What Are the Conditions for Exponential Time-Cubed Echo Decays?

Diffusion of precessing spins through a constant field gradient is well-known to produce two distinctive features: an exp(-bt(3)) decay of the echo amplitude in response to two pulses and a much slower decay of the Carr-Purcell echo train. These features will appear whenever the spin frequency is described by a continuous random-walk. The present work shows that this may also occur in the presence of motions with long correlation times tau(c)-continuous Gaussian frequency noise with an exponential autocorrelation has the correct properties over time durations smaller than tau(c). Thus, time-cubed echo decays will occur in situations other than physical diffusion. The decay rate of the Carr-Purcell echo train is shown to vary with the pulse spacing tau whenever the correlation time tau(c) is long; the slower Carr-Purcell decay compared to the two-pulse echo decay is not unique to diffusion. Simulations are presented that display time-cubed decays. The simulations confirm two important criteria: the echo time must be less than tau(c) and the frequency noise must consist of nearly continuous variations, as opposed to step-like changes. These criteria define the range of physical parameters for which time-cubed decays will be observable.     

Velocity autocorrelation spectra of fluid in porous media measured by the CPMG sequence and constant magnetic field gradient

Carr-Purcell-Meiboom-Gill (CPMG) train of radiofrequency pulses applied to spins in the constant magnetic field gradient is an efficient variant of the modulated magnetic field gradient spin echo method, which provides information about molecular diffusion in the frequency domain instead of in the time domain as with the two-pulse gradient spin echo. The frequency range of this novel technique is broad enough to sample the power spectrum of displacement fluctuation in water-saturated pulverized silica (SiO2) and provides comprehensive information about the molecular restricted motion as well as about the structure of the medium.     

Spectral characterization of diffusion in porous media by the modulated gradient spin echo with CPMG sequence

Carr-Purcell-Meiboom-Gill train of radiofrequency pulses applied to spins in the constant magnetic field gradient is an efficient variant of the modulated magnetic field gradient spin echo method, which provides information about molecular diffusion in the frequency-domain instead in the time-domain as with the two-pulse gradient spin echo. The frequency range of novel technique is broad enough to sample the power spectrum of displacement fluctuation in water-saturated pulverized silica (SiO(2)) and provides comprehensive information about the molecular restricted motion as well as about the structure of medium.     

Computer simulation of the spin-echo spatial distribution in the case of restricted self-diffusion

This article concerns the question of a proper stochastic treatment of the spin-echo self-diffusion attenuation of confined particles that arises when short gradient pulse approximation fails. Diffusion is numerically simulated as a succession of random steps when motion is restricted between two perfectly reflecting parallel planes. With the magnetic field gradient perpendicular to the plane boundaries, the spatial distribution of the spin-echo signal is calculated from the simulated trajectories. The diffusion propagator approach (Callaghan, "Principles of Nuclear Magnetic Resonance Microscopy," Oxford Univ. Press, Oxford, 1991), which is just the same as the evaluation of the spin-echo attenuation by the method of cumulant expansion in the Gaussian approximation, with Einstein's approximation of the velocity correlation function (VCF) (delta function), agrees with the results of simulation only for the particle displacements that are much smaller than the size of the confinement. A strong deviation from the results of the simulation appears when the bouncing rate from the boundaries increases at intermediate and long gradient sequences. A better fit, at least for intermediate particle displacements, was obtained by replacing the VCF with the Oppenheim--Mazur solution of the Langevin equation (Oppenheim and Mazur, Physica 30, 1833--1845, 1964), which is modified in a way to allow for spatial dependence of particle displacements. Clearly, interplay of the correlation dynamics and the boundary conditions is taking place for large diffusion displacements. However, the deviation at long times demonstrates a deficiency of the Gaussian approximation for the spin echo of diffusion inside entirely closed pores. Here, the cumulants higher than the second one might not be negligible. The results are compared with the experiments on the edge enhancement by magnetic resonance imaging of a pore.     

