Scaling of spin-echo amplitudes with frequency, diffusion coefficient, pore size, and susceptibility difference for the NMR of fluids in porous media and biological tissues

Both Carr-Purcell-Meiboom-Gill (CPMG) measurements and single-spin-echo measurements have been made at frequencies of ν=10, 20, and 50 MHz for two relatively homogeneous porous porcelain materials with different pore sizes, both saturated separately with three liquids of different diffusion coefficients. The CPMG transverse relaxation rate is increased by an amount R by diffusion in the inhomogeneous fields caused by susceptibility differences χ; R shows the dependence on τ (half the echo spacing) given by the model of Brown and Fantazzini [Phys. Rev. B 47, 14 823 (1993)] if relaxation is slow enough that there are several CPMG echoes in a transverse relaxation time. For τ values over a range of a factor of about 40, the increase of R with τ is nearly linear, with a slope that is independent of pore dimension a and diffusion coefficient D. For this nearly linear region and a short initial region quadratic in τ, we find R∝(χν${)}^2$. In these regions we can scale and compare measurements of R taken for different values of χ ν, a, and D by plotting RD/(1/3χνa${)}^2$ vs Dτ/${\mathit{a}}^2$. The asymptotic values of R for large τ for CPMG data can be inferred from the asymptotic slope, -${\mathit{R}}_{\mathit{s}}$, of lnM (magnetization) for single spin echoes as a function of echo time t=2τ.