Dependence of the magnetization relaxation time of single-domain ferromagnetic particles on damping in the Brown model

Analytical expressions for the magnetization relaxation time τ of single-domain ferromagnetic particles with cubic or uniaxial anisotropy in a static transverse magnetic field are derived. The derivation is based on calculating the escape rate of a Brownian particle from a potential well; this technique is applicable at any damping and is generalized to the case of magnetic relaxation of superparamagnetic particles. The validity of the expressions obtained for τ is checked against a numerical solution of the Landau-Lifshitz-Gilbert equation over the whole range of damping (very low, intermediate, and high damping and the crossover region between low and intermediate damping).     

Brownian motion of massive skyrmions in magnetic thin films

We report on the thermal effects on the motion of current-driven massive magnetic skyrmions. The reduced equation for the motion of skyrmion has the form of a stochastic generalized Thiele’s equation. We propose an ansatz for the magnetization texture of a non-rigid single skyrmion that depends linearly with the velocity. By using this ansatz it is found that the skyrmion mass tensor is closely related to intrinsic skyrmion parameters, such as Gilbert damping, skyrmion-charge and dissipative force. We have found an exact expression for the average drift velocity as well as the mean-square velocity of the skyrmion. The longitudinal and transverse mobility of skyrmions for small spin-velocity of electrons is also determined and found to be independent of the skyrmion mass.     

Quantitative interpretation of magnetization transfer in spoiled gradient echo MRI sequences

A method for analyzing general pulsed magnetization transfer (MT) experiments in which off-resonance saturation pulses are interleaved with on-resonance excitation pulses is presented. We apply this method to develop a steady-state signal equation for MT-weighted spoiled gradient echo sequences and consider approximations that facilitate its rapid computation. Using this equation, we assess various experimental designs for quantitatively imaging the fractional size of the restricted pool, cross-relaxation rate, and T(1) and T(2) relaxation times of the two pools in a binary spin bath system. From experiments on agar gel, this method is shown to reliably and accurately estimate the exchange and relaxation properties of a material in an imaging context, suggesting the feasibility of using this technique in vivo.     

Dynamics of paramagnetic agents by off-resonance rotating frame technique in the presence of magnetization transfer effect

The simple method for measuring the rotational correlation time of paramagnetic ion chelates via off-resonance rotating frame technique is challenged in vivo by the magnetization transfer effect. A theoretical model for the spin relaxation of water protons in the presence of paramagnetic ion chelates and magnetization transfer effect is described. This model considers the competitive relaxations of water protons by the paramagnetic relaxation pathway and the magnetization transfer pathway. The influence of magnetization transfer on the total residual z-magnetization has been quantitatively evaluated in the context of the magnetization map and various difference magnetization profiles for the macromolecule conjugated Gd-DTPA in cross-linked protein gels. The numerical simulations and experimental validations confirm that the rotational correlation time for the paramagnetic ion chelates can be measured even in the presence of strong magnetization transfer. This spin relaxation model also provides novel approaches to enhance the detection sensitivity for paramagnetic labeling by suppressing the spin relaxations caused by the magnetization transfer. The inclusion of the magnetization transfer effect allows us to use the magnetization map as a simulation tool to design efficient paramagnetic labeling targeting at specific tissues, to design experiments running at low RF power depositions, and to optimize the sensitivity for detecting paramagnetic labeling. Thus, the presented method will be a very useful tool for the in vivo applications such as molecular imaging via paramagnetic labeling.     

Magnetization transfer contrast (MTC) in FLASH MR imaging

Magnetization transfer between bound and free protons was used as a source of contrast in high speed MR imaging using the FLASH technique. Contrast in FLASH MR images was found to depend upon the reduced magnetization and the spin lattice relaxation rate of free protons in the presence of bound proton radio-frequency saturation. MTC FLASH imaging was thus used to estimate the variation with saturation frequency of free proton spin-lattice relaxation during magnetization transfer.     

