Echo attenuation

A measure of the echo attenuation within each pixel of an image created using the pulse sequence of figure Bl.14,9 perhaps by repeating the experiment with different values of and/or 8, gives data from which a true diffusion map can be constructed [37, 38],... 

The situation becomes quite different in heterogeneous systems, such as a fluid filling a porous medium. Restrictions by pore walls and the pore space microstructure become relevant if the root mean squared displacement approaches the pore dimension. The fact that spatial restrictions affect the echo attenuation curves permits one to derive structural information about the pore space [18]. This was demonstrated in the form of diffraction-like patterns in samples with micrometer pores [19]. Moreover, subdiffusive mean squared displacement laws [20], (r2) oc tY with y < 1, can be expected in random percolation clusters in the so-called scaling window,... 

In order to avoid flow artifacts it may be advisable to replace the spatial encoding pulses (right-hand box) by velocity compensated pulses such as shown in Figure 2.9.4(e) for phase encoding. The amplitude of the Hahn spin-echo is attenuated by hydrodynamic dispersion. Evaluation of the echo attenuation curve for fixed intervals but varying preparation gradients (left box) permits the allocation of a hydrodynamic dispersion coefficient to each voxel, so that maps of this parameter can be rendered. 

B. Balinov, B. Jonsson, P. Linse, O. So-derman 1993, (The NMR self-diffusion method applied to restricted diffusion -simulation of echo attenuation from molecules in spheres and between planes),/. Mag. Reson. A 104, 17. 

The diffusion coefficient (D) of ferf-butyloxycarbonyl-L-phenylalanine (Boc-Phe) was determined in a Merrifield network of polystyrene (PS) gels used as a solid phase reaction field.108 When probe molecules have multicomponents in diffusion on the measurement time scale, the total echo attenuation is given by a superposition of contributions from the individual components ... 

Several NMR pulse sequences have been developed for measuring the diffusion coefficient. The most successful and currently used experiment is the modified spin-echo experiment (PGSE) proposed by Stejskal in 1967.62 As the echo signal is due to the transversal components of magnetic moments, the echo attenuation normalized for the attenuation due to relaxation can be written as63 ... 

Considering the probability described in Equation (5), the spherical polar coordinates and setting g = gz and R r, r0, the final expression for the echo attenuation is ... 

This experiment has two limitations (a) nuclei with short T2 are difficult to be detected and (b) multiplets produce a distorted signal phase after n pulse which inverts the coupled spin states. Both limitations are partially overcome by the stimulated echo experiment (STE).64,65 The main difference with the experiment previously described is that the echo attenuation due to the diffusion competes with Tx rather than with T2. The equation analogous to Equation (13) is now ... 

NMR diffusometry Decay of spin echoes in the presence of a magnetic field gradient (echo attenuation curve) Price (1996, 1997, 1998a)... 

Diffusive attenuation is not merely of academic interest because it will probably be an inherent cause of echo attenuation in on-line NMR sensors whenever the sample moves through gradients and inhomogeneities in the main sensor field. It is therefore important to understand their potential effect and possible use in assessing quality of horticultural products. 

Fig. 1. LED pulse sequence for the measurement of stimulated echoes attenuated by diffusion. Typical values for delays and pulse widths are 5 ms for the gradient pulse widths tg, 2-3 ms for the recovery time tr, 0.1 s for the diffusion time td, and 5 ms for the storage time...

Lastly, Callaghan and Pinder believed to have detected cooperative diffusion in their experiment48). Their polystyrene specimen of M = 2x 10 displayed a two-component echo attenuation, interpreted in terms of two time-dependent diffusion coefficients. The more rapid component was not observable at long diffusion times, and at short diffusion time (t = 16 ms) scaled with polymer concentration (2, 4, and 8 %) in the manner... 

Zibrowius et al. (77) used an NMR spin-echo attenuation technique to estimate the ratio of the diffusion tensor element related to motion along the y-axis (the straight channel) and the average of the other two elements. For a methane loading of 8 molecules per unit cell, the ratio was estimated to be less than 5 at room temperature ... 

Tanner [49] measured diffusion coefficients of water in three different types of frog muscle cells. He used a variety of magnetic field gradient techniques so as to cover a wide range of diffusion times A= 1 ms to 1 s. The time dependence of the diffusion coefficient was analyzed to obtain the intracellular diffusion coefficients and estimates of the permeability of the cell membranes. In restricted diffusion studies three 90 degree r.f. pulse sequences (stimulated echo) are often used which provides PG-NMR experiments with long diffusion times to explore the dependence of diffusion time on the echo attenuation [49]. 

