Conditions for initial quasilinearT2−1versus τ for Carr-Purcell-Meiboom-Gill NMR with diffusion and susceptibility differences in porous media and tissues

Porous media containing fluids and subject to the field of an NMR instrument usually have incremental local magnetic fields due to susceptibility differences ${\mathrm{\chi }}_{\mathit{d}}$, such as between a fluid and a solid matrix. If ‖${\mathrm{\chi }}_{\mathit{d}}$‖≪1 the effective variation ω of the local precession angular frequency is limited to ±1/2${\mathrm{\chi }}_{\mathit{d}}$ ${\mathrm{\omega }}_0$, where ${\mathrm{\omega }}_0$ is the mean precession angular frequency. Diffusion of fluid molecules through these local fields leads to a τ-dependent increase ${\mathit{R}}_{\mathit{d}}$ in the value of 1/${\mathit{T}}_2$ obtained from Carr-Purcell-Meiboom-Gill (CPMG) measurements. Many porous media appear likely to have significant ω variation over a substantial range of diffusion time scales, or correlation times. For a sample with a single correlation time ${\mathrm{\tau }}_{\mathit{c}}$ the logarithm of the additional decay of the nth echo amplitude due to diffusion through regions of different ω is ${\mathrm{\Omega }}^2$ ${\mathrm{\tau }}_{\mathit{c}}${2nτf(τ/${\mathrm{\tau }}_{\mathit{c}}$)-${\mathrm{\tau }}_{\mathit{c}}$[(1-x${)}^4$/(1+${\mathit{x}}^2$ ${)}^2$] [1-(-${)}^{\mathit{n}}$ ${\mathit{x}}^{2\mathit{n}}$]}, where f(t)=1-(tanht)/t, ${\mathrm{\Omega }}^2$=〈${\mathrm{\omega }}^2$〉, and x=exp(-τ/${\mathrm{\tau }}_{\mathit{c}}$). The term without n causes only a shift in the relaxation curve, and the terms in ${\mathit{x}}^{2\mathit{n}}$ are small, so ${\mathit{R}}_{\mathit{d}}$≊${\mathrm{\Omega }}^2$ ${\mathrm{\tau }}_{\mathit{c}}$f(τ/${\mathrm{\tau }}_{\mathit{c}}$). The function f(t) starts quadratically at small t, has a nearly linear portion, and then approaches 1-1/t at large t. However, the superposition of terms of the form f(τ/${\mathrm{\tau }}_{\mathit{c}\mathit{i}}$) tends to give a nearly linear portion of the ${\mathit{R}}_{\mathit{d}}$ vs τ curve extending from small values of ${\mathit{R}}_{\mathit{d}}$ to about a third of the asymptotic value if there is a significant range of ${\mathrm{\tau }}_{\mathit{c}\mathit{i}}$.