Self-Diffusion in Restricted Geometries

Time Dependence of Mean-Squared Displacement Consider the evaluation of Eq. (47) in the low q limit, that is, 

Consequently, the slope of the low q echo attenuation data allows (Z (A))/. to be measured directly. This represents the simplest of all possible signal analysis in the case of the narrow gradient pulse PGSE experiment. In the study of hindered and restricted diffusion, such an analysis provides a useful guide to interdependence of length and time 

We now turn our attention to the diffusion of molecules inside a completely enclosing pore. We shall see that signal analysis based on both the q- and A-dependence of the echo will prove particularly illuminating. A number of exact solutions for the propagator are available by solving Pick s law using the standard eigenmode expansion (Arfken, 1970) 

The special cases of plane parallel pores, cylindrical pores, and spherical pores have been solved exactly and we quote only the echo attenuation results here. The pore geometries and applied gradient directions are shown in Fig. 7. Readers seeking more information about these solutions 

This is a one-dimensional problem in which the gradient is applied along the 2-direction normal to a pair of bounding planes and these relaxing planes are separated by a distance 2a and placed at z - a  

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