The most difficult step in the performance of PFG-measurements is the analysis of the experimentally measured R-values. For unrestricted diffusion (i.e. the quantity V2DA is much smaller than the distance between the barriers) the PFG-NMR echo attenuation is given by ... 

In the situation where V2DA is of the same order or larger than the distance between any diffusional barriers in the system, so-called restricted diffusion is observed. In a W/O emulsion, for example, the water molecules are restricted in the extent of their diffusion by the presence of the boundaries of the water droplets. The extent of the restriction of the diffusion of the water molecules is reflected in the ratio R = E /E. An expression for the echo attenuation R-factor as a function of droplet diameter has been derived by Murday and Cotts for uniform spherical droplet sizes [7] ... 

Packer and Rees [3] extended the expression derived by Murday and Cotts [7] to include the effects of a droplet size distribution, assuming a log-normal distribution. By curve fitting they were able to determine the principal parameters of such a distribution from the experimental R-values. In the presence of a distribution of sizes, the observed echo attenuation ratio R is expressed in terms of the calculated attenuation of individual droplets, R ... 

Eq. 1 showed that in the case of unrestricted diffusion the echo attenuation value R depends upon the durations 8 and A. This is also true in the case of restricted diffusion, although in a different manner. The dependence of the R-value upon these two parameters is shown in Fig. 5. This figure clearly shows that the echo attenuation factor R steadily decreases with increasing A in the case of unrestricted diffusion, but becomes independent of this parameter in the case of restricted diffusion. It may be deduced from this figure that it is necessary to determine the parameters of the log-normal droplet size distribution R as a function of A or by measuring R as a function of 8 for a fixed large value of A. Measurement of only one R-value, at a chosen 8 or A, is not sufficient for a careful determination of the droplet size distribution in Fig. 5 a given In R-value can be found on more than one In R versus A-5/3 curve. This means that the In R-values have to be determined for different values of A and/or 8. 

Figure 5. Echo attenuation R versus the time interval A between field gradient pulses for different widths 5 of the field gradient pulses in the case of unrestricted (A) and restricted (B) diffusion.

Fleisher et al. [12] studied the self-diffusion of oil and water in rape seeds. The selfdiffusion of oil was found to be completely restricted. The experiments could be explained in toms of the model of diffusion within spherical droplets and a Gaussian mass distribution of the droplet radii. At the same time Van den Enden et al. [9] introduced the technique described above. It is a rapid method for the determination of water droplet size distributions in spreads by using low resolution pulsed field gradient NMR. Their method was based on the recognition that a set of echo attenuation values (R) as a function of the field gradient pulsed width, obtained under conditions where R is independent of the time allowed for diffusion, contains all the necessary information on the water droplet size distribution (see above). A log-normal distribution of water droplet sizes was assumed. 

Figure 6. Theoretical curves, showing the echo attenuation R versus the time interval 5. Parameters, used in these calculations are A=210 ms, D=1.31-10 9mV, G=2.0T/m.

Figure 1 Diffusional spin echo attenuation of Boc-Phe in MPS1 gels with DMF-d7 as solvent on A at 30 °C by varying field-gradient pulse duration (diffusing time) A, where the Boc-Phe concentration is 10 wt%. (A) A = 10 ms and (B) A=100 ms.

Figure 3 Diffusional stimulated echo attenuations of PSs with different Mws in PMMA gels (PMMA gel samples 1-7) with CDCl3 as solvent, respectively, and PS in CDClj solution by varying field-gradient strength g. (O) PS in CDCl3 solution ( ) in PMMA gel prepared without PS (PMMA gel sample 1) (A) in PMMA gels prepared with PS (PMMA gel samples 2-7) The Mws of PSs are (A) 4000, (B) 19,000, (C) 29,000, (D) 50,000, (E) 170,000 and (F) 400,000.

Figure 4 Diffusional stimulated echo attenuation of PS (Mw = 4000) in PMMA gel sample 2 with CDCl3 as solvent by varying field-gradient strength g at room temperature, where A —40 ms (O), A = 60 ms ( ), A = 100 ms ( ), A = 300 ms ( ) and A = 500 ms (A).

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