ESR and longitudinal response in metals containing localized paramagnetic centers with spin<Emphasis Type="Italic">S</Emphasis>>1/2

The theory describing electron spin resonance (ESR) and the longitudinal magnetization response of coupled spin systems in a metal containing both delocalized conduction electrons (“espins”) and localized paramagnetic centers (“s-spins”) is generalized to the case of arbitrary half-integer spin value,S>1/2, of the s-spins. The consideration is based on the Bloch-Hasegawa equations supplemented by taking into account the coupled evolution of the longitudinal magnetization components and the effect of weak ESR saturation by the microwave field. The ESR transversal susceptibility and longitudinal magnetization response are worked out in terms of normal modes related to the coupled s- and e-spin oscillators taking into account the ESR fine structure (FS) of the s-spins. These modes are characterized by effective (renormalized) frequencies and relaxation rates (decays) which differ from the partial ones. In the specific cases of a well-resolved FS (in the isothermal limit) and of the relaxational collapse of the FS due to strong exchange coupling between the s- and e-spins (in both the isothermal and bottlenecked limits), the analytical expressions are derived which are relevant to the modulation technique of measuring extremely fast spin-lattice relaxation times in metals.     

Investigation of cerebral ischemia using magnetization transfer contrast (MTC) MR imaging

The effects of cerebral ischemia in rat brain were monitored as a function of time using proton MR imaging. Spinspin relaxation time (T₂), proton density, and magnetization transfer contrast (MTC) were measured by MR imaging at various time intervals during a 1-week period following the induction of ischemic damage. Ischemic injury was characterized by a maximization of both T₂ value and MTC appearance at 24 hr postischemic injury. These changes were accompanied by a gradual increase in MR observable water density over the first few days of ischemia. A reduction in the magnetization exchange rate between “free” and “bound” water protons as measured by MTC imaging is at least partially responsible for the elevation in T₂ values observed during ischemia, and may accompany breakdown of cellular structure.     

The effects of morphology and exchange on proton N.M.R. relaxation in agarose gels

The effects of morphology and exchange on N.M.R. relaxation times in agarose gels are interpreted within a unified theoretical framework based on the generalized Bloch equations. By acknowledging the spacial dependence of the N.M.R. parameters it is shown how the relaxation behaviour depends on the distance scale characterizing the heterogeneity of the gel. If this distance scale is sufficiently small to allow complete diffusive averaging we recover the traditional results based on the Bloch-McConnell equations describing relaxation in a homogeneous system. This is the case for fresh agarose gels which show monoexponential relaxation and has been widely interpreted in terms of the rapid exchange of protons between populations of ‘free’ and ‘bound’ water. Conversely, if the distance scale characterizing the heterogeneity is sufficiently large to prevent complete diffusive averaging our model predicts multiexponential relaxation. This is the case with the transverse magnetization in agarose gels that have been slowly frozen then thawed. These results show how it is possible to probe the degree of microheterogeneity in gel samples using N.M.R. For the purpose of deriving simple analytical expressions for the N.M.R. relaxation times we only consider one-dimensional solutions to our model. More realistic morphologies can be treated using numerical methods.     

Evaluation of the role of magnetization transfer imaging in prostate: a preliminary study

Results of the preliminary study on the evaluation of the role of magnetization transfer imaging (MTI) of prostate in men who had raised prostate-specific antigen (PSA) (>4 ng/ml) or abnormal digital rectal examination (DRE) are reported. MT ratio (MTR) was calculated for 20 patients from the hyper- (normal) and hypo-intense regions (area suspicious of malignancy as seen on T2-weighted MRI) of the peripheral zone (PZ) and the central gland (CG) at 1.5 T. In addition, MTR was calculated for three healthy controls. Mean MTR was also calculated for the whole of the PZ (including hyper- and hypo-intense area) in all patients. Out of 20 patients, biopsy revealed malignancy in 12 patients. Mean MTR value (8.29+/-3.49) for the whole of the PZ of patients who were positive for malignancy on biopsy was statically higher than that observed for patients who were negative for malignancy (6.18+/-3.15). The mean MTR for the whole of the PZ of controls was 6.18+/-1.63 and is similar to that of patients who were negative for malignancy. Furthermore, for patients who showed hyper- (normal portion) and hypo-intense (region suspicious of malignancy) regions of the PZ, the MTR was statistically significantly different. These preliminary results reveal the potential role of MT imaging in the evaluation of prostate cancer.     

