Abstract
The magnetic-resonance signal usually recorded with this measuring technique is either a two-pulse Hahn echo or a stimulated echo. The RF pulse sequences (see Chap. 2) are applied in the presence of a strong pulsed or steady main-field gradient
where we have arbitrarily assumed that the gradient is aligned along the z axis of the laboratory frame. Field gradients constant within the sample are particularly favorable. Figure 19.1 shows the time dependences of the field gradients and of the RF pulses which are typically in use.
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Notes
Note that this attenuation factor is analogous to the incoherent scattering function playing a crucial role in quasi-elastic incoherent neutron scattering experiments. Consider a stimulatedecho experiment with a spatially and temporally constant field gradient (Fig. 19.1). The time intervals are supposed to obey τ2 ≫ τ1. Formally introducing the “wave vector” q = γnτ1G suggests the phase shift \( \varphi ({{t}_{e}}) = q \cdot r({{t}_{e}}) - q \cdot r(0) \) so that the echo attenuation factor becomes \( {{A}_{{diff}}}({{t}_{e}}) = \langle {{e}^{{ - {}_{i}q \cdot r(0)}}}{{e}^{{{}_{i}q \cdot r({{t}_{e}})}}}\rangle \) This expression has the structure of the incoherent scattering function. The wave vector q is more than of a formal nature. As will be shown below, the pulse sequences employed in field-gradient NMR diffusometry produce wavelike magnetization grids characterized by just this wavenumber. The effect of translational diffusion is then to smear out the grid. This is the real-space variant for explaining NMR diffusometry. Because of the real and static character of the “wave” we prefer in the following to use the symbol k instead of q (see Chap. 20).
This is a consequence of the central limit theorem [147]. It generally states that sums of the form \(\zeta_{N}=m_{j=1} {N} \xi_{j}\) which are repeatedly formed of N random and statistically independent variables ξj, are distributed according to a Gaussian function when N → ∞. An immensely important characteristic of this theorem is that the random distribution, with which the variables themselves occur, may be arbitrary provided that it is the same for each variable and that its second moment is finite.
An explicit treatment for a non-Gaussian probability density of displacements can be found in Sect. 22.2 for polymer diffusion (“reptation”).
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© 1997 Springer-Verlag Berlin Heidelberg
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Kimmich, R. (1997). Main-Field Gradient NMR Diffusometry. In: NMR. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60582-6_19
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DOI: https://doi.org/10.1007/978-3-642-60582-6_19
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