Diffusion echo intensity

Three analytical expressions for the spin-echo intensity as a function of the gradient in a pulsed field gradient NMR experiment for spins diffusing in a sphere with reflecting walls are reinvestigated. It is found that none of the published formulas are completely correct. By numerical comparisons the correct formula is found. 

Since the Murday-Cotts paper in 1968 much progress has been made toward the theoretical description of the spin-echo intensity E g, A) for spins diffusing in well-defined geometries. Tanner and Stejskal derived already in 1968 the exact expression of ii(g. A) for spins diffusing in a rectangular box. The derivation of an exact expression for E g, A) for diffusion in a sphere with reflecting walls is not a trivial mathematical problem and it took between 1992 and 1994 when three expressions were published. All three expressions are only valid in the short-gradient-pulse approximation (see below). 

For a complete definition of Eq. (53) we need to determine the constants Cnk from the conditions (17)-(19) and then calculate the Fourier integral Eq. (1) for the echo signal. To avoid the tedious algebra we compare the three published solutions numerically, but first reproduce these solutions here using our notation. Two of these solutions resulted from a calculation that included the effect of surface relaxation. To make a correct comparison we eliminate from the equations the terms due to relaxation. Then we have the following formulae for the echo intensity for diffusion in a sphere with radius a and reflecting walls ... 

By chosing short values of x, the effect of diffusion may be eliminated, because the diffusion is effective in reducing the echo intensity only during the 2x period. However, there is a modified version of spin-echo pulse sequences which provides the measurement of the spin diffusion coefficients (D) by NMR, as it will be shown later. 

The effect of curing on the diffusion of polymer and the curing agent is studied for the system of hydroxyl-terminated polybutadiene (R-45-M)/isophorone disso-cyanate (IPDI). Both components contribute to the echo intensity and the plot of In P(x)/I(0)] vs (G5)2 (A — 5/3) consists of two exponentials (Eq. (22)) the fast component (the steep intial slope) is attributed to the IPDI, and the long component to the R-45-M. The dependence of both diffusion constants on the curing time is shown in Fig. 19. The accuracy for Dfast data is less pronounced than for the polymer D(Mn), because only the first few data points are relevant for its determination. Furthermore, the low tail of the R-45-M molecular weight distribution nearly coin-... 

For spins which did diffuse from their original position (or more accurately who changed their x-coordinate value) during the time change, the phase jump their magnetisation vector acquired during the first pulse is not compensated by the phase jump during the second pulse and their contribution to the echo intensity is decreased. 

Figure 6 represents typical plots of the spin-echo intensity in PFG NMR experiments. Comparing the slopes of these representations with those of standard liquids, one obtains the mean self-diffusivities, which are found to decrease with increasing sorbate concentration (5,12,16,59,60). It appears from Fig. 6 that within the accuracy of the measurement no deviation from a single exponential decrease may be observed. A comparison of the experimental spin-echo attenuation (Fig. 6) with the results of numerical calcula-... 

The use of PFGs in combination with echo sequences (case d) allows for the observation of the lateral mobility of any observed component. Experimentally, the echo intensity of any observable resonance in a given echo experiment (most commonly, Hahn echoes or stimulated echoes are used) is studied as a function of the gradient strength and the diffusion... 

Diffusion of nuclear spins in an inhomogeneous magnetic field can cause dephasing and contribute additionally to T2, making it shorter. Thus, the echo intensity observed in SE, as well as GE, methods is sensitive to Ti, T, and diffusion effects. MR images sensitive to diffusion effects, i.e., diffusion-weighted images, can be produced by... 

For this NMR diffusometiy experiment the echo intensities for free diffusion is given by (Balinov et al., 1993) ... 

Determine the self-diffusion coefficients of probes with more complicated NMR spectra by collecting the echo intensity A 2r) as a function oi g (= kl) while keeping all the other variables in eqn (5.1.1) constant. Use this procedure for the calibration also. (The time r between the 90° and the 180° pulse should be around 72.) Set up the pulse sequence for r = T2, and determine the intensity A 2r) for at least ten different pulse currents / then calculate kfrom eqn (5.1.1). 

