## Laplacian Eigenfunctions in NMR

#### Denis GrebenkovÉcole PolytechniqueLaboratoire de Physique de la Matiere Condensee

A geometrical confinement considerably affects the diffusive motion of the nuclei and the consequent signal attenuation under inhomogeneous
magnetic fields. In this talk, we illustrate the use of Laplacian eigenfunctions to describe this effect. 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. We illustrate the efficiency of the matrix technique by considering restricted diffusion in simple domains: a slab, a cylinder, and a sphere.

References:

D. S. Grebenkov, NMR survey of reflected Brownian motion, Rev. Mod. Phys. 79, 1077-1137 (2007).

D. S. Grebenkov, Laplacian Eigenfunctions in NMR I. A Numerical Tool, Conc. Magn. Reson. A 32, 277-301 (2008).

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