X- and Q-Band Electron Spin Echo Study of Stochastic Molecular Librations of Spin Labels in Lipid Bilayers

Electron spin echo (ESE) of nitroxide spin labels allows detecting fast nanosecond stochastic restricted rotations (stochastic molecular librations), which is a common property of molecules in disordered media including biological systems. Under the typical experimental conditions, the anisotropic electron paramagnetic resonance (EPR) spectrum of a nitroxide is only partly excited by microwave pulses, which allows selecting an anisotropic contribution to the transverse spin relaxation by comparing echo decays at different spectral positions. On the other hand, for low-amplitude orientational motion, the excitation bandwidth is large enough to cover the range of spectral diffusion occurring during the echo formation. To verify that the two-pulse echo decay is indeed related to fast motions, the stimulated electron spin echo can be used. In addition, theory predicts an increase of the relaxation rates at higher microwave resonance frequency. To check this prediction, in the present work we performed a comparative study of ESE decays at microwave X- and Q-bands, for spin-labeled lipids in the gel phase of a 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) bilayer. A good agreement found between experimental data and computer simulation provides additional justification for the model of fast stochastic molecular librations.     

Spin Echo Attenuation of Restricted Diffusion as a Discord of Spin Phase Structure

By using the particle probability density we analyze the spin echo attenuation of particles, diffusing in a bounded region. It provides a means to expand a nonuniform spin phase distribution into a series of waves that characterize the geometry and boundary conditions of confinement. Random motion disrupts the initial phase structure created by applied gradients and consequently discords its structure waves. By assuming the spin phase fluctuation and/or the randomness of spin phase distribution in the subensemble as a Gaussian stochastic process, we derive a new analytical expression for the echo attenuation related to the particle velocity correlation. For a diffusion in porous structure we get the expression featuring the same “diffusive diffraction” patterns as those being found and explained by P. T. Callaghan and A. Coy (“Principles of Nuclear Magnetic Resonance Microscopy,” Oxford Univ. Press, Oxford (1991);J. Chem. Phys.101, 4599–4609 (1994)) with the use of propagator theory. With the new approach we cast a new light on the phenomena and derive analitically how the diffusive diffractions appear when the sequence of finite or even modulated gradients are applied. The method takes into account the non-Markovian character of restricted diffusion, and therefore the echo dependence on the diffusion lengths and on the strength of applied gradient differs from the results of authors assuming the Markovian diffusion either by dealing with the diffusion propagators or by the computer simulation of Fick's diffusion.     

Enhanced transverse relaxation in porous media due to internal field gradients

We present a detailed comparison between the theoretically computed spin echo decay of a fluid-saturated periodic porous medium with strong internal field gradients and that obtained from various approximations using the free diffusion formula which is strictly valid only for infinite uniform fluids. The theoretical computation of the spin echo amplitude was carried out by using the diffusion eigenstates in Fourier representation, and the internal field gradients induced by magnetic susceptibility contrast were calculated by using a two-component composite theory. The comparison allows us to have an assessment of the regime of validity of various approximations for a periodic porous medium where a rigorous theoretical computation of the enhanced transverse relaxation due to magnetic susceptibility contrast induced field heterogeneity is possible.     

A modified PGSE for measuring diffusion in the presence of static magnetic field gradients

With a proper timing of pi pulses, it is possible to reduce the effect of the static internal magnetic field gradient on the measurement of diffusion with the pulsed gradient spin echo (PGSE). A pulse sequence that in the first order eliminates the effect of weak internal static gradients in a standard PGSE experiment is introduced. The method should be applied in the cases, where strong and short magnetic gradient pulses are used to investigate the motion of liquid in heterogeneous samples with large susceptibility differences such as porous media.     

NMR diffusometry and the short gradient pulse limit approximation

In NMR diffusometry, one often uses the short gradient pulse (SGP) limit approximation in the interpretation of data from systems with restricted diffusion. The SGP limit approximation means that the gradient pulse length, delta, is so short that the spins do not diffuse during the pulse duration, but this condition is rarely met. If the length scale of the pores corresponds to the molecular mean square displacement during the gradient pulse, the measured echo intensities become a function of the gradient pulse length. Here, we have studied highly concentrated emulsions to show how the length of the gradient pulse influences NMR diffusion experiments. We have focused on molecules confined to one pore and molecules that can migrate through the porous system. For the former the echo decays give smaller pores than the actual case and for the latter we show large changes in echo decay depending on the gradient pulse length, everything else being equal.     

