Anomalous NMR relaxation in cartilage matrix components and native cartilage: fractional-order models

We present a fractional-order extension of the Bloch equations to describe anomalous NMR relaxation phenomena (T(1) and T(2)). The model has solutions in the form of Mittag-Leffler and stretched exponential functions that generalize conventional exponential relaxation. Such functions have been shown by others to be useful for describing dielectric and viscoelastic relaxation in complex, heterogeneous materials. Here, we apply these fractional-order T(1) and T(2) relaxation models to experiments performed at 9.4 and 11.7 Tesla on type I collagen gels, chondroitin sulfate mixtures, and to bovine nasal cartilage (BNC), a largely isotropic and homogeneous form of cartilage. The results show that the fractional-order analysis captures important features of NMR relaxation that are typically described by multi-exponential decay models. We find that the T(2) relaxation of BNC can be described in a unique way by a single fractional-order parameter (α), in contrast to the lack of uniqueness of multi-exponential fits in the realistic setting of a finite signal-to-noise ratio. No anomalous behavior of T(1) was observed in BNC. In the single-component gels, for T(2) measurements, increasing the concentration of the largest components of cartilage matrix, collagen and chondroitin sulfate, results in a decrease in α, reflecting a more restricted aqueous environment. The quality of the curve fits obtained using Mittag-Leffler and stretched exponential functions are in some cases superior to those obtained using mono- and bi-exponential models. In both gels and BNC, α appears to account for micro-structural complexity in the setting of an altered distribution of relaxation times. This work suggests the utility of fractional-order models to describe T(2) NMR relaxation processes in biological tissues.     

Relaxation time distributions for an anomalously diffusing particle

As well known, the generalized Langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle. Here, we focus attention on some new aspects of the dynamics, successively considering the memory kernel, the particle’s mean velocity, and the scattering function. All these quantities are studied from a unique angle, namely, the discussion of the possible existence of a distribution of relaxation times characterizing their time decay. Although a very popular concept, a relaxation time distribution cannot be associated with any time-decreasing quantity (from a mathematical point of view, the decay has to be described by a completely monotonic function).Technically, we use a memory kernel decaying as a Mittag-Leffler function (the Mittag-Leffler functions interpolate between stretched or compressed exponential behaviour at short times and inverse power law behaviour at large times). We show that, in the case of a subdiffusive motion, relaxation time distributions can be defined for the memory kernel and for the scattering function, but not for the particle’s mean velocity. The situation is opposite in the superdiffusive case.     

Anomalous magnetic relaxation in thin Pd<Subscript>0.99</Subscript>Fe<Subscript>0.01</Subscript> films

The magnetic relaxation in Pd_0.99Fe_0.01 films, which have the thicknesses that are practically important for cryoelectronics (25 and 40 nm), is detected and experimentally studied. The relaxation is shown to be substantial only in thin films. The magnetization relaxation is found to be well described by the sum of two exponential functions with characteristic times that differ by an order of magnitude from each other. The characteristic relaxation time and the ratio of the contributions of two relaxations depend on temperature. The activation energies of the relaxation processes are determined. The activation volume is shown to correspond to a 20-nm ferromagnetic cluster. The results obtained agree with the model of two-component magnetization in thin PdFe films [6].     

Numerical Determination of the Relaxation Parameters of NMR in Complex Heterogeneous Systems

The methods of mathematical processing of the envelopes of spin-echo signals have been considered within the framework of the multiphase relaxation theory. A mathematical model for separation of multiexponential relaxation curves into individual exponential components of spin-spin relaxation times T _2i and amplitudes I _i is described. The multiphase nature of the relaxation of protons in complex heterogeneous systems has been revealed, and the relaxation characteristics of individual components — spin-spin relaxation times and amplitudes — have been determined.     