Gauge-free electromagnetic gyrokinetic theory

A new gauge-free electromagnetic gyrokinetic theory is developed, in which the gyrocenter equations of motion and the gyrocenter phase-space transformation are expressed in terms of the perturbed electromagnetic fields, instead of the usual perturbed potentials. Gyrocenter polarization and magnetization are derived explicitly from the gyrocenter Hamiltonian, up to first order in the gyrocenter perturbation expansion. Expressions for the sources in Maxwell's equations are derived in a form that is suitable for simulation studies, as well as kinetic-gyrokinetic hybrid modeling.     

Nonlinear response of superparamagnetic particles to a sudden change of a high constant magnetic field

For a system of superparamagnetic particles in a high external constant magnetic field, a technique for calculating the nonlinear response to a sudden change in the field direction and magnitude is proposed. A set of momentary equations for the averaged spherical harmonics, which is derived from the Fokker-Planck equation for the magnetization-orientation distribution function is the basis of this technique. As an example, the nonlinear response of a system of particles with anisotropy of the easy-axis type is examined. For this case, a solution to the momentary equations is obtained by using matrix continued fractions. The magnetization relaxation time and the spectrum of the relaxation function are calculated for typical values of anisotropy, dissipation, and nonlinearity parameters. It is shown that the magnetization kinetics is essentially dependent on these parameters.     

Langevin description of markovian integro-differential master equations

For a given master equation of a discontinuous irreversible Markov process, we present the derivation of stochastically equivalent Langevin equations in which the noise is either multiplicative white generalized Poisson noise or a spectrum of multiplicative white Poisson noise. In order to achieve this goal, we introduce two new stochastic integrals of the Ito type, which provide the corresponding interpretation of the Langevin equations. The relationship with other definitions for stochastic integrals is discussed. The results are elucidated by two examples of integro-master equations describing nonlinear relaxation.     

Nakajima-Zwanzig's generalized master equation: Evaluation of the kernel of the integro-differential equation

A method is presented, which allows the exact elimination of the projection operator from the kernel of the Nakajima-Zwanzig generalized master equation without using perturbational expansions. Expressions for kernels of generalized master equations using several frequently occuring types of projection operators are derived explicitly. The application of this method for the exact derivation of generalized master equations describing the coherent and the coupled coherent and incoherent exciton motion is proposed. As another application, the derivation of the Smoluchowski equation is suggested.     

Generalization of Biot’s equations with allowance for shear relaxation of a fluid

On the basis of the generalized variational principle for dissipative continuum mechanics, a system of generalized Biot’s equations is derived to describe the wave propagation in a two-phase porous permeable medium in the presence of shear relaxation in the pore-filling fluid. It was shown that the inclusion of shear viscoelasticity of the fluid leads to the appearance of two transverse modes in addition to two longitudinal modes described by the Biot theory. One of the transverse modes is an acoustic mode, whereas the other is a diffusion mode characterized by the linear frequency dependence of phase velocity and attenuation coefficient in the low-frequency region.     

Quantitative cerebral magnetic resonance imaging during ACTH treatment of multiple sclerosis

Serial MR scans were performed with the 2DFT imaging method and the filtered backprojection imaging method on 12 patients with multiple sclerosis in acute phase, 4 in a relapsing/remitting form, and 8 in a progressive form, before, during and after ACTH treatment. Both T₁ and T_2mono relaxation times, obtained by fitting transverse magnetization decay curves with a monoexponential function within the apparently normal white matter and the areas of increased signal, were measured. With the backprojection method it was possible to fit the transverse magnetization decay curve with a biexponential function and obtain T_2long and T_2short relaxation times. The T_2mono and T₁ relaxation times of the apparently normal white matter were significantly different from those obtained for volunteers, but no significant differences were found before, during, or after treatment. The transverse magnetization decay curves of the areas of increased signal were better fitted by a biexponential function. No significant changes in these relaxation times were observed after ACTH treatment. These results argue against an anti-oedematous action of ACTH and may suggest that it has an immunosuppressant effect.     

