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222 PRICE Figure 17 Sequences for removal of background gradients. Ž A. The Karlicek and Lowe sequence Ž 91 . , Ž B. the nine pulse sequence of Cotts et al. Ž 92 . , Ž C. the improved stimulated echo sequence of Latour et al. Ž 77 . , and Ž D. the PFG multiple-spin-echo Ž PFG MSE. pulse sequence of Van Dusschoten et al. Ž 86 . . the T2 relaxation time of the macromolecule is much less than that of the water so that by the end of the 1 period the macromolecules are fully relaxed, whereas the relaxation of the water magnetization is insignificant. After the application of the second 2 pulse, the z magnetization of the macromolecule will be zero, since it was entirely aligned along the z axis prior to the pulse. For the water magnetization, the situation is entirely different; after the pulse, the water magnetization Ž . Ž . 1 is proportional to cos qz , where q 2 g recall that the gradient pulse creates a helix along the direction of the gradient with a period of 2Ž g .. However, as qz ranges over many periods, the net z magnetization over the sample is zero. Thus, the local normalized deviation from equilibrium in the macromolecule phase will be 1, and for the water phase, cosŽ qz. 1. Thus, during 2, cross-relaxation results from the equilibrium differences in both phases. Consequently the cross-relaxation rate will depend on q. Equations have been derived to account for this crossrelaxation in a two-phase system Ž 93 . . If significant cross-relaxation occurs, it can affect the measured signal intensities, thereby complicating diffusion measurements. Under limited conditions, it is possible to determine the exchange parameters to allow D to be calculated correctly. Importantly, the problem of cross-relaxation does not apply to the Stejskal and Tanner sequence. Multiple Quantum and Heteronuclear Experiments It is often desirable to work with heteronuclei, especially when measuring the diffusion coefficient of nuclei in a complex mixture such as a biological fluid. However, heteronuclei generally have a sensitivity far beneath that of protons. Further, because of the low gyromagnetic ratios of heteronuclei, larger gradients must be used. The most straightforward means of alleviating the signal-to-noise problem is through the use of specifically labeledenriched probe molecules. Large gains in sensitivity can be made through using pulse sequences to generate polarization transfer from protons to the heteronuclei Ž94, 95 . . This approach has the advantage of generating multiple quantum transitions. Multiple quantum transitions can, of course, also be used in homonuclear work Ž 96, 97 . . Multiple quantum spectra are also generally simpler and better resolved than the corresponding single quantum spectra; and for n coupled protons, the n-quantum transition is free of dipolar couplings. This may also allow an increase in the possible observation time owing to decreased relaxation Ž 98 . . Furthermore, in the case of homonuclear studies, multiple quantum spectra have the added benefit of providing solvent suppression. However, there are some restrictions on the applicability of multiple quantum PFG experiments, since the spectrum of the species in question must have either a Ž scalar, dipolar, or quadrupolar coupling e.g., Refs.
9499 . . If the attenuation of the multiple quantum coherence can be studied Žinstead of the single quantum coherence . , the same degree of attenuation can be achieved, but with smaller gradients and therefore smaller eddy current problems. In multiple quantum experiments, it is the effective sum of the values of the nuclei involved in the coherence which is relevant to the attenuation. Thus, for the Stejskal and Tanner pulse sequence, in the case of free diffusion and neglecting the effects of background gradients, the formula relating echo signal attenuation to diffusion can be written as Ž . Ž Ž . 2 2 E q, exp f g D Ž 3 .. . 41 For the normal Ž i.e., single quantum. experiment, Ž . 2 f . For homonuclear multiple quantum Ž . Ž . 2 experiments, f n . Examples of multiple quantum pulse sequences based on the Stejskal and Tanner and STE sequences are presented in Fig. 18. For heteronuclear multiple quantum experiments, the definition of fŽ . is not as straightforward. For heteronuclear double quantum experiments with an IS spin system, where I Ž not to be confused with I, the current. is the Ž . ŽŽ . . 2 observed nucleus, f Ž 95 . 1 s s 1 . The use of heteronuclear inverse DEPT- Ži.e., inverse detection. and IHETCOR-based sequences has been investigated Ž 94, 95 . . The DEPT-based sequence has significant advantages for working with low nuclei owing to the polar- 1 ization transfer from the I spin Žusually H. to the S spin, whereas the IHETCOR pulse sequence is more suited to the observation of protons as the unfavorable polarization transfer from the less abundant heteronuclear population to the proton population. Coherence-order selection may be incorporated into the diffusion experiment to provide solvent suppression. In the case of quadrupolar nuclei, if the spectrum contains a static quadrupolar splitting, the PFG experiment can be performed using a quadrupolar echo instead of a spin-echo Ži.e., the pulse would be replaced by a 2 pulse in Fig. 1 and appropriate phase cycling. Ž80, 100, 101 . . GRADIENT CALIBRATION So far in our discussion, we have implicitly assumed that the value of the gradient is known. The reality is that before we can determine the diffusion coefficient, we need to determine the PULSED-FIELD GRADIENT NMR: EXPERIMENTAL ASPECTS 223 Figure 18 Some representative multiple quantum PFG sequences based on Ž A. the Stejskal and Tanner, and Ž B. the STE sequence. The phase cycling for these sequences can be found elsewhere Ž 9699 . . gradient strength very accurately Žn.b., the gradient is squared in Eq. . 2 . We now consider the different ways in which the gradient can be calibrated, although it should be noted that even without calibration the relative diffusion coefficients can be measured. The methods of gradient calibration are summarized in Table 7. Theoretical Coil Calculation In theory, the applied gradient could be calculated from the known dimensions, geometry, number of turns of wire in the coil, and current applied Ž see Gradient Coils . . In practice, this method should give an estimate with an error of 10%. The major reason is interaction with nearby metal in the probe and nonideal gradient pulse generation problems. Thus, other methods are needed to get accurate gradient calibrations. A Standard Sample with Known Diffusion Coefficient The simplest way of calibrating a gradient is to use a standard sample of known diffusion coefficient Ž e.g., pure water . . Ideally, a reference compound should have a diffusion coefficient and T2 that are not strongly temperature dependent. Some suitable standard samples and their diffu-
Page 1 and 2: Pulsed-Field Gradient Nuclear Magne
Page 3 and 4: PULSED-FIELD GRADIENT NMR: EXPERIME
Page 5 and 6: winding can be estimated from the B
Page 7 and 8: dient pulses might change during th
Page 9 and 10: prior to the start of the rf pulse
Page 11 and 12: where 1 Ž . 2 2 kŽ. f t 2 e
Page 13 and 14: Pulse Sequences and Postprocessing
Page 15 and 16: Figure 10 Example of the Stejskal a
Page 17 and 18: Figure 11 The sequence used in the
Page 19 and 20: a short sample to stay within the c
Page 21 and 22: nulling methods and flow compensati
Page 23 and 24: gradients for the Stejskal and Tann
Page 25: quence that uses only pulses of one
Page 29 and 30: PULSED-FIELD GRADIENT NMR: EXPERIME
Page 31 and 32: We use the integral representation
Page 33 and 34: The integral in Eq. 70 over 1 is gi
Page 35 and 36: A Practical Calibration Procedure I
Page 37 and 38: l4t1 + Delta; > g4g; > F4subs(tf =
Page 39 and 40: 51. M. R. Merrill, ‘‘NMR Diffus
Page 41: ments Using Bessel Function Fits to

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