14N核四极共振自旋系统自旋-晶格弛豫时间的测量

¹⁴N核四极共振自旋系统的自旋-晶格弛豫是一种双指数弛豫。本文介绍了¹⁴N核四极共振自旋系统的自旋-晶格弛豫时间的3种测量方法,利用可变多面体方法对实验数据进行拟合,获得了¹⁴N核四极共振自旋系统的自旋-晶格弛豫时间T_1s和T₁₁,对有关文献中关于核四极共振弛豫时间的测量的3个观点提出了质疑。     

Magnetic relaxation in superconductors with varying creep behavior

The relaxation of an electromagnetic field inside low-and high-temperature superconductors in the case when the magnetic flux occupies the cross section incompletely is studied theoretically in the self-similar approximation. On the macroscopic level, creep is described by phenomenological equations for exponential and power I-V characteristics. A relation between the nonlinearity of the I-V curves and the magnetic relaxation process is found. The range of low electric fields is shown to be of considerable importance for the relaxation in high-temperature superconductors. In this range, the equations for exponential and power I-V curves may differ substantially from those at high fields. Complete analytical solutions are contrasted with a numerical solution to the problem.     

Application of trilinear SLICING to analyse a single relaxation curve

Determining the time constants and amplitudes of exponential decays from relaxation data is a common task in LF-NMR. In this communication, we present an application of the SLICING algorithm to evaluate its possibilities for solving this problem. The method, originally introduced to compare different samples, is applied here to analyse a single relaxation curve, using the embedding technique. To test this procedure, we acquired data sets from samples of liquids properly separated, and characterized by different relaxation times. The results show a good estimation of parameters, comparable with those obtained applying Marquardt's algorithm, when the components have sufficiently different relaxation times.     

Optical Kerr effect in supercooled water

We present molecular dynamics simulations of the optical Kerr effect in liquid and supercooled water and compare with recent time-resolved Kerr spectroscopy measurements [R. Torre, Nature (London) 428, 296 (2004)]. The short time features of the Kerr response, characterized by peaks near 15, 60, and 160 fs, are weakly temperature dependent. The long-time decay is well described by a stretched exponential with a nearly constant stretch parameter and relaxation times that follow a power law approximately (T-T(S))(-gamma), with T(S)=198.3 K and gamma=2.35. Our findings are discussed in the light of the spectroscopy data and previous simulation analyzes of the structural relaxation in supercooled water.     

Nuclear magnetic relaxation in ferrimagnetic Y3Fe5−x SixO12 films

The effect of silicon impurities on the damping of spin-echo signals from the ⁵⁷Fe nuclei of tetrahedral Fe³⁺ ions in epitaxial yttrium-iron-garnet films was investigated. It was found that for silicon concentrations 0.015⩽x⩽0.037 the damping of the spin echo is a two-component process, which made it possible to separate nuclei into two types, differing by both the longitudinal and transverse magnetic relaxation times. For silicon concentrations 0.044⩽x⩽0.073 the decay of the echo can be described by one exponential and all nuclei in the sample have the same transverse relaxation times and the same longitudinal relaxation times. The experimental results are interpreted on the basis of the supposition that impurity “macromolecules” form around the Si⁴⁺ ions. The relaxation times of the iron nuclei in a “macromolecule” are much shorter than the relaxation times of iron nuclei belonging to the matrix ions. The radius of a “macromolecule” is estimated on the basis of percolation theory. Fiz. Tverd. Tela (St. Petersburg) 40, 1494–1497 (August 1998)     

Structure of Polyethylene from X-Ray Powder Diffraction: Influence of the Amorphous Fraction on Data Analysis