Electron spin coherence in Si

We discuss pulsed electron spin resonance measurements of electrons in Si and determine the spin coherence from the decay of the spin echo signals. Tightly bound donor electrons in isotopically enriched ²⁸Si are found to have exceptionally long spin coherence. Placing the donors near a surface or interface is found to decrease the spin coherence time, but it is still in the range of milliseconds. Unbound two-dimensional electrons have shorter coherence times of a few microseconds, though still long compared to the Zeeman frequency or the typical time to manipulate a spin with microwave pulses. Longer spin coherence is expected in two-dimensional systems patterned into quantum dots, but relatively small dots will be required. Data from dots with a lithographic size of 400 nm do not yet show longer spin coherence.     

Multiexponential attenuation of the CPMG spin echoes due to a geometrical confinement

The CPMG multi-echo technique is often used to investigate the translational motion of diffusing nuclei in a confining medium. Henceforth, periodically repeated RF pulses with a diffusion-sensitizing gradient yield a formation of spin echoes of gradually decreasing amplitudes. The parameters of their exponential fits may characterize the structure of porous materials or biological tissue. In this paper, a multiexponential character of the CPMG measurements is rigorously demonstrated, once a geometrical confinement is present. Based on the multiple propagator approach, we derived a spectral representation for the echo amplitudes under external magnetic field of an arbitrary gradient profile. The multiple relaxation times and their spectral weights were found in a general form. The study of simple restrictive media allowed to obtain a quantitative condition under which the multiexponential attenuation is reduced to a monoexponential one.     

MRI edge enhancement as a diffusive discord of spin phase structure

The enhancement of magnetic resonance image intensity near impermeable boundaries can be nicely described by a new approach where the diffusional spin echo attenuation is linked to the correlation function of molecular motion. In this method the spin phase structure created by the applied gradient is considered to be a composition of plane waves with the wave vectors representing feasible momentum states of a particle in confinement. The enhancement of edges on the magnetic resonance images (MRI) comes out as a discord of plane waves due to particle motion. It results from the average of the wave phase by using the cumulant expansion in the Gaussian approximation. The acquired analytical expression describes the MRI signal space distribution where the enhancement of edges depends on the intensity and the duration of gradient sequence as well as on the length of the mean squared particle displacement in restricted geometry. This new method works well with gradients of general waveform and is, therefore, suitable for imaging sequences where finite or even modulated gradients are usually used.     

Multi-dimensional inverse Laplace spectroscopy in the NMR of porous media

Multi-dimensional NMR methods based on Inverse Laplace Transformations (ILT) may be used to examine the behavior of liquid state molecules in a porous matrix. The ILT is particularly useful when the signal is characterized by multi-exponential decay, for example in spin relaxation or in the dephasing of the NMR spin echo signal associated with molecular diffusion under the influence of pulsed magnetic or internal field gradients. Both correlation and exchange experiments are possible, the latter providing insight regarding the migration of molecules between regions characterized by different local dynamics.     

Cross-relaxation effects in stimulated-echo-type PGSE NMR experiments by bipolar and monopolar gradient pulses

Exchange of longitudinal spin polarization by dipolar cross relaxation between nonequivalent spins results in a modulation of the stimulated echo signal on increasing the encoding/decoding delays and in a multiexponential decay on increasing the diffusion time. These artifacts are suppressed by 180° pulses inserted in the middle of the gradient encoding/decoding periods. The efficiency of the gradient encoding is preserved if bipolar gradient pulses are used instead of monopolar pulses. The behavior of the different pulse sequences is demonstrated by ¹⁹F PGSE NMR experiments in a lyotropic liquid crystal in both isotropic micellar and oriented nematic phases.     

Modified oscillating gradient pulses for direct sampling of the diffusion spectrum suitable for imaging sequences

A variation of the oscillating gradient spin echo method had been developed, which isolates temporal frequencies of the dephasing spectrum. This allows sampling of the diffusion spectrum, the Fourier transform of the velocity correlation function (VCF). It has been shown that restriction and flow alter this function in ways that can be mathematically characterized, yielding quantitative information on restriction geometry and flow parameters. It is demonstrated that in many systems of interest, dispersion of velocity will produce a peak in the VCF spectrum near omega=0, while restricted diffusion will manifest itself in the spectrum at higher frequencies. The method, therefore, may be useful for decoupling their effects on the apparent diffusion coefficient (ADC), as well as in revealing the physics of both phenomena. This method has been implemented in model systems of packed beads, yielding data consistent with theoretical models of restricted diffusion spectra and data from one previous study. The method may have significant application to biology and medicine, as well as the study of transport phenomena in porous media and complex flow.     