It follows from Eq. (25) that any periodicity (/) in the propagator should give rise to a coherence peak, that is, to a local maximum for q = ybg = Iti/I in the representation of the NMR spin echo intensity versus the intensity of the field gradient pulses (78 ). Such behavior has indeed been observed in recent PEG NMR studies of water diffusion through the free space within an array of loosely packed monodisperse polysterene spheres [78],... 

This provides a means of measuring the diffusion coefficient of a fluid. The spin-echo intensity is measured while q is varied (by changing g or S). A plot of ( (9, A)) against 47r (A - 6/3) will be linear with gradient -D. Some... 

Calculations of the spin-echo intensity are complicated by the fact that surface relaxation may play a significant role. A general formalism for calculating PFG spin-echo attenuation for restricted diffusion in isolated pores has recently been proposed that allows for wall relaxation effects. Expressions have been obtained for the cases of diffusion within a sphere, and for planar and cylindrical geometries.These show that diffraction effects are still apparent even when surface relaxation is rapid. Also, the locations of the minima in the spin-echo intensities are not particularly affected by varying the surface relaxation parameter, Analysis of PFG spin-... 

Fig. 8. Plot of spin-echo intensity (on a logarithmic scale) against the gradient wave vector q for water in a randomly packed bed of polystyrene spheres with average diameter 15.8 pm. The diffusion time A was 20 ms (squares), 40 ms (triangles), 70 ms (circles) and 110 ms (diamonds). A coherence peak is observed at a position corresponding to the average interpore spacing. (Reproduced with permission from ref. 147, 1992, American Institute of Physics.)...

The diffusion coefficients that can be measured with the PGSE method cover the range from fast diffusion of small molecules in solutions with D values typically around 10" m /s to very slow diffusion of, for instance, polymers in the semidilute concentration regime, where D values down to 10 m /s can be measured [19]. Measurements of such very slow diffusion requires gradients of extreme magnitudes and places severe demands on the actual experimental setup [9]. What often limits the lowest value of D that can be measured is the value of spin-spin relaxation time, T2. As a general rule, slow diffusion is often found in systems that also show rapid transverse relaxation. As a consequence, the echo intensity gets severely damped by T2 relaxation in such systems. For microemulsion systems, such problems are virtually nonexistent for the solvents, while for the surfactant molecules the accuracy is often reduced because of T2 effects. 

When the network structure of a gel is heterogeneous and the gel consists of small cells with microspace, the diffusion of the solvent is restricted spatially and will not follow Eq. (19). Thus, the observed diffusion coefficient reduces over time. Tanner [7] and Meerwall [18] studied such a limited diffusion using a model with parallel blocking walls with distance a and permeability p. They derived the relationship between the spin echo intensity A t) when the pulsed magnetic gradient with... 

Figure 2 Plots of the calculated normalized echo intensity in the absence of relaxation and diffusion for the ODD sequence (top, circles) and the EVEN sequence (bottom, triangles) versus echo number following a train of 90° pulses. The first echo intensity is the same for each sequence and is governed only by T2 in the absence of diffusion. Subsequent echoes have contributions from T,. The EVEN repeating pattern has two points up and two points down for 90°. Reproduced with permission from Bain AD and Randall EW (1996) Spin echoes in static gradients following a series of 90 degree pulses. Journal of Magnetic Resonance A123 49-55.

As evidenced from the above discussion, vibrational line shapes provide information mostly about intermolecular structure. Transient hole burning and more recently echo experiments, on the other hand, can provide information about the dynamics of spectral diffusion. The first echo experiments on the HOD/ D2O system involved two excitation pulses, and the signal was detected either by integrating the intensity [20] or by heterodyning [22]. The experiments were analyzed with the standard model assuming Gaussian frequency fluctuations. The data were consistent with a spectral diffusion TCF that was bi-exponential, involving fast and slow times of about 100 fs and 1 ps, respectively. 

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