the magnetization as it filled the depleted slice. A diffusion coefficient was determined by the fit of these data. Second, one-dimensional diffusion images were made using a Stejskal–Tanner PGSE method. This was done with and without a temperature gradient present, showing that the effect of temperature can be dynamically monitored by such diffusion images.     

Many-body theory of magnetization of Bloch electrons in the presence of spin-orbit interactions

We derive a theory of magnetization of an interacting electron system in the presence of a periodic potential, spin-orbit interaction and an applied magnetic field in the paramagnetic limits. Starting from a thermodynamic potential, which includes both the quasi-particle and correlation contributions, we show that modifications brought about by the electron-electron interactions for the magnetization in the quasi-particle approximation is precisely cancelled by the contributions due to electron correlations.This is in contrast to the explicit many-body effects seen in case of the magnetic susceptibility and the Knight-shift. The magnetization is expressed as a product of the spin-density and the effective g-factor, mainly due to the spin-orbit interaction. We show the importance of self-energy corrections on the single-particle energy spectrum by considering a variant of the Hubbard Hamiltonian in momentum space.     

Modeling of Self-Diffusion and Relaxation Time NMR in Multi-Compartment Systems

The theory of pulsed field gradient (pfg) NMR applied to molecules in cellular systems which contain different subcellular compartments separated by permeable membranes, acting as diffusion barriers, has been extended. A numerical model of restricted diffusion and magnetization relaxation behavior in pfg-CPMG NMR experiments, based on the Fick's second law of diffusion, is presented. This model is applicable to a wide range of systems and allows the exploration of temporal and spatial behavior of the magnetization with and without the influence of gradient pulses. Results of the numerical experiments show their correspondence to the previously observed ones and demonstrate the importance of the inclusion of the time domain data in analyzing diffusion measurements.     

Theory of Spin Echo in Restricted Geometries under a Step-wise Gradient Pulse Sequence

A closed matrix form solution of the Bloch-Torrey equation is presented for the magnetization density of spins diffusing in a bounded region under a steady gradient field and for the Stejskal-Tanner gradient pulse sequence, assuming straightforward generalization to any step-wise gradient profile. The solution is expressed in terms of the eigenmodes of the diffusion propagator in a given geometry with appropriate boundary conditions (perfectly reflecting or relaxing walls). Applications to rectangular, cylindrical, and spherical geometries are discussed. The relationship with the multiple propagator approach is established and an alternative step-wise gradient discretization procedure is suggested to handle arbitrary gradient waveforms.     

A Pulsed Field Gradient Spin-Echo Method for Diffusion Measurements in the Presence of Internal Gradients

Over the past decade several pulsed field gradient stimulated-echo methods have been presented for diffusion measurements in heterogeneous media. These methods have reduced or eliminated the coupling between the applied magnetic field gradient and a constant internal magnetic field gradient caused by susceptibility changes throughout the sample. For many research purposes thez-storage delay between the second and third π/2 RF pulse has been included in order to increase the decay of the echo attenuation to an appropriate level and to increase the signal-to-noise ratio by avoidingT₂relaxation of the magnetization in parts of the pulse sequence. For these reasons a stimulated-echo method has been applied instead of a spin-echo method. When studying systems where it is necessary to keep the duration of the pulse sequence at a minimum, and one is not dependent on usingz-storage time to increase the echo attenuation or to study diffusion as a function of observation time, a spin-echo method should be chosen. Here we propose a bipolar pulsed field gradient spin-echo method which is well suited to this purpose, and preliminary diffusion measurements are presented as illustration.     

Precessional switching of thin nanomagnets: analytical study

We study analytically the precessional switching of the magnetization of a thin macrospin. We analyze its response when subjected to an external field along its in-plane hard axis. We derive the exact trajectories of the magnetization. The switching versus non switching behavior is delimited by a bifurcation trajectory, for applied fields equal to half of the effective anisotropy field. A magnetization going through this bifurcation trajectory passes exactly along the hard axis and exhibits a vanishing characteristic frequency at that unstable point, which makes the trajectory noise sensitive. Attempting to approach the related minimal cost in applied field makes the magnetization final state unpredictable. We add finite damping in the model as a perturbative, energy dissipation factor. For a large applied field, the system switches several times back and forth. Several trajectories can be gone through before the system has dissipated enough energy to converge to one attracting equilibrium state. For some moderate fields, the system switches only once by a relaxation dominated precessional switching. We show that the associated switching field increases linearly with the damping parameter. The slope scales with the square root of the effective anisotropy. Our simple concluding expressions are useful to assess the potential application of precessional switching in magnetic random access memories.Received: 2 October 2003, Published online: 19 November 2003PACS: 75.40.Gb Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic scaling, etc.) - 75.60.Jk Magnetization reversal mechanisms - 75.75. + a Magnetic properties of nanostructures     

Phase coherent precessional magnetization reversal in microscopic spin valve elements

We evidence multiple coherent precessional magnetization reversal in microscopic spin valves. Stable, reversible, and highly efficient magnetization switching is triggered by transverse field pulses as short as 140 ps with energies down to 15 pJ. At high fields a phase coherent reversal is found revealing periodic transitions from switching to nonswitching under variation of pulse parameters. At the low field limit the existence of a relaxation dominated regime is established allowing switching by pulse amplitudes below the quasistatic switching threshold.