Steady-state free precession with hyperpolarized 3He: experiments and theory

The magnetization response of hyperpolarized 3He gas to a steady-state free precession (SSFP) sequence was simulated using matrix product operators. The simulations included the effects of flip angle (alpha), sequence timings, resonant frequency, gas diffusion coefficient, imaging gradients, T1 and T2. Experiments performed at 1.5 T, on gas phantoms and with healthy human subjects, confirm the predicted theory, and indicate increased SNR with SSFP through use of higher flip angles when compared to optimized spoiled gradient echo (SPGR). Simulations and experiments show some compromise to the SNR and some point spread function broadening at high alpha due to the incomplete refocusing of transverse magnetization, caused by diffusion dephasing from the readout gradient. Mixing of gas polarization levels by diffusion between slices is also identified as a source of signal loss in SSFP at higher alpha through incomplete refocusing. Nevertheless, in the sample experiments, a SSFP sequence with an optimized flip angle of alpha=20 degrees, and 128 sequential phase encoding views, showed a higher SNR when compared to SPGR (alpha=7.2 degrees) with the same bandwidth. Some of the gas sample experiments demonstrated a transient signal response that deviates from theory in the initial phase. This was identified as being caused by radiation damping interactions between the large initial transverse magnetization and the high quality factor (Q=250) birdcage resonator. In 3He NMR experiments, performed without imaging gradients, diffusion dephasing can be mitigated, and the effective T2 is relatively long (1 s). Under these circumstances the SSFP sequence behaves like a CPMG sequence with sinalpha/2 weighting of SNR. Experiments and simulations were also performed to characterize the off-resonance behaviour of the SSFP HP 3He signal. Characteristic banding artifacts due to off-resonance harmonic beating were observed in some of the in vivo SSFP images, for instance in axial slices close to the diaphragm where B0 inhomogeneity is highest. Despite these artifacts, a higher SNR was observed with SSFP in vivo when compared to the SPGR sequence. The trends predicted by theory of increasing SSFP SNR with increasing flip angle were observed in the range alpha=10-20 degrees without compromise to image quality through blurring caused by excessive k-space filtering.     

Mechanically driven domain wall movement in magnetoelastic nanomagnets

Magnetic domain walls are fundamental objects arising in ferromagnetic materials, largely investigated both through micromagnetic simulations and experiments. While current- and field-based techniques for inducing domain wall propagation have been widely studied for fundamental understanding and application-oriented purposes, the possibility to manipulate domain walls using mechanical stress in magnetoelastic materials has only recently drawn interest. Here, a complete analytical model describing stress-induced transverse domain wall movement in ferromagnetic nanostripe with variable cross-section is presented. This approach yields a nonlinear integro-differential equation describing the magnetization field. Its numerical implementation, based on the nonlinear relaxation method, demonstrates the possibility to precisely control the position of a domain wall through mechanical action.     

NQR Spin Diffusion in an Inhomogeneous Internal Field

The theory of NQR spin diffusion is extended to the case of spin lattice relaxation and spin diffusion in an inhomogeneous field. Two coupled equations describing the mutual relaxation and the spin diffusion of the nuclear magnetization and dipolar energy were obtained by using the method of nonequilibrium state operator. The equations were solved for short and long times approximation corresponding to the direct and diffusion relaxation regimes.     

A statistical derivation of the hydrodynamic equations of change for a system of ionized molecules. I. General equation of change and the Maxwell equations

Nonrelativistic, classical statistical mechanics is used to describe a dense fluid of molecules composed of nuclei and electrons with purely Coulomb interaction potentials. A general equation of change is derived for the time rate of change of any macroscopic (ensemble averaged) dynamical variable. From this general equation, Maxwell's equations in a medium are derived and expressed in terms of molecular properties, e.g., polarization and magnetization densities.This research was carried out in part under Grant NsG-275-62 from the National Aeronautics and Space Administration and in part under a grant from the National Science Foundation. This paper is based on a thesis submitted by R. J. B. to the Graduate School of the University of Wisconsin in partial fulfillment of the requirements for the Ph.D. degree.     

Magnetic properties of a mixed spin-1/2 and spin-3/2 transverse Ising model

A theoretical study of a mixed spin-1/2 and spin-3/2 Ising system with independent transverse fields is presented using an effective field method within the framework of a single-site cluster theory. In this approach the effective field equations are derived using a probability distribution method based on the use of generalized van der Waerden identities accounting exactly for the single-site kinematic relations. The effect of the transverse fields on the critical behaviour is studied. The thermal dependence of the longitudinal and transverse components of the magnetization and its higher moments is also studied.