Decay of nuclear magnetization by diffusion in a parabolic magnetic field: An exactly solvable model

Abstract

An exact expression for the decay of the transverse magnetization of spins diffusing in a field (${\mathit{B}}_0$+${\mathit{g}}_1$z+${\mathit{g}}_2$ ${\mathit{z}}^2$)z^ reveals that a natural length scale ${\mathit{l}}_{\mathit{c}}$=(8D/γ${\mathit{g}}_2$ ${)}^{1/4}$, and a frequency associated with it ${\mathrm{\Omega }}_0$=4D/${\mathit{l}}_{\mathit{c}}^2$ govern the problem. Here D is the diffusion constant and γ is the gyromagnetic ratio. ${\mathit{l}}_{\mathit{c}}$ is the size of the packet of magnetization at long times, i.e., ${\mathrm{\Omega }}_0$t≫1. For porous media we estimate ${\mathit{l}}_{\mathit{c}}$∼(${\mathit{R}}_{\mathit{p}}$ ${\mathit{l}}^{\mathrm{*}}$ ${)}^{1/2}$ where ${\mathit{R}}_{\mathit{p}}$ is the pore size and ${\mathit{l}}^{\mathrm{*}2}$=D/Δω, where Δω is the spread in Larmor frequency (inhomogeneous broadening). For typical experimental conditions, ${\mathit{l}}^{\mathrm{*}}$∼3 μm; therefore in rocks the effects of the extrema of the magnetic field can be as important as the wall effects. To estimate finite-pore-size effects we localize the spins in a potential well of size ${\mathit{R}}_{\mathit{p}}$.