It is now well established that the relaxation of volume of frozen-in liquids cannot be interpreted by single relaxation mechanisms. However, it remains obscure whether the decomposition of the volume relaxation curves into a sum of exponential terms has a physical sense or is merely a mathematical curve-fitting model. The fact that experimental data can be represented by a spectrum of relaxation times does not imply that there is physically such a spectrum. In particular, the experimental curves can also be fitted by nonlinear mathematical expressions. Kovacs has suggested that if a distribution of relaxation times exists, it should depend not only on the equilibrium temperature but also on the actual state of the glass [1]. This statement suggests that both the nonlinearity of the relaxation mechanism and the presence of relaxation times have to be expected. According to Kovacs et al. [2], the reason for the dependence of the relaxation times on the state of the glass is particularly demonstrated in the experiment of the “paradox of nonsymmetry”: Two specimens held at the same temperature are brought to the same final volume from a higher and a lower initial volume but with the same initial departure from the equilibrium value. Their contraction and dilatation are then observed to be asymmetrical, which cannot, according to Kovacs, be explained by only a distribution of relaxation times. Contrasting with Kovacs's approach, we resolve the “paradox” mathematically starting from the classical assumption of a distribution of relaxation times independent of the actual state of the glass. In other words, in our interpretation, the relaxation times vary with temperature only. The preexponential volume terms V_ij vary with the thermal past and can even assume negative values. The implications of our new analysis in regard to the concept of a “spectrum of relaxation times” for a glass are also discussed.     

Intrinsic spin fluctuations reveal the dynamical response function of holes coupled to nuclear spin baths in (In,Ga)As quantum dots

The problem of how single central spins interact with a nuclear spin bath is essential for understanding decoherence and relaxation in many quantum systems, yet is highly nontrivial owing to the many-body couplings involved. Different models yield widely varying time scales and dynamical responses (exponential, power-law, gaussian, etc.). Here we detect the small random fluctuations of central spins in thermal equilibrium [holes in singly charged (In,Ga)As quantum dots] to reveal the time scales and functional form of bath-induced spin relaxation. This spin noise indicates long (400 ns) spin correlation times at a zero magnetic field that increase to ～5 μs as dominant hole-nuclear relaxation channels are suppressed with small (100 G) applied fields. Concomitantly, the noise line shape evolves from Lorentzian to power law, indicating a crossover from exponential to slow [～1/log(t)] dynamics.     

Theoretical and experimental investigation of fluid rheology effects on modulated ultrasound propagation

A mathematical model is developed and presented to capture the effect of viscoelastic nature of a material on modulated ultrasound (US) pulses. The model is established by considering perturbation of material elements subject to modulated US pulses and by introducing the exponential relaxation of the perturbed fluid elements with a spectrum of time constants. Both the model and experimental findings revealed that consecutive perturbation of a material via the modulated US pulses enabled to probe the relaxation times of similar order of magnitudes to the frequency of the US modulation while filtering out the impact of other relaxation times on the US measurement. The US experimental results were verified by those of a conventional rheometer. Hence carrying out measurements at different US modulation frequencies in the Hz ranges seems to allow one to obtain the relaxation time spectrum of the investigated material in the time scales of milliseconds to seconds.     

Spin dynamics in the regime of hopping conductivity

Spin dynamics in the impurity band of a semiconductor with the spin-split spectrum is considered. Due to the splitting, phonon-assisted hops from one impurity to another axe accompanied by rotation of the electron spin, which leads to spin relaxation. The system is strongly inhomogeneous because of exponential variation of hopping times. However, at very small coupling an electron diffuses over a distance exceeding the characteristic scale of the inhomogeneity during the time of spin relaxation, so one can introduce an averaged spin relaxation rate. At larger values of coupling, the system is effectively divided into two subsystems: one where relaxation is very fast and another where relaxation is rather slow. In this case, spin decays due to the escape of the electrons from one subsystem to another. As a result, the spin dynamics is nonexponential and hardly depends on spin-orbit coupling. The text was submitted by the authors in English.     

The influence of the inertial medium relaxation component on the kinetics of “hot” transitions

The probabilities of “hot” electronic transitions for inertial (Gauss correlation function) and diffusion (exponential correlation function) polar solvent relaxation modes were calculated. It was shown that the probabilities of transitions for inertial and diffusion relaxation could differ by two-three times. The possibility and conditions of the replacement of the inertial by diffusion relaxation component were studied in the simulation of electronic transition kinetics in real solvents.     

