Conditions for initial quasilinear T2-1 versus tau for Carr-Purcell-Meiboom-Gill NMR with diffusion and susceptibility differences in porous media and tissues.

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Porous media containing fluids and subject to the field of an NMR instrument usually have incremental local magnetic fields due to susceptibility differences ${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$, such as between a fluid and a solid matrix. If \ensuremath{\Vert}${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$\ensuremath{\Vert}\ensuremath{\ll}1 the effective variation \ensuremath{\omega} of the local precession angular frequency is limited to \ifmmode\pm\else\textpm\fi{}1/2${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$${\mathrm{\ensuremath{\omega}}}_{0}$, where ${\mathrm{\ensuremath{\omega}}}_{0}$ is the mean precession angular frequency. Diffusion of fluid molecules through these local fields leads to a \ensuremath{\tau}-dependent increase ${\mathit{R}}_{\mathit{d}}$ in the value of 1/${\mathit{T}}_{2}$ obtained from Carr-Purcell-Meiboom-Gill (CPMG) measurements. Many porous media appear likely to have significant \ensuremath{\omega} variation over a substantial range of diffusion time scales, or correlation times. For a sample with a single correlation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$ the logarithm of the additional decay of the nth echo amplitude due to diffusion through regions of different \ensuremath{\omega} is ${\mathrm{\ensuremath{\Omega}}}^{2}$${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}${2n\ensuremath{\tau}f(\ensuremath{\tau}/${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$)-${\mathrm{\ensuremath{\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, ${\mathrm{\ensuremath{\Omega}}}^{2}$=〈${\mathrm{\ensuremath{\omega}}}^{2}$〉, and x=exp(-\ensuremath{\tau}/${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$). The term without n causes only a shift in the relaxation curve, and the terms in ${\mathit{x}}^{2\mathit{n}}$ are small, so ${\mathit{R}}_{\mathit{d}}$\ensuremath{\approxeq}${\mathrm{\ensuremath{\Omega}}}^{2}$${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$f(\ensuremath{\tau}/${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$). The function f(t) starts quadratically at small t, has a nearly linear portion, and then approaches 1-1/t at large t. However, the superposition of terms of the form f(\ensuremath{\tau}/${\mathrm{\ensuremath{\tau}}}_{\mathit{c}\mathit{i}}$) tends to give a nearly linear portion of the ${\mathit{R}}_{\mathit{d}}$ vs \ensuremath{\tau} 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 ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}\mathit{i}}$.For this linear portion the effect of a shift in all the ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}\mathit{i}}$, such as from a change of temperature or from a liquid with a different diffusion coefficient, is small. Examples of measurements of ${\mathit{T}}_{1}$, of ${\mathit{T}}_{2}$ by Hahn single echoes, and of ${\mathit{T}}_{2}$ by CPMG measurements for a porous porcelain sample and a natural porous chalk sample illustrate this nearly linear \ensuremath{\tau} dependence, which is quite different from the quadratic dependence for unrestricted diffusion in a uniform field gradient. The CPMG data were fit very well over the entire range by a function of the form ${\mathit{R}}_{\mathit{a}}$+${\mathit{R}}_{\mathit{b}}$ ${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$(${\mathit{R}}_{\mathit{c}}$\ensuremath{\tau}), and the computed asymptotes match the Hahn single-echo results for \ensuremath{\tau}\ensuremath{\gg}${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$ surprisingly well. Our result depends on the limited range of field variation and does not apply to the case of ferromagnetic or superparamagnetic grains in close contact with diffusing fluid molecules. Our approach can also be applied, under particular circumstances, to biological tissues.