Restricted diffusion and exchange of water in porous media: average structure determination and size distribution resolved from the effect of local field gradients on the proton NMR spectrum

NMR Pulsed field gradient measurements of the restrained diffusion of confined fluids constitute an efficient method to probe the local geometry in porous media. In most practical cases, the diffusion decay, when limited to its principal part, can be considered as Gaussian leading to an apparent diffusion coefficient. The evolution of the latter as a function of the diffusion interval yields average information on the surface/volume ratio of porosities and on the tortuosity of the network. In this paper, we investigate porous model systems of packed spheres (polystyrene and glass) with known mean diameter and polydispersity, and, in addition, a real porous polystyrene material. Applying an Inverse Laplace Transformation in the second dimension reveals an evolution of the apparent diffusion coefficient as a function of the resonance frequency. This evolution is related to a similar evolution of the transverse relaxation time T2. These results clearly show that each resonance frequency in the water proton spectrum corresponds to a particular magnetic environment produced by a given pore geometry in the porous media. This is due to the presence of local field gradients induced by magnetic susceptibility differences at the liquid/solid interface and to slow exchange rates between different pores as compared to the frequency differences in the spectrum. This interpretation is nicely confirmed by a series of two-dimensional exchange experiments.     

Double PGSE NMR with Stimulated Echoes: Phase Cycles for the Selection of Desired Encoding

When two pairs of position-encoding pulses are used in a pulsed gradient spin echo (PGSE) NMR experiment, it is possible to examine velocity fluctuations. The one-dimensional version of double PGSE NMR uses identical pulse pairs whose amplitudes are stepped simultaneously. In the two-dimensional version (VEXSY) the pulse pairs are stepped independently, resulting in a velocity exchange spectrum. A key limitation in such experiments is transverse relaxation, so that stimulated echoes are often used as the method of choice. It is shown here that the use of stimulated echoes results in a superposition of signals arising from different magnetization pathways such that the spin phases may reflect both the sum and difference of displacements over the pulse pair encoding times, as well as the displacement over the exchange time between the pulse pairs. A phase cycle scheme that selects desired encodings as required is demonstrated.     

A mixed basis approach in the SGP-limit

A perturbation method for computing quick estimates of the echo decay in pulsed spin echo gradient NMR diffusion experiments in the short gradient pulse limit is presented. The perturbation basis involves (relatively few) dipole distributions on the boundaries generating a small perturbation matrix in O(s²) time, where s denotes the number of boundary elements. Several approximate eigenvalues and eigenfunctions to the diffusion operator are retrieved. The method is applied to 1D and 2D systems with Neumann boundary conditions.     

Determination of pore space shape and size in porous systems using NMR diffusometry. Beyond the short gradient pulse approximation

The influence of finite length gradient pulses on NMR diffusion experiments on liquids confined to diffuse between two parallel planes is investigated. It is experimentally verified that the pore size decreases when determined using finite gradient pulses if the results are analyzed within the short gradient pulse approximation. The results are analyzed using the matrix formulation. The observed minima in the echo decay profiles are considerably less sharp than theoretical analysis would indicate and we suggest that this is due to the presence of a distribution of pore sizes in the sample. In addition, effects due to the presence of background gradients are discussed. It is argued that effects due to the finite length gradient pulses are relatively minor and in realistic applications the effects due to inhomogeneities in pore sizes and effects due to background gradients will constitute more serious problems in pore size determinations by means of NMR diffusometry.     

Electron spin-lattice relaxation time and spectral diffusion in γ-irradiated l-alanine

We performed continuous (CW) wave and pulsed ESR experiments to obtain information on the relaxation behavior of the l-alanine radical in an irradiated single crystal. The analysis of the CW saturation behavior gives a relaxation time of 2.8 μs. The echo detected saturation recovery was obtained for a number of different experimental conditions. In any case only a portion of the 120 G wide ESR spectrum can be affected by the microwave (MW) pulses, spectral diffusion is active and a multi-exponential decay is therefore obtained. We measured characteristic spectral diffusion times of 1–10 and 20–50 μs. We found that a long time of about 200 μs can be measured only by using a train of long selective saturating pulses and short detecting pulses. The stimulated echo decay is bi-exponential, and the characteristic times are very short. A variable temperature investigation in the range 200 to 290 K showed that the decay is governed by the spectral diffusion and by the transverse nuclear spin relaxation timeT _2n of the methyl protons.