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312 PRICE complicated; also the obstructing particles are not Ž necessarily. distributed in space in a totally ordered or totally random way. CORRELATING SIGNAL ATTENUATION WITH DIFFUSION Introduction We will now discuss the mathematical formulations necessary to relate the signal attenuation to the diffusion coefficient and boundary conditions in the PFG experiment. Starting from the Bloch equations modified to include the diffusion of magnetization Ž 75, 76. it is possible to derive the necessary relationships analytically for free diffusion, as we shall show below. However, in the case of restricted diffusion this macroscopic approachbecomes mathematically intractable. Thus, in general case one is forced to use different approximations to find formulae relating E to the diffusion coefficient, boundary, and experimental conditions. There are two common approximations, namely: the GPD approximation and the SGP approximation. However, even using these approximations, analytic solutions are generally not possible and numerical methods must be used. In this section, we will only consider the case of free diffusion and describe the macroscopic approach and the SGP and GPD approximations in this case. It is assumed that the gradient pulses are rectangular. Detailed discussion of the signal attenuation of spins undergoing restricted diffusion will be deferred until later in this section. The Macroscopic Approach Bloch Equations Including the Effects of Diffusion. The Bloch equations for the macroscopic nuclear magnetization, Mr,t Ž . MxMyM, z including the diffusion of magnetization, are given by Ž 75, 76 . , Mr,t Ž . MxiMyj MBr,t Ž . t T2 Ž M M . k D M. 35 T1 z 0 2 In the case of anisotropic diffusion, the last term in Eq. 35 would be replaced by DM.Ifwe now take Ž as is usually the case. B to be oriented 0 along the z-axis and that this is superposed by a gradient g vanishing at the origin which is parallel to B Ž 0 we assume that the inhomogeneities caused by g are much smaller than B . 0 , and thus we can write B 0, B 0, x y Ž . B B gr B g xg yg z 36 z 0 0 x y z If Eq. 36 is then substituted into Eq. 35 , noting that MB Ž M B MB. Ž MBM B . y y z z y x z x x z Ž . M B M B 37 x y y x z and defining the Ž complex. transverse magnetization as mM iM 38 x y we obtain m Ž . 2 i0mi gr mmT2D m. t 39 The Stejskal and Tanner Pulse Sequence in the Absence of Diffusion. In the absence of diffusion Ž i.e., D 0,mrelaxes . exponentially with a time constant T 2, and thus we set i 0 tt T 2 me 40 where represents the amplitude of the precessing magnetization unaffected by the effects of relaxation. If we substitute Eq. 40 into 39 , we obtain t Ž . 2 igr D . 41 In the absence of diffusion, Eq. 41 is a first-order ordinary differential equation with solution Ž . Ž . r,t Sexp ir F 42 where S is a constant and t H 0 FŽ t. gŽ t. dt. 43 Now, if we consider the case of the PFG pulse sequence, then during the period from the 2 pulse to the pulse, we have Ži.e., Eq. 42. Ž . Ž . r,t Sexp ir F , 44
and S corresponds to the value of immediately after the 2 pulse. After the pulse Žneglect ing the phase angle of the pulse, which is of no consequence here . , we have where Ž . Ž Ž .. r,t Sexp ir F 2f , 45 Ž . fF . 46 Thus, from Eq. 45 we can see that the effect of the pulse is to set back the phase of by twice the amount that it had advanced up until the pulse Žsee the first series of phase diagrams in Fig. 2 . . Equations 44 and 45 can then be combined into Ž . Ž Ž Ž . .. r,t Sexp ir F 2 H t f 47 where Ht Ž. is the Heaviside step function. We note here that Eq. 47 is valid for the Hahn spin-echo pulse sequence. The Stejskal and Tanner Pulse Sequence in the Presence of Diffusion. In the previous section, we considered the solution to Eq. 41 in the absence of the diffusion. In this section, we derive a solution to Eq. 41 including the effects of diffusion. We assume a solution to Eq. 41 , including the diffusion term, to be of the form of Eq. 47 but allow S to be a function of t i.e., St Ž..By substituting Eq. 47 into Eq. 41 , we obtain dSŽ t. 2 2 DF2HŽ t. f SŽ t. 48 dt Now we integrate Eq. 48 from t 0tot2 SŽ 2. ln lnŽEŽ 2 .. SŽ 0. H 2 H 2 0 2 2 ½H 2 2 H 2 5 2 2 2 DF dt DF2f dt D F dt4f Fdt 4 f 0 49 The application of Eq. 49 to the calculation of the echo attenuation resulting from the effects of diffusion and the application of gradients is quite straightforward but rather tedious. If we apply the gradient pulses as shown in the pulse sequences in Fig. 2 and neglect the effects of any back- PULSED-FIELD GRADIENT NMR 313 ground gradients, then we can define gŽ. t and the effective field gradient, g Ž. eff t , as in Table 1. It is important to note that the lower limit of integration in Eq. 43 refers to the start of the sequence. For example, using the above definition of gŽ. t , Ft Ž. for t1tt1is calculated as follows, H H t1 t1 FŽ t. 0 dt gdt 0 t 1 t1 t H H 0 dt gdt t1 t1 Ž . g tt . 50 1 An example of the use of the symbolic algebra package Maple Ž 77 . to calculate Eq. 49 is given in the Appendix, and from this we obtain the result Ž 54. Ž . 2 2 2 Ž . ln E g D 3. 51 The term 3 accounts for the finite width of the gradient pulse. Equation 51 is not a function of t 1, and thus the placement of the gradient pulses in the sequence is of no consequence; for example, there is no requirement that the gradient pulses be symmetrically placed around the pulse. If instead we had imposed a steady gradient throughout the echo sequence Ži.e., . , we would have reproduced the well-known diffusion term in the expression for the intensity of the Hahn spin-echo sequence Žsee Eq. 17 . , as expected. It is instructive to consider Eq. 51 in some detail. Let us suppose that we do an experiment at, say 298 K on a sample containing a small molecule, such as water, which has a diffusion 9 2 1 coefficient of about 2.3 10 m s Ž 68, 69. and a protein with a diffusion coefficient of 1 1010 m2s1 . From our discussion above and also Eq. 51 , it is apparent that after we have cali- Table 1 g( t) and g ( t) eff for the Stejskal and Tanner Pulse Sequence Subinterval of Pulse Sequence gŽ. t g Ž. t 0tt 0 0 1 t tt g g 1 1 t tt 0 0 1 1 t tt g g 1 1 t t2 0 0 1 eff
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