Propagators, Gaussian

The propagator for normal displacements, Z, along the gradient direction with the diffusion coefficient D is given by the Gaussian function 

Displacements along other space directions are not directly probed in this experiment and are therefore irrelevant. Evaluating Eq. 10 based on this propagator results in (e- )=exp -fc DA so that Eq. 9 becomes 

Inserting (Z )=2DA in Eq. 13 provides a direct relation between echo attenuation and the mean squared displacement in the diffusion time A  

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PES evaluations. In either case (coupled or split-operator with frozen Gaussian propagator)... 

In contrast to center-of-mass diffusion, segment diffusion is characterized by non-Gaussian propagators. The attenuation of spin echoes is then no longer governed by unspecific equations like Eqs. 13 or 14. However, in special cases such as the reptation model, a formalism for the evaluation of the echo attenuation can be set up [3]. 

Figure 3b shows the analogous variant for the stimulated echo. This is the preferential method for extremely short transverse relaxation times. In this case the pulsed-gradient parameters turn into S=ti and A=Ti-i-T2, so that Eq. 20 for Gaussian propagators becomes... 

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