Dynamics of magnetic defects in heavy fermion LiV2O4 from stretched exponential 7Li NMR relaxation

7Li NMR measurements on LiV2O4 from 0.5 to 4.2 K are reported. A small concentration of magnetic defects within the structure drastically changes the nuclear magnetization relaxation versus time from a pure exponential as in pure LiV2O4 to a stretched exponential, indicating glassy behavior of the magnetic defects. The stretched exponential function is described as arising from a distribution of 7Li nuclear spin-lattice relaxation rates and we present a model for the distribution in terms of the dynamics of the magnetic defects. Our results explain the origin of recent puzzling 7Li NMR literature data on LiV2O4 and our model is likely applicable to other glassy systems.     

Atomic short-range order in CuMn alloys

The establishment of atomic short-range order (SRO) has been investigated in CuMn alloys (5, 8,13,16,20 at. % Mn) by measurement of the electrical resistivity during isochronal and isothermal annealing. An increasing degree of SRO is accompanied by a reduction of resistivity; this effect increases with Mn concentration. For 8,13 and 16 at. % Mn SRO kinetics turn out to deviate significantly from single exponential behavior with a maximum at 13 at. % Mn, whereas at 5 and 20 at. % Mn SRO is adjusted in a single exponential process. For data analysis three methods are used: a sum of two and three exponentials as well as a log-normal spectrum of relaxation times. The strong interaction between second-nearest-neighbor atoms in CuMn seems to be essential for SRO kinetics but of minor importance for the value of SRO-induced resistivity change.     

Nonexponential quasiparticle decay and phase relaxation in low-dimensional conductors

We show that in low-dimensional disordered conductors, the quasiparticle decay and the relaxation of the phase are not exponential processes. In the quasi-one-dimensional case, both behave at small time as e(-(t/tau(in))3/2) where the inelastic time, tau(in), identical for both processes, is a power T-2/3 of the temperature. The nonexponential quasiparticle decay results from a modified derivation of the Fermi golden rule. This result implies the existence of an unusual distribution of relaxation times.     

The polymer mat: arrested rebound of a compressed polymer layer

Compression of an adsorbed polymer layer distorts its relaxed structure. Surface force measurements from different laboratories show that the return to this relaxed structure after the compression is released can require tens of minutes and that the recovery time can grow rapidly with molecular weight. We argue that the arrested state of the free layer before relaxation can be described as a Guiselin brush structure (O. Guiselin, Europhys. Lett. 17, 225 (1992)), in which the monomer density falls off only weakly with distance from the surface. This brush structure predicts an exponential falloff of the force at large distance with a decay length that varies as the initial compression distance to the 6/5 power. This exponential falloff is consistent with surface force measurements. We propose a relaxation mechanism that accounts for the increase in relaxation time with chain length.     

A Robust Method for Estimating Cross-Relaxation Rates from Simultaneous Fits to Build-Up and Decay Curves

This paper introduces a novel computational method for estimating relaxation rates among pairs of spin orders. This method simultaneously estimates all the auto- and cross-relaxation rates from the same measurements, and avoids the ill-conditioning problems associated with multiexponential fits. The method models the relaxation dynamics by a system of linear differential equations, and assumes that measurements of the spin orders have been made at an equally spaced sequence of time points. It computes a nonlinear least-squares fit of the exponential of the rate matrix at the shortest time point to these measurements. Preliminary estimates of the exponential matrix and initial spin orders from which to start the computations are obtained by solving simpler linear-least-squares problems. The performance of the method on simulated 2 × 2 test problems indicates that when measurements at eight or more equally spaced times spanning the maximum and inflection points of the build-up curves are available, the relative errors in the rates are usually less than the relative errors in the measurements. The method is further demonstrated by applying it to the problem of determining the cross correlation-induced cross-relaxation rates between the in-phase and antiphase coherence of the amide groups in the¹⁵N-labeled protein oxidized flavodoxin. Finally, the possibility of extending the method to other kinds of relaxation measurements and larger spin systems is discussed.