Pulsed-field gradient nuclear magnetic resonance as a tool for ...

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316 PRICE ing displacement in the direction of the gradient, Ž . which can be written most generally as 79 ² Ž . Ž .: z t z t a a b HHH 1 0 z 2 0 z 0 0 1 a Ž r r . Ž r r . Ž r . PŽ r ,r ,t . PŽ r ,r ,t t . dr dr dr . 60 1 2 b a 0 1 2 It should be noted that Eq. 60 holds only when t t . We will now evaluate Eq. 60 b a for the Ž particularly simple. case of PŽ r , r , t. 0 1 as given by Eq. 27 Ž i.e., free diffusion . . Since we are only interested in motion in one dimension, we can simplify our task by using the one-dimensional version of Eq. 27 , 12 PŽ z , z ,t. Ž 4Dt. exp 0 1 ž 2 Ž z z . 1 0 4Dt / 61 and making obvious changes to Eq. 60 thus obtain ² Ž . Ž .: z t z t a a b H H H 0 1 0 2 0 Ž z .Ž z z .Ž z z . 12 Ž . 4Dt a ž / ž Ž z z . / 4DŽ t t . 2 1 0 12 Ž z z . exp Ž4DŽ t t .. b a 4Dta 2 1 exp dz dz dz . 2 b a 0 1 2 Now we let Z z z and Z z z , and 1 1 0 2 2 0 thus, H Ž z . 0 dz0 ž / 2 12 Z1 H Z Ž 4Dt . 1 a exp dZ1 4Dta H Z24D tbta Ž Ž .. ž / 12 Ž Z . 2Z1 exp dZ 2 . 4DŽ t t . b a 2 By noting Eq. 29 , we can remove the integral over z , and making the substitution Z 0 2Z2 Z 1, we then get ž / 2 12 Z1 H Z Ž 4Dt . 1 a exp dZ1 4Dta H Z2Z1 4D tbta Z 2 ž b a / Ž .Ž Ž .. 12 2 exp dZ . 62 2 4DŽ t t . We now consider the integral over Z in Eq. 62 2 , which we rewrite as 12 Ž4DŽ t t .. b a ½ 2 Z2 Z1Hexp dZ 4DŽ t t . b a ž / 2 Z2 ž 4DŽ t t . / H 5 2 b a Z exp dZ . 63 The first integral in Eq. 63 can be evaluated with the standard integral e.g., integral 3.323 2. in Ref. Ž 53 ., H e dxe . 64 p 2 2 2 2 p x qx q 4p ' Ž Ž .. 12 2 a b by setting x Z , p 4D t t and Ž Ž .. 12 q0, to give 4D tbt a . The second integral in Eq. 63 can be evaluated using the standard integral Eq. 3.462 6. in Ref. Ž 53 ., H ( 2 q 2 2 px 2qx q p xe dx e Re p 0. p p 65 In our case, x Z , p 4DŽ t t. 2 b a and q 0, and so this integral equals 0. Hence, Eq. 63 reduces to simply Z and now Eq. 62 becomes 1 2 12 ² zŽ t . zŽ t .: Z Ž 4Dt . a b 1 a Z2 1 1 ž 4Dt / a a H exp dZ . 66
Finally, noting the standard integral given in Eq. 32 , we obtain the final result of ² zŽ t . zŽ t .: 2Dt 67 a b a a which is of course equal to the mean-squared displacement for the one-dimensional diffusion equation Žsee Eq. 30 . . We now come to a subtle point in evaluating Eqs. 59 and 60 . In performing the integrals, we need to consider the range of integration when inputting ² zt Ž . zt Ž .: a b a; that is, we have to inter- change t for t in Eq. 60 a b depending on whether tatb or tat b. This can be understood by noting that the exponentials in Eq. 60 must have negative exponents Žrecall the validity of Eq. 60 . ; hence, we get Ž 60, 80. ² Ž . Ž .: ² 2 z t z t z Ž t .: 2Dt if t t a b a a a a a b ² Ž . Ž .: ² 2 z t z t z Ž t .: a b a b a 2Dtb if tat b. 68 Continuing on with our derivation of Eq. 58 , we have ½ t t1 t 2 2 2 a a H H b b 1 t1 ² : g 2Dt dt t1 H a b a ta 2Dt dt dt t1 t1 H H a b a t1 t1 2 2Dt dt dt t1 ta H H b b t1 t1 t1 H a b a5 ta 2Dt dt 2Dt dt dt 2 2 2 Ž . g 2 3 69 If we evaluate Eq. 12 using the distribution of phases as given in Eq. 57 , we find that the echo attenuation is given by Ž ² 2 : . Eexp 2 . 70 Finally, if we substitute Eq. 69 into Eq. 70 , we get our final result Ži.e., Eq. 51. as before, Ž . 2 2 2 Ž . ln E g D 3. PULSED-FIELD GRADIENT NMR 317 As expected, if we modify the limits of integration in Eq. 59 , then the GPD approximation can be used to calculate the diffusion term in the expression for the intensity of the Hahn spin-echo sequence Žsee Eq. 17.Ž 60, 80 . . We have shown in this section that since the Ž ² 2 mean-squared phase change i.e., :. can be calculated exactly for unrestricted diffusion, the GPD approximation gives the same result as the macroscopic approach Ži.e., Eq. 51 . . Subsequently, the validity of the GPD approximation represented by Eq. 57 and its ramifications when the diffusion is bounded will be considered further. SGP Approximation To understand the SGP approximation, we start back at Eq. 7 but ignore the effects of motion during the gradient pulse Žrigorously, one assumes that the gradient pulse is like a delta function, that is, 0 and g, while their product remains finite . . Experimentally, this condition is approximated by keeping . Hence, the effect of a gradient pulse of duration on a spin at position r is given by, neglecting the effect of the static field, Ž. r g r. 71 The scalar product arises because only motion parallel to the direction of the gradient will cause a change in the phase of the spin. Hence, if we consider the phase change of a spin which was at position r during the first gradient pulse and at 0 position r during the second, then the change in 1 phase in moving from r to r is given by 0 1 Ž . Ž . r r g r r . 72 1 0 1 0 Žn.b., the in Eq. 72 represents the difference in , not the duration between gradient pulses . . Now we need to consider the probability of a spin starting at r Ž i.e., the starting spin density. 0 at t0, Ž r , 0 . 0 , which is usually taken as being equal to the equilibrium spin density Ž r . Ž 0 see Eq. 28 . . This assumption requires that insignificant relaxation occur between the first rf pulse Ž i.e., excitation. and the first gradient pulse. In practice this is normally the case. Now the probability of moving from r to r in time Ž 0 1 i.e., the separation between the gradient pulses. is, of course, given by PŽ r , r , t. as before. Thus, the 0 1
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: every time a pulse is applied. Thu
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 and 26: In the next section, we will illust
Page 29 and 30: sor, D app. When a single effectiv
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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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