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

An exact expression for the decay of the transverse magnetization of spins diffusing in a field (B₀+g₁z+g₂{z}²)z^ reveals that a natural length scale{l}c=(8D/γ{g}₂)1/4, and a frequency associated with it Ω₀=4D/lc² govern the problem. Here D is the diffusion constant and {γ} is the gyromagnetic ratio. lc is the size of the packet of magnetization at long times, i.e., Ω₀t{}1. For porous media we estimate lc{~}(Rₚ{l}*)1/2 where Rₚ is the pore size and l*2=D/{Δ}{ω}, where {Δ}{ω} is the spread in Larmor frequency (inhomogeneous broadening). For typical experimental conditions, l*{~}3 {μ}m; 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 Rₚ.We find that the effective pore size is lc or Rₚ, whichever is smaller. At short times, the magnetization density {}M(z,t){}{~}exp[-Dγ²(g₁+2g₂z)²{t}³$/3], i.e., it is permissible to use an effective local gradient. The magnetization decays rapidly where the magnetic field varies rapidly---thus the magnetization accumulates at the extremum of the field. At long times, the magnetization decays as exp(-Ω₀t/2), as opposed to exp(-t³), in a uniform gradient. The phase distribution is not Gaussian, which leads the decay rate Ω₀{~} {}g₂ to be a nonanalytic function of g₂. There is an overall shift Ω₀/2 (the ``g-shift'') in the effective Larmor frequency, due to diffusion. The signal from a pulse-field-gradient experiment is similar to that of an isolated pore of size lc. We compute the Hahn- and Carr-Purcell-Meiboom-Gill-(CPMG-) echo envelopes and find qualitative agreement with experimental data on porous media. Extracting g₂ from the observed inhomogeneous broadening gives correct crossover times toward the linear regime. The slopes of the CPMG envelopes depend linearly on pulse spacing, as observed experimentally.