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324 PRICE Figure 7 A plot of simulated echo attenuation in the case of free diffusion Ž . and diffusion in a sphere Ž . based on the GPD approximation Ži.e., Eq. 99. versus . The parameters used in the simulation were 1 ms, D 5 1010 m2s1 , g 1Tm1 , R8m, and 1H 8 1 1 2.6571 10 rad T s . The echo attenuation in the case of diffusion in the sphere can be seen to go through three stages: Ž. i when 1, the diffusion appears unrestricted and the result is the same as that of free diffusion, Ž ii. as increases the spins begin to feel the effects of the surface, and Ž iii. when 1, the diffusion is fully restricted and the attenuation curve plateaus out. Ž 2 exp . term leaving the trigonometric Ž nm i.e., periodic. function 9Ž 2qR. cosŽ 2 qR. sinŽ 2 qR. EŽ q,. . 6 Ž 2qR. 102 Obviously, as q increases, the denominator of Eq. 102 increases Žsuch that Eq. 102 as a whole decreases . , but the trigonometric functions in the numerator result in the function having an infinite series of maxima and minima. The minima occur when q takes a value such that Ž 2qR. cosŽ 2 qR. sinŽ 2 qR. 0, for the first minima, this occurs when q 0.71R. The simulated echo intensity calculated using both the GPD and SGP approximations versus qR is shown in Fig. 8. We can see that at small attenuation values, the GPD and SGP approximations agree very well Ž n.b., . , but at larger attenuation values, the SGP approximation gives diffractive minima, whereas as expected, the GPD approximation does not. In well-chosen systems where the signal-to-noise ratio is sufficient and the sample geometry is monodisperse Žor at least not too 2 polydisperse . , it is possible to observe such minima e.g., Ž 105 .. The diffractive minima are an additional source of information and their position is R dependent. Anisotropic Diffusion Earlier, it was noted that isotropic diffusion is really just a special case, and more generally, we must consider anisotropic diffusion resulting from either the physical arrangement of the medium or anisotropic Ž i.e., nonspherical. restriction. Such situations commonly arise in biological Že.g., cells, skeletal muscle. and liquid crystals systems e.g., Refs. Ž 112, 113. and references therein , thus the diffusion process is represented by a Cartesian tensor, D Ž see Free and Restricted Diffusion . . In such systems, the echo-signal attenuation will have an orientational dependence with respect to the measuring gradient. For example, for anisotropic free diffusion, the g 2 D term in Eq. 51 must be replaced by g D g, where ÝÝ gDg D g g ,x, y, z 103
PULSED-FIELD GRADIENT NMR 325 Figure 8 A plot of the simulated echo-attenuation data for diffusion within a sphere calculated using the SGP approximation Ž . and the GPD approximation Ž . versus qR. The parameters used in the simulation were 1 ms, 100 ms, D 1 109 m2s1 , g 1 T m1 , R8 m, and 1H 8 1 1 2.6571 10 rad T s . The minima in the SGP plot occur when q takes a value such that the numerator in Eq. 102 equates to 0. Ž . and so we obtain 55 Ž . 2 2 Ž . ln E gDg 3 . 104 We remark that Eq. 104 can be rewritten in a form similar to Eq. 53 , i.e., ÝÝ lnŽ E. b D b:D 105 where ‘‘:’’ is the generalized dot product and b is Ž . now a symmetric matrix given by 114 2 H 0 2 b ŽFŽ t. 2HŽ t. f. ŽFŽ t. 2HŽ t. f. dt 2 2 Ž . g g 3 . 106 The first line in Eq. 106 is a general definition and the second line is the specific solution for the PFG sequence. Alternatively and totally equivalently, we can rewrite Eq. 52 as t 2H 0 lnŽEŽ t.. FDFdt. 107 T From Eq. 104 , it can be seen that the direction in which the diffusion is measured is determined by the gradient, and it is actually a diagonal element of D, the diffusion tensor in the gradient frame, that is measured. Thus, the equation relating the echo attenuation due to free diffusion when measured using a z-gradient Ži.e., Eq. 51. written in tensor notation is Ž . 2 2 2 Ž . ln E g D 3 b D . z zz zz zz 108 The diffusion tensor in the molecular frame, D, can be transformed to the laboratory Ži.e., gradient frame.Ž Fig. 9. by using rotation matrices e.g., Ref. Ž 115. 1 Ž . Ž . DR , DR , 109 Ž . where R , is the relevant rotation matrix and and are the polar and azimuthal angles between the director and gradient frames, respectively. Thus, the off-diagonal elements of D will vanish only when the director and laboratory frames of reference coincide. Thus, in the general case, both diagonal and off-diagonal elements of
Page 1 and 2: Pulsed-Field Gradient Nuclear Magne
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Page 5 and 6: the gradient oriented along the z-a
Page 7 and 8: ent pulses cancel and all spins ref
Page 9 and 10: ent echo, on the other hand, refocu
Page 11 and 12: For the case of Ž three-dimensiona
Page 15 and 16: and S corresponds to the value of
Page 17 and 18: every time a pulse is applied. Thu
Page 19 and 20: Finally, noting the standard integr
Page 21 and 22: Now, if we integrate Eq. 83 over al
Page 23 and 24: In the long time limit 1 Ž i.e.,
Page 25: In the next section, we will illust
Page 29 and 30: sor, D app. When a single effectiv
Page 33 and 34: Experimental aspects of PFG NMR wil
Page 35 and 36: 32. P. C. M. Van Zijl and C. T. W.
Page 37 and 38: 99. A. Caprihan, L. Z. Wang, and E.

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