Diffusion in porous media

A geometrical confinement considerably affects the diffusive motion of
the nuclei and the consequent signal attenuation under inhomogeneous
magnetic fields.  In the tutorial lecture, we focus on theoretical
and numerical aspects of resticted diffusion in NMR.  Starting from 
the classical Bloch-Torrey equation, we obtain the free induction 
decay (FID) and the spin-echo or gradient-echo signal in a compact 
matrix form.  Each attenuation mechanism (restricted diffusion, 
gradient dephasing, surface or bulk relaxation) is represented by 
a matrix which is constructed from the Laplace operator eigenbasis
and thus depending only on the geometry of the confinement.  In turn, 
the physical parameters (free diffusion coefficient, gradient intensity, 
surface or bulk relaxivity) characterize the "strengths" of the 
underlying attenuation mechanisms and naturally appear as coefficients 
in front of these matrices.  Once the Laplacian eigenfunctions for a 
given confinement are found (analytically or numerically), further
computation of the macroscopic signal is more accurate and much faster
than by using conventional simulation methods.  The matrix technique
is actually a simple numerical tool to deal with arbitrary gradient
waveforms, including simple or stimulated, single or multiple spin
echoes.  We illustrate its efficiency by considering restricted
diffusion in simple domains: a slab, a cylinder, and a sphere.  
Classical and recent theoretical advances achieved by using 
Laplacian eigenfunctions are overviewed.